UniformSampleCone, x

Percentage Accurate: 57.4% → 99.0%

Time: 15.3s

Alternatives: 22

Speedup: 2.1×

Specification

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (* 2.0 PI)))
  (sqrt
   (*
    ux
    (+
     (+ (* (+ maxCos -1.0) (* ux (- 1.0 maxCos))) (- 1.0 maxCos))
     (- 1.0 maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((((maxCos + -1.0f) * (ux * (1.0f - maxCos))) + (1.0f - maxCos)) + (1.0f - maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * Float32(Float32(1.0) - maxCos))) + Float32(Float32(1.0) - maxCos)) + Float32(Float32(1.0) - maxCos)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((((maxCos + single(-1.0)) * (ux * (single(1.0) - maxCos))) + (single(1.0) - maxCos)) + (single(1.0) - maxCos))));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   3. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
7. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   2. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

9. Final simplification 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \]
10. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right) + maxCos \cdot -2\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (* 2.0 PI)))
  (sqrt
   (*
    ux
    (+ (+ 2.0 (* ux (* (+ maxCos -1.0) (- 1.0 maxCos)))) (* maxCos -2.0))))))

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((2.0f + (ux * ((maxCos + -1.0f) * (1.0f - maxCos)))) + (maxCos * -2.0f))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))) + Float32(maxCos * Float32(-2.0))))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((single(2.0) + (ux * ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))) + (maxCos * single(-2.0)))));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in ux around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 + \left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 - ux \cdot {\left(maxCos - 1\right)}^{2}\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   6. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \left({\left(maxCos - 1\right)}^{2}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(\mathsf{neg}\left(maxCos \cdot 2\right)\right)\right)\right)\right)\right) \]

   17. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(maxCos \cdot \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(maxCos \cdot -2\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)\right)\right)\right) \]

7. Simplified 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right)\right) + maxCos \cdot -2\right)}} \]

8. Final simplification 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right) + maxCos \cdot -2\right)} \]
9. Add Preprocessing

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;uy \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot t\_0 + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux - ux \cdot ux\right) + t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 1.0 maxCos))))
   (if (<= uy 0.014000000432133675)
     (*
      (sqrt
       (* ux (+ (+ (* (+ maxCos -1.0) t_0) (- 1.0 maxCos)) (- 1.0 maxCos))))
      (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0))
     (* (cos (* uy (* 2.0 PI))) (sqrt (+ (- ux (* ux ux)) t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - maxCos);
	float tmp;
	if (uy <= 0.014000000432133675f) {
		tmp = sqrtf((ux * ((((maxCos + -1.0f) * t_0) + (1.0f - maxCos)) + (1.0f - maxCos)))) * (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f);
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((ux - (ux * ux)) + t_0));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - maxCos))
	tmp = Float32(0.0)
	if (uy <= Float32(0.014000000432133675))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(Float32(Float32(maxCos + Float32(-1.0)) * t_0) + Float32(Float32(1.0) - maxCos)) + Float32(Float32(1.0) - maxCos)))) * Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(ux - Float32(ux * ux)) + t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux * (single(1.0) - maxCos);
	tmp = single(0.0);
	if (uy <= single(0.014000000432133675))
		tmp = sqrt((ux * ((((maxCos + single(-1.0)) * t_0) + (single(1.0) - maxCos)) + (single(1.0) - maxCos)))) * (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt(((ux - (ux * ux)) + t_0));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0140000004

   1. Initial program 59.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
   7. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      2. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

   9. Taylor expanded in uy around 0

      \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\color{blue}{\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \color{blue}{\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \color{blue}{\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{maxCos}, -1\right)\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 99.1%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right)\right)\right) \]

   11. Simplified 99.1%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)} \]

   if 0.0140000004 < uy

   1. Initial program 66.2%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 97.3%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\color{blue}{\left(ux - {ux}^{2}\right)}, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left({ux}^{2}\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(ux \cdot ux\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 94.1%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{*.f32}\left(ux, ux\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

   9. Simplified 94.1%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - ux \cdot ux\right)} - ux \cdot \left(maxCos + -1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux - ux \cdot ux\right) + ux \cdot \left(1 - maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{maxCos \cdot \left(ux \cdot \left(-2 + 2 \cdot ux\right)\right) + ux \cdot \left(2 - ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (* 2.0 PI)))
  (sqrt (+ (* maxCos (* ux (+ -2.0 (* 2.0 ux)))) (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf(((maxCos * (ux * (-2.0f + (2.0f * ux)))) + (ux * (2.0f - ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(maxCos * Float32(ux * Float32(Float32(-2.0) + Float32(Float32(2.0) * ux)))) + Float32(ux * Float32(Float32(2.0) - ux)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt(((maxCos * (ux * (single(-2.0) + (single(2.0) * ux)))) + (ux * (single(2.0) - ux))));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   3. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
7. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   2. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

9. Taylor expanded in maxCos around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(maxCos \cdot \left(ux \cdot \left(2 \cdot ux - 2\right)\right) + ux \cdot \left(2 + -1 \cdot ux\right)\right)}\right)\right) \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(maxCos \cdot \left(ux \cdot \left(2 \cdot ux - 2\right)\right) + ux \cdot \left(2 + \left(\mathsf{neg}\left(ux\right)\right)\right)\right)\right)\right) \]

    2. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(maxCos \cdot \left(ux \cdot \left(2 \cdot ux - 2\right)\right) + ux \cdot \left(2 - ux\right)\right)\right)\right) \]

    3. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\left(maxCos \cdot \left(ux \cdot \left(2 \cdot ux - 2\right)\right)\right), \left(ux \cdot \left(2 - ux\right)\right)\right)\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \left(ux \cdot \left(2 \cdot ux - 2\right)\right)\right), \left(ux \cdot \left(2 - ux\right)\right)\right)\right)\right) \]

    5. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \left(2 \cdot ux - 2\right)\right)\right), \left(ux \cdot \left(2 - ux\right)\right)\right)\right)\right) \]

    6. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \left(2 \cdot ux + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right), \left(ux \cdot \left(2 - ux\right)\right)\right)\right)\right) \]

    7. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \left(2 \cdot ux + -2\right)\right)\right), \left(ux \cdot \left(2 - ux\right)\right)\right)\right)\right) \]

    8. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 \cdot ux\right), -2\right)\right)\right), \left(ux \cdot \left(2 - ux\right)\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(ux \cdot 2\right), -2\right)\right)\right), \left(ux \cdot \left(2 - ux\right)\right)\right)\right)\right) \]

    10. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(ux, 2\right), -2\right)\right)\right), \left(ux \cdot \left(2 - ux\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f32 N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(ux, 2\right), -2\right)\right)\right), \mathsf{*.f32}\left(ux, \left(2 - ux\right)\right)\right)\right)\right) \]

    12. --lowering--.f32 98.1%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(ux, 2\right), -2\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(2, ux\right)\right)\right)\right)\right) \]

11. Simplified 98.1%

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{maxCos \cdot \left(ux \cdot \left(ux \cdot 2 + -2\right)\right) + ux \cdot \left(2 - ux\right)}} \]

12. Final simplification 98.1%

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{maxCos \cdot \left(ux \cdot \left(-2 + 2 \cdot ux\right)\right) + ux \cdot \left(2 - ux\right)} \]
13. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(-2 + 2 \cdot ux\right) - ux\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (cos (* uy (* 2.0 PI)))
  (sqrt (* ux (+ 2.0 (- (* maxCos (+ -2.0 (* 2.0 ux))) ux))))))

