UniformSampleCone, y

Percentage Accurate: 58.0% → 98.3%

Time: 14.3s

Alternatives: 15

Speedup: 2.0×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 58.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* 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 sinf(((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(sin(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 = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 98.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. distribute-lft-in 98.3%

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

   2. cancel-sign-sub-inv 98.3%

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

   3. fma-define 98.3%

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

   4. metadata-eval 98.3%

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

7. Applied egg-rr 98.3%

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

8. Final simplification 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. distribute-lft-in 98.3%

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

   2. cancel-sign-sub-inv 98.3%

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

   3. fma-define 98.3%

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

   4. metadata-eval 98.3%

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in maxCos around 0 98.3%

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

9. Final simplification 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around 0 98.2%

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

7. Final simplification 98.2%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in ux around 0 55.7%

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

4. Taylor expanded in maxCos around 0 97.3%

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

   1. associate-*r* 97.3%

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

   2. mul-1-neg 97.3%

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

   3. sub-neg 97.3%

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

   4. metadata-eval 97.3%

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

6. Simplified 97.3%

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

7. Final simplification 97.3%

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

Alternative 5: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

   1. distribute-lft-in 98.3%

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

   2. cancel-sign-sub-inv 98.3%

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

   3. fma-define 98.3%

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

   4. metadata-eval 98.3%

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

7. Applied egg-rr 98.3%

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

8. Taylor expanded in maxCos around 0 97.3%

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

9. Final simplification 97.3%

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

Alternative 6: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.5%

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

3. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r* 98.2%

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

   3. mul-1-neg 98.2%

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

   4. sub-neg 98.2%

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

   5. metadata-eval 98.2%

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

   6. +-commutative 98.2%

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

5. Simplified 98.2%

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

6. Taylor expanded in maxCos around -inf 49.4%

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

7. Taylor expanded in maxCos around 0 97.3%

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

   1. fma-define 97.3%

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

   2. mul-1-neg 97.3%

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

   3. fmm-undef 97.3%

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

   4. neg-mul-1 97.3%

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

9. Simplified 97.3%

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

10. Final simplification 97.3%

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

Alternative 7: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(maxCos \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\frac{\left(\frac{2 - ux}{maxCos} - ux \cdot -2\right) - 2}{maxCos} - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999987376214e-7)
   (* (sin (* (* uy 2.0) PI)) (sqrt (- (* 2.0 ux) (* ux ux))))
   (*
    2.0
    (*
     (* maxCos (* uy PI))
     (sqrt
      (*
       ux
       (- (/ (- (- (/ (- 2.0 ux) maxCos) (* ux -2.0)) 2.0) maxCos) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.999999987376214e-7f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((2.0f * ux) - (ux * ux)));
	} else {
		tmp = 2.0f * ((maxCos * (uy * ((float) M_PI))) * sqrtf((ux * ((((((2.0f - ux) / maxCos) - (ux * -2.0f)) - 2.0f) / maxCos) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999987376214e-7))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(2.0) * ux) - Float32(ux * ux))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(maxCos * Float32(uy * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(Float32(Float32(Float32(Float32(Float32(2.0) - ux) / maxCos) - Float32(ux * Float32(-2.0))) - Float32(2.0)) / maxCos) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.999999987376214e-7))
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt(((single(2.0) * ux) - (ux * ux)));
	else
		tmp = single(2.0) * ((maxCos * (uy * single(pi))) * sqrt((ux * ((((((single(2.0) - ux) / maxCos) - (ux * single(-2.0))) - single(2.0)) / maxCos) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999999e-7

   1. Initial program 53.7%

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

   3. Taylor expanded in ux around 0 98.2%

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

      1. associate--l+ 98.2%

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

      2. associate-*r* 98.2%

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

      3. mul-1-neg 98.2%

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

      4. sub-neg 98.2%

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

      5. metadata-eval 98.2%

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

      6. +-commutative 98.2%

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

   5. Simplified 98.2%

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

      1. distribute-lft-in 98.2%

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

      2. cancel-sign-sub-inv 98.2%

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

      3. fma-define 98.2%

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

      4. metadata-eval 98.2%

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

   7. Applied egg-rr 98.2%

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

   8. Taylor expanded in maxCos around 0 98.2%

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

      1. neg-mul-1 98.2%

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

   10. Simplified 98.2%

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

   if 4.99999999e-7 < maxCos

   1. Initial program 52.4%

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

   3. Taylor expanded in ux around 0 98.3%

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

      1. associate--l+ 98.4%

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

      2. associate-*r* 98.4%

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

      3. mul-1-neg 98.4%

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

      4. sub-neg 98.4%

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

      5. metadata-eval 98.4%

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

      6. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in maxCos around -inf 98.4%