C

float code(float ux, float uy, float maxCos) {
	return cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f + ((maxCos * (-2.0f + (2.0f * ux))) - ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(Float32(-2.0) + Float32(Float32(2.0) * ux))) - ux)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) + ((maxCos * (single(-2.0) + (single(2.0) * ux))) - ux))));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   3. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
7. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   2. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

9. Taylor expanded in maxCos around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\left(2 + \left(-1 \cdot ux + maxCos \cdot \left(2 \cdot ux - 2\right)\right)\right)}\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(-1 \cdot ux + maxCos \cdot \left(2 \cdot ux - 2\right)\right)\right)\right)\right)\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(\left(\mathsf{neg}\left(ux\right)\right) + maxCos \cdot \left(2 \cdot ux - 2\right)\right)\right)\right)\right)\right) \]

    3. +-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(maxCos \cdot \left(2 \cdot ux - 2\right) + \left(\mathsf{neg}\left(ux\right)\right)\right)\right)\right)\right)\right) \]

    4. unsub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(maxCos \cdot \left(2 \cdot ux - 2\right) - ux\right)\right)\right)\right)\right) \]

    5. --lowering--.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\left(maxCos \cdot \left(2 \cdot ux - 2\right)\right), ux\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(maxCos, \left(2 \cdot ux - 2\right)\right), ux\right)\right)\right)\right)\right) \]

    7. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(maxCos, \left(2 \cdot ux + \left(\mathsf{neg}\left(2\right)\right)\right)\right), ux\right)\right)\right)\right)\right) \]

    8. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(maxCos, \left(2 \cdot ux + -2\right)\right), ux\right)\right)\right)\right)\right) \]

    9. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{+.f32}\left(\left(2 \cdot ux\right), -2\right)\right), ux\right)\right)\right)\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{+.f32}\left(\left(ux \cdot 2\right), -2\right)\right), ux\right)\right)\right)\right)\right) \]

    11. *-lowering-*.f32 98.0%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(maxCos, \mathsf{+.f32}\left(\mathsf{*.f32}\left(ux, 2\right), -2\right)\right), ux\right)\right)\right)\right)\right) \]

11. Simplified 98.0%

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + \left(maxCos \cdot \left(ux \cdot 2 + -2\right) - ux\right)\right)}} \]

12. Final simplification 98.0%

    \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(-2 + 2 \cdot ux\right) - ux\right)\right)} \]
13. Add Preprocessing

Alternative 6: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux + ux \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.014000000432133675)
   (*
    (sqrt
     (*
      ux
      (+
       (+ (* (+ maxCos -1.0) (* ux (- 1.0 maxCos))) (- 1.0 maxCos))
       (- 1.0 maxCos))))
    (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0))
   (* (cos (* uy (* 2.0 PI))) (sqrt (+ ux (* ux (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.014000000432133675f) {
		tmp = sqrtf((ux * ((((maxCos + -1.0f) * (ux * (1.0f - maxCos))) + (1.0f - maxCos)) + (1.0f - maxCos)))) * (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f);
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux + (ux * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.014000000432133675))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * Float32(Float32(1.0) - maxCos))) + Float32(Float32(1.0) - maxCos)) + Float32(Float32(1.0) - maxCos)))) * Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux + Float32(ux * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.014000000432133675))
		tmp = sqrt((ux * ((((maxCos + single(-1.0)) * (ux * (single(1.0) - maxCos))) + (single(1.0) - maxCos)) + (single(1.0) - maxCos)))) * (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((ux + (ux * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0140000004

   1. Initial program 59.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
   7. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      2. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

   9. Taylor expanded in uy around 0

      \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\color{blue}{\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \color{blue}{\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \color{blue}{\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{maxCos}, -1\right)\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 99.1%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right)\right)\right) \]

   11. Simplified 99.1%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)} \]

   if 0.0140000004 < uy

   1. Initial program 66.2%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 97.3%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 93.7%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 93.7%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - \left(\mathsf{neg}\left(ux\right)\right)\right) - ux \cdot ux\right)\right)\right) \]

       2. associate--l- N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot ux\right)\right)\right)\right) \]

       3. --lowering--.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(ux, \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot ux\right)\right)\right)\right) \]

       4. neg-mul-1 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux + ux \cdot ux\right)\right)\right)\right) \]

       5. distribute-rgt-out N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(ux, \left(ux \cdot \left(-1 + ux\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{*.f32}\left(ux, \left(-1 + ux\right)\right)\right)\right)\right) \]

       7. +-lowering-+.f32 93.9%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(-1, ux\right)\right)\right)\right)\right) \]

   11. Applied egg-rr 93.9%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \color{blue}{\sqrt{ux - ux \cdot \left(-1 + ux\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux + ux \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.014000000432133675)
   (*
    (sqrt
     (*
      ux
      (+
       (+ (* (+ maxCos -1.0) (* ux (- 1.0 maxCos))) (- 1.0 maxCos))
       (- 1.0 maxCos))))
    (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0))
   (* (sqrt (* ux (- 2.0 ux))) (cos (* 2.0 (* uy PI))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.014000000432133675f) {
		tmp = sqrtf((ux * ((((maxCos + -1.0f) * (ux * (1.0f - maxCos))) + (1.0f - maxCos)) + (1.0f - maxCos)))) * (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f);
	} else {
		tmp = sqrtf((ux * (2.0f - ux))) * cosf((2.0f * (uy * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.014000000432133675))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * Float32(Float32(1.0) - maxCos))) + Float32(Float32(1.0) - maxCos)) + Float32(Float32(1.0) - maxCos)))) * Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.014000000432133675))
		tmp = sqrt((ux * ((((maxCos + single(-1.0)) * (ux * (single(1.0) - maxCos))) + (single(1.0) - maxCos)) + (single(1.0) - maxCos)))) * (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0));
	else
		tmp = sqrt((ux * (single(2.0) - ux))) * cos((single(2.0) * (uy * single(pi))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0140000004

   1. Initial program 59.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 59.8%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
   7. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      2. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

   9. Taylor expanded in uy around 0

      \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\color{blue}{\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \color{blue}{\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \color{blue}{\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{maxCos}, -1\right)\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 99.1%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right)\right)\right) \]

   11. Simplified 99.1%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)} \]

   if 0.0140000004 < uy

   1. Initial program 66.2%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 66.2%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in ux around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)\right)}\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)\right)\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      3. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 + \left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      5. unsub-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 - ux \cdot {\left(maxCos - 1\right)}^{2}\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \left({\left(maxCos - 1\right)}^{2}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(\mathsf{neg}\left(maxCos \cdot 2\right)\right)\right)\right)\right)\right) \]

      17. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(maxCos \cdot \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(maxCos \cdot -2\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f32 97.5%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)\right)\right)\right) \]

   7. Simplified 97.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right)\right) + maxCos \cdot -2\right)}} \]

   8. Taylor expanded in maxCos around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\left(\sqrt{ux \cdot \left(2 - ux\right)}\right), \color{blue}{\cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      2. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\left(ux \cdot \left(2 - ux\right)\right)\right), \cos \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)}\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(2 - ux\right)\right)\right), \cos \left(\color{blue}{2} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(2, ux\right)\right)\right), \cos \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. cos-lowering-cos.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(2, ux\right)\right)\right), \mathsf{cos.f32}\left(\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(2, ux\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(2, ux\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI}\left(\right)\right)\right)\right)\right) \]

      8. PI-lowering-PI.f32 93.9%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(2, ux\right)\right)\right), \mathsf{cos.f32}\left(\mathsf{*.f32}\left(2, \mathsf{*.f32}\left(uy, \mathsf{PI.f32}\left(\right)\right)\right)\right)\right) \]

   10. Simplified 93.9%

       \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \cos \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 94.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \mathbf{if}\;uy \leq 0.032999999821186066:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{\left(t\_0 + ux \cdot \left(\left(maxCos + -1\right) \cdot t\_0\right)\right) + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 1.0 maxCos))))
   (if (<= uy 0.032999999821186066)
     (*
      (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0)
      (sqrt (+ (+ t_0 (* ux (* (+ maxCos -1.0) t_0))) t_0)))
     (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - maxCos);
	float tmp;
	if (uy <= 0.032999999821186066f) {
		tmp = (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf(((t_0 + (ux * ((maxCos + -1.0f) * t_0))) + t_0));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - maxCos))
	tmp = Float32(0.0)
	if (uy <= Float32(0.032999999821186066))
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(Float32(t_0 + Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * t_0))) + t_0)));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux * (single(1.0) - maxCos);
	tmp = single(0.0);
	if (uy <= single(0.032999999821186066))
		tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt(((t_0 + (ux * ((maxCos + single(-1.0)) * t_0))) + t_0));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0329999998

   1. Initial program 60.6%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 60.7%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.3%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in uy around 0

      \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

      8. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

      9. PI-lowering-PI.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\color{blue}{maxCos}, -1\right)\right)\right)\right)\right) \]

      10. PI-lowering-PI.f32 97.5%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right)\right)\right) \]

   9. Simplified 97.5%

      \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)} \]

   if 0.0329999998 < uy

   1. Initial program 63.4%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 63.4%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 96.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 91.6%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 91.6%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in ux around 0

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(2 \cdot ux\right)}\right)\right) \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot 2\right)\right)\right) \]

       2. *-lowering-*.f32 68.7%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, 2\right)\right)\right) \]

   12. Simplified 68.7%

       \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.032999999821186066:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right) + ux \cdot \left(1 - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(1 - maxCos\right)\\ \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{\left(t\_0 + ux \cdot \left(\left(maxCos + -1\right) \cdot t\_0\right)\right) + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- 1.0 maxCos))))
   (*
    (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0)
    (sqrt (+ (+ t_0 (* ux (* (+ maxCos -1.0) t_0))) t_0)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (1.0f - maxCos);
	return (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf(((t_0 + (ux * ((maxCos + -1.0f) * t_0))) + t_0));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(Float32(1.0) - maxCos))
	return Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(Float32(t_0 + Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * t_0))) + t_0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	t_0 = ux * (single(1.0) - maxCos);
	tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt(((t_0 + (ux * ((maxCos + single(-1.0)) * t_0))) + t_0));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   3. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

7. Taylor expanded in uy around 0

   \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]
8. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\color{blue}{maxCos}, -1\right)\right)\right)\right)\right) \]

   10. PI-lowering-PI.f32 89.6%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(1, \mathsf{neg.f32}\left(maxCos\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right)\right)\right) \]

9. Simplified 89.6%

   \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)} \]

10. Final simplification 89.6%

    \[\leadsto \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{\left(ux \cdot \left(1 - maxCos\right) + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right)\right)\right) + ux \cdot \left(1 - maxCos\right)} \]
11. Add Preprocessing

Alternative 10: 88.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (*
    ux
    (+
     (+ (* (+ maxCos -1.0) (* ux (- 1.0 maxCos))) (- 1.0 maxCos))
     (- 1.0 maxCos))))
  (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * ((((maxCos + -1.0f) * (ux * (1.0f - maxCos))) + (1.0f - maxCos)) + (1.0f - maxCos)))) * (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(ux * Float32(Float32(Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * Float32(Float32(1.0) - maxCos))) + Float32(Float32(1.0) - maxCos)) + Float32(Float32(1.0) - maxCos)))) * Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * ((((maxCos + single(-1.0)) * (ux * (single(1.0) - maxCos))) + (single(1.0) - maxCos)) + (single(1.0) - maxCos)))) * (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   3. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
7. Step-by-step derivation

   1. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   2. distribute-lft-out-- N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

8. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

9. Taylor expanded in uy around 0

   \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]
10. Step-by-step derivation

    1. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right)\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\color{blue}{\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \color{blue}{\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

    6. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \color{blue}{\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)}\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

    7. unpow2 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

    8. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \color{blue}{\mathsf{+.f32}\left(maxCos, -1\right)}\right)\right)\right)\right) \]

    9. PI-lowering-PI.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(\color{blue}{maxCos}, -1\right)\right)\right)\right)\right) \]

    10. PI-lowering-PI.f32 89.6%

        \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\mathsf{+.f32}\left(\mathsf{\_.f32}\left(1, maxCos\right), \mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(maxCos, \color{blue}{-1}\right)\right)\right)\right)\right) \]

11. Simplified 89.6%

    \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)} \]

12. Final simplification 89.6%

    \[\leadsto \sqrt{ux \cdot \left(\left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right) + \left(1 - maxCos\right)\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \]
13. Add Preprocessing

Alternative 11: 88.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right) + maxCos \cdot -2\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (*
    ux
    (+ (+ 2.0 (* ux (* (+ maxCos -1.0) (- 1.0 maxCos)))) (* maxCos -2.0))))
  (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * ((2.0f + (ux * ((maxCos + -1.0f) * (1.0f - maxCos)))) + (maxCos * -2.0f)))) * (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f);
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))) + Float32(maxCos * Float32(-2.0))))) * Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * ((single(2.0) + (ux * ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))) + (maxCos * single(-2.0))))) * (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in ux around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)\right)}\right)\right) \]
6. Step-by-step derivation

   1. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)\right)\right)\right) \]

   2. sub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) + \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   3. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 + \left(\mathsf{neg}\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   5. unsub-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 - ux \cdot {\left(maxCos - 1\right)}^{2}\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   6. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \left({\left(maxCos - 1\right)}^{2}\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(\mathsf{neg}\left(2 \cdot maxCos\right)\right)\right)\right)\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(\mathsf{neg}\left(maxCos \cdot 2\right)\right)\right)\right)\right)\right) \]

   17. distribute-rgt-neg-in N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(maxCos \cdot \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right)\right) \]

   18. metadata-eval N/A

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \left(maxCos \cdot -2\right)\right)\right)\right)\right) \]

   19. *-lowering-*.f32 99.0%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)\right)\right)\right) \]

7. Simplified 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right)\right) + maxCos \cdot -2\right)}} \]

8. Taylor expanded in uy around 0

   \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)\right)\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)\right)}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)}\right)\right)\right) \]

   3. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \color{blue}{\mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)}\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\color{blue}{\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)}, \mathsf{*.f32}\left(maxCos, -2\right)\right)\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \color{blue}{\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \color{blue}{\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)}\right), \mathsf{*.f32}\left(maxCos, -2\right)\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \color{blue}{\mathsf{*.f32}\left(maxCos, -2\right)}\right)\right)\right)\right) \]

   8. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \color{blue}{\mathsf{*.f32}\left(maxCos, -2\right)}\right)\right)\right)\right) \]