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

   7. Taylor expanded in uy around 0 87.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux - ux \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(maxCos \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\frac{\left(\frac{2 - ux}{maxCos} - ux \cdot -2\right) - 2}{maxCos} - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 95.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(maxCos \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\frac{\left(\frac{2 - ux}{maxCos} - ux \cdot -2\right) - 2}{maxCos} - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 4.999999987376214e-7)
   (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux))))
   (*
    2.0
    (*
     (* maxCos (* uy PI))
     (sqrt
      (*
       ux
       (- (/ (- (- (/ (- 2.0 ux) maxCos) (* ux -2.0)) 2.0) maxCos) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 4.999999987376214e-7f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = 2.0f * ((maxCos * (uy * ((float) M_PI))) * sqrtf((ux * ((((((2.0f - ux) / maxCos) - (ux * -2.0f)) - 2.0f) / maxCos) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(4.999999987376214e-7))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(maxCos * Float32(uy * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(Float32(Float32(Float32(Float32(Float32(2.0) - ux) / maxCos) - Float32(ux * Float32(-2.0))) - Float32(2.0)) / maxCos) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(4.999999987376214e-7))
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((ux * (single(2.0) - ux)));
	else
		tmp = single(2.0) * ((maxCos * (uy * single(pi))) * sqrt((ux * ((((((single(2.0) - ux) / maxCos) - (ux * single(-2.0))) - single(2.0)) / maxCos) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999999e-7

   1. Initial program 53.7%

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

   3. Taylor expanded in ux around 0 98.2%

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

      1. associate--l+ 98.2%

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

      2. associate-*r* 98.2%

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

      3. mul-1-neg 98.2%

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

      4. sub-neg 98.2%

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

      5. metadata-eval 98.2%

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

      6. +-commutative 98.2%

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

   5. Simplified 98.2%

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

      1. distribute-lft-in 98.2%

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

      2. cancel-sign-sub-inv 98.2%

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

      3. fma-define 98.2%

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

      4. metadata-eval 98.2%

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

   7. Applied egg-rr 98.2%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot 2 + ux \cdot \mathsf{fma}\left(-ux, {\left(-1 + maxCos\right)}^{2}, -2 \cdot maxCos\right)}} \]
   8. Step-by-step derivation

      1. pow1/2 98.2%

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(ux \cdot 2 + ux \cdot \mathsf{fma}\left(-ux, {\left(-1 + maxCos\right)}^{2}, -2 \cdot maxCos\right)\right)}^{0.5}} \]

      2. distribute-lft-in 98.2%

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot {\color{blue}{\left(ux \cdot \left(2 + \mathsf{fma}\left(-ux, {\left(-1 + maxCos\right)}^{2}, -2 \cdot maxCos\right)\right)\right)}}^{0.5} \]

      3. *-commutative 98.2%

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot {\color{blue}{\left(\left(2 + \mathsf{fma}\left(-ux, {\left(-1 + maxCos\right)}^{2}, -2 \cdot maxCos\right)\right) \cdot ux\right)}}^{0.5} \]

      4. unpow-prod-down 98.2%

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{\left({\left(2 + \mathsf{fma}\left(-ux, {\left(-1 + maxCos\right)}^{2}, -2 \cdot maxCos\right)\right)}^{0.5} \cdot {ux}^{0.5}\right)} \]

      5. pow1/2 98.2%

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\color{blue}{\sqrt{2 + \mathsf{fma}\left(-ux, {\left(-1 + maxCos\right)}^{2}, -2 \cdot maxCos\right)}} \cdot {ux}^{0.5}\right) \]

      6. *-commutative 98.2%

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \left(\sqrt{2 + \mathsf{fma}\left(-ux, {\left(-1 + maxCos\right)}^{2}, \color{blue}{maxCos \cdot -2}\right)} \cdot {ux}^{0.5}\right) \]

      7. pow1/2 98.3%

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

   9. Applied egg-rr 98.3%

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

   10. Taylor expanded in maxCos around 0 98.2%

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

       1. neg-mul-1 98.2%

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

       2. unsub-neg 98.2%

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

   12. Simplified 98.2%

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

   if 4.99999999e-7 < maxCos

   1. Initial program 52.4%

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

   3. Taylor expanded in ux around 0 98.3%

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

      1. associate--l+ 98.4%

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

      2. associate-*r* 98.4%

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

      3. mul-1-neg 98.4%

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

      4. sub-neg 98.4%

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

      5. metadata-eval 98.4%

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

      6. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in maxCos around -inf 98.4%