   9. PI-lowering-PI.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(\color{blue}{maxCos}, -2\right)\right)\right)\right)\right) \]

   10. PI-lowering-PI.f32 89.5%

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{\_.f32}\left(2, \mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right), \mathsf{*.f32}\left(maxCos, \color{blue}{-2}\right)\right)\right)\right)\right) \]

10. Simplified 89.5%

    \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right)\right) + maxCos \cdot -2\right)} \]

11. Final simplification 89.5%

    \[\leadsto \sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right) + maxCos \cdot -2\right)} \cdot \left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \]
12. Add Preprocessing

Alternative 12: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\pi \cdot \left(-2 \cdot \left(\pi \cdot \left(uy \cdot uy\right)\right)\right) + 1\right) \cdot \sqrt{\left(ux + ux\right) - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.0000000467443897e-7)
   (* (+ (* PI (* -2.0 (* PI (* uy uy)))) 1.0) (sqrt (- (+ ux ux) (* ux ux))))
   (sqrt
    (*
     (* ux ux)
     (+ (/ (+ 2.0 (* maxCos -2.0)) ux) (* (+ maxCos -1.0) (- 1.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.0000000467443897e-7f) {
		tmp = ((((float) M_PI) * (-2.0f * (((float) M_PI) * (uy * uy)))) + 1.0f) * sqrtf(((ux + ux) - (ux * ux)));
	} else {
		tmp = sqrtf(((ux * ux) * (((2.0f + (maxCos * -2.0f)) / ux) + ((maxCos + -1.0f) * (1.0f - maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(Float32(Float32(pi) * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(uy * uy)))) + Float32(1.0)) * sqrt(Float32(Float32(ux + ux) - Float32(ux * ux))));
	else
		tmp = sqrt(Float32(Float32(ux * ux) * Float32(Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) / ux) + Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.0000000467443897e-7))
		tmp = ((single(pi) * (single(-2.0) * (single(pi) * (uy * uy)))) + single(1.0)) * sqrt(((ux + ux) - (ux * ux)));
	else
		tmp = sqrt(((ux * ux) * (((single(2.0) + (maxCos * single(-2.0))) / ux) + ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.00000005e-7

   1. Initial program 62.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 98.9%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 98.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 98.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 88.9%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   12. Simplified 88.9%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]
   13. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \mathsf{PI}\left(\right)\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. associate-*l* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot \left(\left(uy \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(\left(uy \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\left(uy \cdot uy\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, uy\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, uy\right), \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{PI}\left(\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       8. PI-lowering-PI.f32 88.9%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(\mathsf{*.f32}\left(uy, uy\right), \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{PI.f32}\left(\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

   14. Applied egg-rr 88.9%

       \[\leadsto \left(1 + \color{blue}{\left(-2 \cdot \left(\left(uy \cdot uy\right) \cdot \pi\right)\right) \cdot \pi}\right) \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]

   if 4.00000005e-7 < maxCos

   1. Initial program 53.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\left(1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left({\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left(1 + ux \cdot \left(maxCos - 1\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f32 47.0%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right)\right) \]

   7. Simplified 47.0%

      \[\leadsto \color{blue}{\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]

   8. Taylor expanded in ux around inf

      \[\leadsto \mathsf{sqrt.f32}\left(\color{blue}{\left({ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left({ux}^{2}\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left(ux \cdot ux\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(2 \cdot \frac{1}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2 \cdot 1}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2}{ux} - \frac{2 \cdot maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      8. div-sub N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 - 2 \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 + -2 \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      11. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\left(\frac{2 + -2 \cdot maxCos}{ux}\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(2 + -2 \cdot maxCos\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      13. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(-2 \cdot maxCos\right)\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      20. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

      22. +-lowering-+.f32 83.8%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

   10. Simplified 83.8%

       \[\leadsto \sqrt{\color{blue}{\left(ux \cdot ux\right) \cdot \left(\frac{2 + -2 \cdot maxCos}{ux} - \left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\pi \cdot \left(-2 \cdot \left(\pi \cdot \left(uy \cdot uy\right)\right)\right) + 1\right) \cdot \sqrt{\left(ux + ux\right) - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{\left(ux + ux\right) - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.0000000467443897e-7)
   (* (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0) (sqrt (- (+ ux ux) (* ux ux))))
   (sqrt
    (*
     (* ux ux)
     (+ (/ (+ 2.0 (* maxCos -2.0)) ux) (* (+ maxCos -1.0) (- 1.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.0000000467443897e-7f) {
		tmp = (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf(((ux + ux) - (ux * ux)));
	} else {
		tmp = sqrtf(((ux * ux) * (((2.0f + (maxCos * -2.0f)) / ux) + ((maxCos + -1.0f) * (1.0f - maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(Float32(ux + ux) - Float32(ux * ux))));
	else
		tmp = sqrt(Float32(Float32(ux * ux) * Float32(Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) / ux) + Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.0000000467443897e-7))
		tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt(((ux + ux) - (ux * ux)));
	else
		tmp = sqrt(((ux * ux) * (((single(2.0) + (maxCos * single(-2.0))) / ux) + ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.00000005e-7

   1. Initial program 62.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 98.9%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 98.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 98.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 88.9%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   12. Simplified 88.9%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]
   13. Step-by-step derivation

       1. --lowering--.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - \left(\mathsf{neg}\left(ux\right)\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

       2. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux\right)\right)\right)\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

       3. remove-double-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux + ux\right), \left(ux \cdot ux\right)\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(ux, ux\right), \left(ux \cdot ux\right)\right)\right)\right) \]

       5. *-lowering-*.f32 88.9%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{+.f32}\left(ux, ux\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   14. Applied egg-rr 88.9%

       \[\leadsto \left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux + ux\right) - ux \cdot ux}} \]

   if 4.00000005e-7 < maxCos

   1. Initial program 53.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\left(1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left({\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left(1 + ux \cdot \left(maxCos - 1\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f32 47.0%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right)\right) \]

   7. Simplified 47.0%

      \[\leadsto \color{blue}{\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]

   8. Taylor expanded in ux around inf

      \[\leadsto \mathsf{sqrt.f32}\left(\color{blue}{\left({ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left({ux}^{2}\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left(ux \cdot ux\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(2 \cdot \frac{1}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2 \cdot 1}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2}{ux} - \frac{2 \cdot maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      8. div-sub N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 - 2 \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 + -2 \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      11. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\left(\frac{2 + -2 \cdot maxCos}{ux}\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(2 + -2 \cdot maxCos\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      13. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(-2 \cdot maxCos\right)\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      20. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

      22. +-lowering-+.f32 83.8%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

   10. Simplified 83.8%

       \[\leadsto \sqrt{\color{blue}{\left(ux \cdot ux\right) \cdot \left(\frac{2 + -2 \cdot maxCos}{ux} - \left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{\left(ux + ux\right) - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{ux + \left(ux - ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.0000000467443897e-7)
   (* (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0) (sqrt (+ ux (- ux (* ux ux)))))
   (sqrt
    (*
     (* ux ux)
     (+ (/ (+ 2.0 (* maxCos -2.0)) ux) (* (+ maxCos -1.0) (- 1.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.0000000467443897e-7f) {
		tmp = (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf((ux + (ux - (ux * ux))));
	} else {
		tmp = sqrtf(((ux * ux) * (((2.0f + (maxCos * -2.0f)) / ux) + ((maxCos + -1.0f) * (1.0f - maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(ux + Float32(ux - Float32(ux * ux)))));
	else
		tmp = sqrt(Float32(Float32(ux * ux) * Float32(Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) / ux) + Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.0000000467443897e-7))
		tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt((ux + (ux - (ux * ux))));
	else
		tmp = sqrt(((ux * ux) * (((single(2.0) + (maxCos * single(-2.0))) / ux) + ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.00000005e-7

   1. Initial program 62.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 98.9%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 98.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 98.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 88.9%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   12. Simplified 88.9%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]
   13. Step-by-step derivation

       1. sub-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(ux\right)\right)\right)\right)\right) - ux \cdot ux\right)\right)\right) \]

       2. remove-double-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux + ux\right) - ux \cdot ux\right)\right)\right) \]

       3. associate--l+ N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux + \left(ux - ux \cdot ux\right)\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(ux, \left(ux - ux \cdot ux\right)\right)\right)\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(ux, \mathsf{\_.f32}\left(ux, \left(ux \cdot ux\right)\right)\right)\right)\right) \]

       6. *-lowering-*.f32 88.8%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{+.f32}\left(ux, \mathsf{\_.f32}\left(ux, \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right)\right) \]

   14. Applied egg-rr 88.8%

       \[\leadsto \left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux + \left(ux - ux \cdot ux\right)}} \]

   if 4.00000005e-7 < maxCos

   1. Initial program 53.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\left(1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left({\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left(1 + ux \cdot \left(maxCos - 1\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f32 47.0%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right)\right) \]

   7. Simplified 47.0%

      \[\leadsto \color{blue}{\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]

   8. Taylor expanded in ux around inf

      \[\leadsto \mathsf{sqrt.f32}\left(\color{blue}{\left({ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left({ux}^{2}\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left(ux \cdot ux\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(2 \cdot \frac{1}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2 \cdot 1}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2}{ux} - \frac{2 \cdot maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      8. div-sub N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 - 2 \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 + -2 \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      11. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\left(\frac{2 + -2 \cdot maxCos}{ux}\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(2 + -2 \cdot maxCos\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      13. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(-2 \cdot maxCos\right)\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      20. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

      22. +-lowering-+.f32 83.8%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

   10. Simplified 83.8%

       \[\leadsto \sqrt{\color{blue}{\left(ux \cdot ux\right) \cdot \left(\frac{2 + -2 \cdot maxCos}{ux} - \left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{ux + \left(ux - ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.0000000467443897e-7)
   (* (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0) (sqrt (* ux (- 2.0 ux))))
   (sqrt
    (*
     (* ux ux)
     (+ (/ (+ 2.0 (* maxCos -2.0)) ux) (* (+ maxCos -1.0) (- 1.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.0000000467443897e-7f) {
		tmp = (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf(((ux * ux) * (((2.0f + (maxCos * -2.0f)) / ux) + ((maxCos + -1.0f) * (1.0f - maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(Float32(ux * ux) * Float32(Float32(Float32(Float32(2.0) + Float32(maxCos * Float32(-2.0))) / ux) + Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.0000000467443897e-7))
		tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt(((ux * ux) * (((single(2.0) + (maxCos * single(-2.0))) / ux) + ((maxCos + single(-1.0)) * (single(1.0) - maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.00000005e-7

   1. Initial program 62.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 98.9%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 98.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 98.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 88.9%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   12. Simplified 88.9%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]

   13. Taylor expanded in ux around 0

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux \cdot \left(2 + -1 \cdot ux\right)\right)}\right)\right) \]
   14. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(2 + -1 \cdot ux\right)\right)\right)\right) \]

       2. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(-1 \cdot ux\right)\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(ux\right)\right)\right)\right)\right)\right) \]

       4. neg-lowering-neg.f32 88.8%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{neg.f32}\left(ux\right)\right)\right)\right)\right) \]

   15. Simplified 88.8%

       \[\leadsto \left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + \left(-ux\right)\right)}} \]

   if 4.00000005e-7 < maxCos

   1. Initial program 53.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\left(1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left({\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left(1 + ux \cdot \left(maxCos - 1\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f32 47.0%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right)\right) \]

   7. Simplified 47.0%

      \[\leadsto \color{blue}{\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]

   8. Taylor expanded in ux around inf

      \[\leadsto \mathsf{sqrt.f32}\left(\color{blue}{\left({ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)}\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left({ux}^{2}\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left(ux \cdot ux\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(2 \cdot \frac{1}{ux} - \left(2 \cdot \frac{maxCos}{ux} + {\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      4. associate--r+ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(2 \cdot \frac{1}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2 \cdot 1}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2}{ux} - 2 \cdot \frac{maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\left(\frac{2}{ux} - \frac{2 \cdot maxCos}{ux}\right) - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      8. div-sub N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 - 2 \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      9. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \left(\frac{2 + -2 \cdot maxCos}{ux} - {\left(maxCos - 1\right)}^{2}\right)\right)\right) \]

      11. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\left(\frac{2 + -2 \cdot maxCos}{ux}\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      12. /-lowering-/.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\left(2 + -2 \cdot maxCos\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      13. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \left(-2 \cdot maxCos\right)\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      14. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \left({\left(maxCos - 1\right)}^{2}\right)\right)\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      16. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      17. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      18. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      19. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right)\right)\right) \]

      20. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

      22. +-lowering-+.f32 83.8%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, ux\right), \mathsf{\_.f32}\left(\mathsf{/.f32}\left(\mathsf{+.f32}\left(2, \mathsf{*.f32}\left(-2, maxCos\right)\right), ux\right), \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right) \]

   10. Simplified 83.8%

       \[\leadsto \sqrt{\color{blue}{\left(ux \cdot ux\right) \cdot \left(\frac{2 + -2 \cdot maxCos}{ux} - \left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2 + maxCos \cdot -2}{ux} + \left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 87.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - 2 \cdot maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.0000000467443897e-7)
   (* (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0) (sqrt (* ux (- 2.0 ux))))
   (sqrt
    (*
     ux
     (+ 2.0 (- (* (+ maxCos -1.0) (* ux (- 1.0 maxCos))) (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.0000000467443897e-7f) {
		tmp = (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux * (2.0f + (((maxCos + -1.0f) * (ux * (1.0f - maxCos))) - (2.0f * maxCos)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.0000000467443897e-7))
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * Float32(Float32(1.0) - maxCos))) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.0000000467443897e-7))
		tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = sqrt((ux * (single(2.0) + (((maxCos + single(-1.0)) * (ux * (single(1.0) - maxCos))) - (single(2.0) * maxCos)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.00000005e-7

   1. Initial program 62.8%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 62.8%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 98.9%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 98.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 98.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 88.9%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   12. Simplified 88.9%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]

   13. Taylor expanded in ux around 0

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux \cdot \left(2 + -1 \cdot ux\right)\right)}\right)\right) \]
   14. Step-by-step derivation

       1. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(2 + -1 \cdot ux\right)\right)\right)\right) \]

       2. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(-1 \cdot ux\right)\right)\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(\mathsf{neg}\left(ux\right)\right)\right)\right)\right)\right) \]

       4. neg-lowering-neg.f32 88.8%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{neg.f32}\left(ux\right)\right)\right)\right)\right) \]

   15. Simplified 88.8%

       \[\leadsto \left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 + \left(-ux\right)\right)}} \]

   if 4.00000005e-7 < maxCos

   1. Initial program 53.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 53.4%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.5%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
   7. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      2. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 99.2%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

   9. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right) - 2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right) - 2 \cdot maxCos\right)\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right) - 2 \cdot maxCos\right)\right)\right) \]

       3. associate--l+ N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(2 + \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right) - 2 \cdot maxCos\right)\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right) - 2 \cdot maxCos\right)\right)\right)\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos - 1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(ux \cdot \left(1 - maxCos\right)\right), \left(maxCos - 1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \left(1 - maxCos\right)\right), \left(maxCos - 1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \left(maxCos - 1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \left(maxCos + -1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f32 83.7%

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right), \mathsf{*.f32}\left(2, maxCos\right)\right)\right)\right)\right) \]

   11. Simplified 83.7%

       \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + \left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right) - 2 \cdot maxCos\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.0000000467443897 \cdot 10^{-7}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - 2 \cdot maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 83.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - 2 \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0034000000450760126)
   (sqrt
    (*
     ux
     (+ 2.0 (- (* (+ maxCos -1.0) (* ux (- 1.0 maxCos))) (* 2.0 maxCos)))))
   (* (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0034000000450760126f) {
		tmp = sqrtf((ux * (2.0f + (((maxCos + -1.0f) * (ux * (1.0f - maxCos))) - (2.0f * maxCos)))));
	} else {
		tmp = (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0034000000450760126))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(Float32(maxCos + Float32(-1.0)) * Float32(ux * Float32(Float32(1.0) - maxCos))) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0034000000450760126))
		tmp = sqrt((ux * (single(2.0) + (((maxCos + single(-1.0)) * (ux * (single(1.0) - maxCos))) - (single(2.0) * maxCos)))));
	else
		tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00340000005

   1. Initial program 60.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 60.1%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]
   7. Step-by-step derivation

      1. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      2. distribute-lft-out-- N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(\left(\left(1 + \left(\mathsf{neg}\left(maxCos\right)\right)\right) - \left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(maxCos + -1\right)\right)\right)\right)\right) \]

   8. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\left(1 - maxCos\right) + \left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right)\right) - \left(maxCos + -1\right)\right)}} \]

   9. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right) - 2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\left(ux \cdot \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right) - 2 \cdot maxCos\right)\right)\right) \]

       2. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(2 + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right) - 2 \cdot maxCos\right)\right)\right) \]

       3. associate--l+ N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \left(2 + \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right) - 2 \cdot maxCos\right)\right)\right)\right) \]

       4. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right) - 2 \cdot maxCos\right)\right)\right)\right) \]

       5. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       6. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos - 1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\left(ux \cdot \left(1 - maxCos\right)\right), \left(maxCos - 1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \left(1 - maxCos\right)\right), \left(maxCos - 1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       9. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \left(maxCos - 1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       10. sub-neg N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \left(maxCos + -1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       12. +-lowering-+.f32 N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right), \left(2 \cdot maxCos\right)\right)\right)\right)\right) \]

       13. *-lowering-*.f32 95.1%

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(2, \mathsf{\_.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{\_.f32}\left(1, maxCos\right)\right), \mathsf{+.f32}\left(maxCos, -1\right)\right), \mathsf{*.f32}\left(2, maxCos\right)\right)\right)\right)\right) \]

   11. Simplified 95.1%

       \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 + \left(\left(ux \cdot \left(1 - maxCos\right)\right) \cdot \left(maxCos + -1\right) - 2 \cdot maxCos\right)\right)}} \]

   if 0.00340000005 < uy

   1. Initial program 63.7%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 97.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 91.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 91.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 59.3%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   12. Simplified 59.3%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]

   13. Taylor expanded in ux around 0

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(2 \cdot ux\right)}\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot 2\right)\right)\right) \]

       2. *-lowering-*.f32 51.4%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, 2\right)\right)\right) \]

   15. Simplified 51.4%

       \[\leadsto \left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(1 - maxCos\right)\right) - 2 \cdot maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 83.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{\left(ux \cdot maxCos\right) \cdot \left(\left(-2\right) - ux \cdot -2\right) - ux \cdot \left(ux + -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0034000000450760126)
   (sqrt (- (* (* ux maxCos) (- (- 2.0) (* ux -2.0))) (* ux (+ ux -2.0))))
   (* (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0034000000450760126f) {
		tmp = sqrtf((((ux * maxCos) * (-2.0f - (ux * -2.0f))) - (ux * (ux + -2.0f))));
	} else {
		tmp = (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0034000000450760126))
		tmp = sqrt(Float32(Float32(Float32(ux * maxCos) * Float32(Float32(-Float32(2.0)) - Float32(ux * Float32(-2.0)))) - Float32(ux * Float32(ux + Float32(-2.0)))));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0034000000450760126))
		tmp = sqrt((((ux * maxCos) * (-single(2.0) - (ux * single(-2.0)))) - (ux * (ux + single(-2.0)))));
	else
		tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00340000005

   1. Initial program 60.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 60.1%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing

   5. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\sqrt{1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}}} \]
   6. Step-by-step derivation

      1. sqrt-lowering-sqrt.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\left(1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left({\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      4. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left(1 + ux \cdot \left(maxCos - 1\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      5. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      14. +-lowering-+.f32 59.3%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right)\right) \]

   7. Simplified 59.3%

      \[\leadsto \color{blue}{\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]

   8. Taylor expanded in ux around 0

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{\left(1 + ux \cdot \left(\left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \left(ux \cdot \left(\left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2\right)\right)\right)\right)\right) \]

      2. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(\left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2\right)\right)\right)\right)\right) \]

      3. associate--l+ N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(2 \cdot maxCos + \left(ux \cdot {\left(maxCos - 1\right)}^{2} - 2\right)\right)\right)\right)\right)\right) \]

      4. +-lowering-+.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 \cdot maxCos\right), \left(ux \cdot {\left(maxCos - 1\right)}^{2} - 2\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \left(ux \cdot {\left(maxCos - 1\right)}^{2} - 2\right)\right)\right)\right)\right)\right) \]

      6. --lowering--.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right), 2\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \left({\left(maxCos - 1\right)}^{2}\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f32 N/A

         \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      10. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      12. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f32 61.9%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   10. Simplified 61.9%

       \[\leadsto \sqrt{1 - \color{blue}{\left(1 + ux \cdot \left(2 \cdot maxCos + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right) - 2\right)\right)\right)}} \]

   11. Taylor expanded in maxCos around 0

       \[\leadsto \mathsf{sqrt.f32}\left(\color{blue}{\left(-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right) - ux \cdot \left(ux - 2\right)\right)}\right) \]
   12. Step-by-step derivation

       1. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\mathsf{neg}\left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       3. neg-lowering-neg.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\left(maxCos \cdot \left(ux \cdot \left(2 + -2 \cdot ux\right)\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       4. associate-*r* N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\left(\left(maxCos \cdot ux\right) \cdot \left(2 + -2 \cdot ux\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       5. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(maxCos \cdot ux\right), \left(2 + -2 \cdot ux\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       6. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\left(ux \cdot maxCos\right), \left(2 + -2 \cdot ux\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       7. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, maxCos\right), \left(2 + -2 \cdot ux\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       8. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, maxCos\right), \mathsf{+.f32}\left(2, \left(-2 \cdot ux\right)\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, maxCos\right), \mathsf{+.f32}\left(2, \left(ux \cdot -2\right)\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       10. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, maxCos\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(ux, -2\right)\right)\right)\right), \left(ux \cdot \left(ux - 2\right)\right)\right)\right) \]

       11. *-lowering-*.f32 N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, maxCos\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(ux, -2\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \left(ux - 2\right)\right)\right)\right) \]

       12. sub-neg N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, maxCos\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(ux, -2\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \left(ux + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, maxCos\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(ux, -2\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \left(ux + -2\right)\right)\right)\right) \]

       14. +-lowering-+.f32 94.5%

           \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{neg.f32}\left(\mathsf{*.f32}\left(\mathsf{*.f32}\left(ux, maxCos\right), \mathsf{+.f32}\left(2, \mathsf{*.f32}\left(ux, -2\right)\right)\right)\right), \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(ux, -2\right)\right)\right)\right) \]

   13. Simplified 94.5%

       \[\leadsto \sqrt{\color{blue}{\left(-\left(ux \cdot maxCos\right) \cdot \left(2 + ux \cdot -2\right)\right) - ux \cdot \left(ux + -2\right)}} \]

   if 0.00340000005 < uy

   1. Initial program 63.7%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 97.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 91.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 91.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 59.3%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   12. Simplified 59.3%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]

   13. Taylor expanded in ux around 0

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(2 \cdot ux\right)}\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot 2\right)\right)\right) \]

       2. *-lowering-*.f32 51.4%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, 2\right)\right)\right) \]

   15. Simplified 51.4%

       \[\leadsto \left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{\left(ux \cdot maxCos\right) \cdot \left(\left(-2\right) - ux \cdot -2\right) - ux \cdot \left(ux + -2\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 79.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0034000000450760126)
   (sqrt (- (* 2.0 ux) (* ux ux)))
   (* (+ (* (* -2.0 (* uy uy)) (* PI PI)) 1.0) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0034000000450760126f) {
		tmp = sqrtf(((2.0f * ux) - (ux * ux)));
	} else {
		tmp = (((-2.0f * (uy * uy)) * (((float) M_PI) * ((float) M_PI))) + 1.0f) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0034000000450760126))
		tmp = sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux)));
	else
		tmp = Float32(Float32(Float32(Float32(Float32(-2.0) * Float32(uy * uy)) * Float32(Float32(pi) * Float32(pi))) + Float32(1.0)) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0034000000450760126))
		tmp = sqrt(((single(2.0) * ux) - (ux * ux)));
	else
		tmp = (((single(-2.0) * (uy * uy)) * (single(pi) * single(pi))) + single(1.0)) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00340000005

   1. Initial program 60.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 60.1%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.4%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 92.7%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 92.7%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \color{blue}{\sqrt{2 \cdot ux - {ux}^{2}}} \]
   11. Step-by-step derivation

       1. sqrt-lowering-sqrt.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\left(2 \cdot ux - {ux}^{2}\right)\right) \]

       2. --lowering--.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(2 \cdot ux\right), \left({ux}^{2}\right)\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux \cdot 2\right), \left({ux}^{2}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, 2\right), \left({ux}^{2}\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, 2\right), \left(ux \cdot ux\right)\right)\right) \]

       6. *-lowering-*.f32 89.2%

          \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, 2\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right) \]

   12. Simplified 89.2%

       \[\leadsto \color{blue}{\sqrt{ux \cdot 2 - ux \cdot ux}} \]

   if 0.00340000005 < uy

   1. Initial program 63.7%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      4. distribute-lft1-in N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      5. associate-+l+ N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

      7. +-commutative N/A

         \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      2. *-lft-identity N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

      3. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

      7. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

      8. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 97.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
   8. Step-by-step derivation

      1. associate--r+ N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

      2. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

      3. --lowering--.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      5. neg-lowering-neg.f32 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

      7. *-lowering-*.f32 91.8%

         \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   9. Simplified 91.8%

      \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

   10. Taylor expanded in uy around 0

       \[\leadsto \mathsf{*.f32}\left(\color{blue}{\left(1 + -2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)}, \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]
   11. Step-by-step derivation

       1. +-lowering-+.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(-2 \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)}\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(\left(-2 \cdot {uy}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       3. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\left(-2 \cdot {uy}^{2}\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \color{blue}{\mathsf{*.f32}\left(ux, ux\right)}\right)\right)\right) \]

       4. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left({uy}^{2}\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(\color{blue}{ux}, ux\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \left(uy \cdot uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       6. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left({\mathsf{PI}\left(\right)}^{2}\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       8. *-lowering-*.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, \color{blue}{ux}\right)\right)\right)\right) \]

       9. PI-lowering-PI.f32 N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

       10. PI-lowering-PI.f32 59.3%

           \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

   12. Simplified 59.3%

       \[\leadsto \color{blue}{\left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right)} \cdot \sqrt{\left(ux - \left(-ux\right)\right) - ux \cdot ux} \]

   13. Taylor expanded in ux around 0

       \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(2 \cdot ux\right)}\right)\right) \]
   14. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(ux \cdot 2\right)\right)\right) \]

       2. *-lowering-*.f32 51.4%

          \[\leadsto \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(\mathsf{*.f32}\left(-2, \mathsf{*.f32}\left(uy, uy\right)\right), \mathsf{*.f32}\left(\mathsf{PI.f32}\left(\right), \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(ux, 2\right)\right)\right) \]

   15. Simplified 51.4%

       \[\leadsto \left(1 + \left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0034000000450760126:\\ \;\;\;\;\sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-2 \cdot \left(uy \cdot uy\right)\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 75.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux - ux \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (- (* 2.0 ux) (* ux ux))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((2.0f * ux) - (ux * ux)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(((2.0e0 * ux) - (ux * ux)))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt(((single(2.0) * ux) - (ux * ux)));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(1 \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   2. *-lft-identity N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(1 - \left(\left(1 + ux \cdot \left(maxCos + -1\right)\right) + \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   3. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   5. +-commutative N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right) + 1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   6. distribute-lft1-in N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(\left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right) + ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

   7. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right) - ux \cdot \left(maxCos + -1\right)\right)\right)\right) \]

   8. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(\left(1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right)\right) - \left(ux \cdot \left(maxCos + -1\right)\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right), \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot \left(1 + \left(-maxCos\right)\right) - ux \cdot \left(\left(maxCos + -1\right) \cdot \left(ux \cdot \left(maxCos + -1\right)\right)\right)\right) - ux \cdot \left(maxCos + -1\right)}} \]

7. Taylor expanded in maxCos around 0

   \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\color{blue}{\left(ux - \left(-1 \cdot ux + {ux}^{2}\right)\right)}\right)\right) \]
8. Step-by-step derivation

   1. associate--r+ N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\left(\left(ux - -1 \cdot ux\right) - {ux}^{2}\right)\right)\right) \]

   2. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux - -1 \cdot ux\right), \left({ux}^{2}\right)\right)\right)\right) \]

   3. --lowering--.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(-1 \cdot ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

   4. mul-1-neg N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \left(\mathsf{neg}\left(ux\right)\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

   5. neg-lowering-neg.f32 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left({ux}^{2}\right)\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \left(ux \cdot ux\right)\right)\right)\right) \]

   7. *-lowering-*.f32 92.4%

      \[\leadsto \mathsf{*.f32}\left(\mathsf{cos.f32}\left(\mathsf{*.f32}\left(uy, \mathsf{*.f32}\left(2, \mathsf{PI.f32}\left(\right)\right)\right)\right), \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{\_.f32}\left(ux, \mathsf{neg.f32}\left(ux\right)\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right)\right) \]

9. Simplified 92.4%

   \[\leadsto \cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux - \left(-ux\right)\right) - ux \cdot ux}} \]

10. Taylor expanded in uy around 0

    \[\leadsto \color{blue}{\sqrt{2 \cdot ux - {ux}^{2}}} \]
11. Step-by-step derivation

    1. sqrt-lowering-sqrt.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(2 \cdot ux - {ux}^{2}\right)\right) \]

    2. --lowering--.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(2 \cdot ux\right), \left({ux}^{2}\right)\right)\right) \]

    3. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\left(ux \cdot 2\right), \left({ux}^{2}\right)\right)\right) \]

    4. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, 2\right), \left({ux}^{2}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, 2\right), \left(ux \cdot ux\right)\right)\right) \]

    6. *-lowering-*.f32 75.5%

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, 2\right), \mathsf{*.f32}\left(ux, ux\right)\right)\right) \]

12. Simplified 75.5%

    \[\leadsto \color{blue}{\sqrt{ux \cdot 2 - ux \cdot ux}} \]

13. Final simplification 75.5%

    \[\leadsto \sqrt{2 \cdot ux - ux \cdot ux} \]
14. Add Preprocessing

Alternative 21: 75.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 (sqrt (* ux (- (- ux) -2.0))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (-ux - -2.0f)));
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (-ux - (-2.0e0))))
end function

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(ux * Float32(Float32(-ux) - Float32(-2.0))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * (-ux - single(-2.0))));
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}}} \]
6. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\left(1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right) \]

   2. --lowering--.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left({\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left(1 + ux \cdot \left(maxCos - 1\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f32 52.3%

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right)\right) \]

7. Simplified 52.3%

   \[\leadsto \color{blue}{\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]

8. Taylor expanded in ux around 0

   \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{\left(1 + ux \cdot \left(\left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \left(ux \cdot \left(\left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2\right)\right)\right)\right)\right) \]

   2. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(\left(2 \cdot maxCos + ux \cdot {\left(maxCos - 1\right)}^{2}\right) - 2\right)\right)\right)\right)\right) \]

   3. associate--l+ N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(2 \cdot maxCos + \left(ux \cdot {\left(maxCos - 1\right)}^{2} - 2\right)\right)\right)\right)\right)\right) \]

   4. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\left(2 \cdot maxCos\right), \left(ux \cdot {\left(maxCos - 1\right)}^{2} - 2\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \left(ux \cdot {\left(maxCos - 1\right)}^{2} - 2\right)\right)\right)\right)\right)\right) \]

   6. --lowering--.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\left(ux \cdot {\left(maxCos - 1\right)}^{2}\right), 2\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \left({\left(maxCos - 1\right)}^{2}\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos - 1\right), \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right), \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\left(maxCos + -1\right), \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   12. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos - 1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \left(maxCos + -1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

   15. +-lowering-+.f32 54.5%

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(\mathsf{*.f32}\left(2, maxCos\right), \mathsf{\_.f32}\left(\mathsf{*.f32}\left(ux, \mathsf{*.f32}\left(\mathsf{+.f32}\left(maxCos, -1\right), \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), 2\right)\right)\right)\right)\right)\right) \]

10. Simplified 54.5%

    \[\leadsto \sqrt{1 - \color{blue}{\left(1 + ux \cdot \left(2 \cdot maxCos + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(maxCos + -1\right)\right) - 2\right)\right)\right)}} \]

11. Taylor expanded in maxCos around 0

    \[\leadsto \mathsf{sqrt.f32}\left(\color{blue}{\left(-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)\right)}\right) \]
12. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\left(-1 \cdot ux\right) \cdot \left(ux - 2\right)\right)\right) \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(ux - 2\right)\right)\right) \]

    3. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\left(\mathsf{neg}\left(ux\right)\right), \left(ux - 2\right)\right)\right) \]

    4. neg-lowering-neg.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(ux\right), \left(ux - 2\right)\right)\right) \]

    5. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(ux\right), \left(ux + \left(\mathsf{neg}\left(2\right)\right)\right)\right)\right) \]

    6. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(ux\right), \left(ux + -2\right)\right)\right) \]

    7. +-lowering-+.f32 75.4%

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{*.f32}\left(\mathsf{neg.f32}\left(ux\right), \mathsf{+.f32}\left(ux, -2\right)\right)\right) \]

13. Simplified 75.4%

    \[\leadsto \sqrt{\color{blue}{\left(-ux\right) \cdot \left(ux + -2\right)}} \]

14. Final simplification 75.4%

    \[\leadsto \sqrt{ux \cdot \left(\left(-ux\right) - -2\right)} \]
15. Add Preprocessing

Alternative 22: 6.6% accurate, 223.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (ux uy maxCos) :precision binary32 0.0)

C

float code(float ux, float uy, float maxCos) {
	return 0.0f;
}

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = 0.0e0
end function

Julia

function code(ux, uy, maxCos)
	return Float32(0.0)
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = single(0.0);
end

Derivation

1. Initial program 61.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 + \left(\mathsf{neg}\left(ux\right)\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   3. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) + 1\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   4. distribute-lft1-in N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   5. associate-+l+ N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   6. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)} \]

   7. +-commutative N/A

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right) + \left(\mathsf{neg}\left(ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

3. Simplified 61.1%

   \[\leadsto \color{blue}{\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]
4. Add Preprocessing

5. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\sqrt{1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}}} \]
6. Step-by-step derivation

   1. sqrt-lowering-sqrt.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\left(1 - {\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right) \]

   2. --lowering--.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left({\left(1 + ux \cdot \left(maxCos - 1\right)\right)}^{2}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \left(\left(1 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   4. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\left(1 + ux \cdot \left(maxCos - 1\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   5. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   6. *-lowering-*.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   7. sub-neg N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f32 N/A

      \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \left(1 + ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \left(ux \cdot \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f32 N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos - 1\right)\right)\right)\right)\right)\right) \]

   12. sub-neg N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \left(maxCos + -1\right)\right)\right)\right)\right)\right) \]

   14. +-lowering-+.f32 52.3%

       \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \mathsf{*.f32}\left(\mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right), \mathsf{+.f32}\left(1, \mathsf{*.f32}\left(ux, \mathsf{+.f32}\left(maxCos, -1\right)\right)\right)\right)\right)\right) \]

7. Simplified 52.3%

   \[\leadsto \color{blue}{\sqrt{1 - \left(1 + ux \cdot \left(maxCos + -1\right)\right) \cdot \left(1 + ux \cdot \left(maxCos + -1\right)\right)}} \]

8. Taylor expanded in ux around 0

   \[\leadsto \mathsf{sqrt.f32}\left(\mathsf{\_.f32}\left(1, \color{blue}{1}\right)\right) \]
9. Step-by-step derivation

10. Simplified 6.6%

    \[\leadsto \sqrt{1 - \color{blue}{1}} \]
11. Step-by-step derivation

    1. pow1/2 N/A

       \[\leadsto {\left(1 - 1\right)}^{\color{blue}{\frac{1}{2}}} \]

    2. metadata-eval N/A

       \[\leadsto {0}^{\frac{1}{2}} \]

    3. metadata-eval 6.6%

       \[\leadsto 0 \]

12. Applied egg-rr 6.6%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024138 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))