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

   7. Taylor expanded in uy around 0 87.0%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999987376214 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(maxCos \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(\frac{\left(\frac{2 - ux}{maxCos} - ux \cdot -2\right) - 2}{maxCos} - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 82.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.001500000013038516:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.001500000013038516)
   (* (sin (* (* uy 2.0) PI)) (sqrt (* 2.0 ux)))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ -1.0 maxCos))) (+ -1.0 (- ux (* ux maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.001500000013038516f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((2.0f * ux));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (-1.0f + maxCos))) * (-1.0f + (ux - (ux * maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.001500000013038516))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(2.0) * ux)));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(Float32(-1.0) + Float32(ux - Float32(ux * maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.001500000013038516))
		tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt((single(2.0) * ux));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (single(-1.0) + maxCos))) * (single(-1.0) + (ux - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 0.00150000001

   1. Initial program 40.8%

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

   3. Taylor expanded in ux around 0 41.5%

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

   4. Taylor expanded in maxCos around 0 84.9%

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

   if 0.00150000001 < ux

   1. Initial program 91.5%

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

      1. associate-*l* 91.5%

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

      2. sub-neg 91.5%

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

      3. +-commutative 91.5%

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

      4. distribute-rgt-neg-in 91.5%

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

      5. fma-define 92.1%

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

   3. Simplified 92.2%

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

   5. Taylor expanded in uy around 0 79.9%

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

   7. Simplified 80.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.001500000013038516:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{2 \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 76.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00015300000086426735:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00015300000086426735)
   (* 2.0 (* (* uy PI) (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (+
       1.0
       (* (+ 1.0 (* ux (+ -1.0 maxCos))) (+ -1.0 (- ux (* ux maxCos))))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00015300000086426735f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f + (ux * (-1.0f + maxCos))) * (-1.0f + (ux - (ux * maxCos)))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00015300000086426735))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(Float32(-1.0) + Float32(ux - Float32(ux * maxCos))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00015300000086426735))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) + (ux * (single(-1.0) + maxCos))) * (single(-1.0) + (ux - (ux * maxCos)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.53e-4

   1. Initial program 36.2%

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

      1. associate-*l* 36.2%

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

      2. sub-neg 36.2%

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

      3. +-commutative 36.2%

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

      4. distribute-rgt-neg-in 36.2%

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

      5. fma-define 36.4%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in uy around 0 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in ux around 0 78.4%

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

   if 1.53e-4 < ux

   1. Initial program 88.1%

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

      1. associate-*l* 88.1%

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

      2. sub-neg 88.1%

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

      3. +-commutative 88.1%

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

      4. distribute-rgt-neg-in 88.1%

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

      5. fma-define 88.7%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in uy around 0 73.9%

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

   7. Simplified 74.1%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00015300000086426735:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(-1 + \left(ux - ux \cdot maxCos\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00015300000086426735:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(\left(ux \cdot maxCos + 1\right) - ux\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00015300000086426735)
   (* 2.0 (* (* uy PI) (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (-
       1.0
       (* (+ 1.0 (* ux (+ -1.0 maxCos))) (- (+ (* ux maxCos) 1.0) ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00015300000086426735f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f - ((1.0f + (ux * (-1.0f + maxCos))) * (((ux * maxCos) + 1.0f) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00015300000086426735))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) + Float32(ux * Float32(Float32(-1.0) + maxCos))) * Float32(Float32(Float32(ux * maxCos) + Float32(1.0)) - ux))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00015300000086426735))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) - ((single(1.0) + (ux * (single(-1.0) + maxCos))) * (((ux * maxCos) + single(1.0)) - ux)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.53e-4

   1. Initial program 36.2%

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

      1. associate-*l* 36.2%

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

      2. sub-neg 36.2%

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

      3. +-commutative 36.2%

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

      4. distribute-rgt-neg-in 36.2%

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

      5. fma-define 36.4%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in uy around 0 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in ux around 0 78.4%

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

   if 1.53e-4 < ux

   1. Initial program 88.1%

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

      1. associate-*l* 88.1%

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

      2. sub-neg 88.1%

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

      3. +-commutative 88.1%

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

      4. distribute-rgt-neg-in 88.1%

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

      5. fma-define 88.7%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in uy around 0 73.9%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in uy around 0 73.9%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00015300000086426735:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 - \left(1 + ux \cdot \left(-1 + maxCos\right)\right) \cdot \left(\left(ux \cdot maxCos + 1\right) - ux\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00015300000086426735:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00015300000086426735)
   (* 2.0 (* (* uy PI) (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))
   (* 2.0 (* (* uy PI) (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00015300000086426735f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00015300000086426735))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))))));
	else
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00015300000086426735))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.53e-4

   1. Initial program 36.2%

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

      1. associate-*l* 36.2%

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

      2. sub-neg 36.2%

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

      3. +-commutative 36.2%

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

      4. distribute-rgt-neg-in 36.2%

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

      5. fma-define 36.4%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in uy around 0 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in ux around 0 78.4%

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

   if 1.53e-4 < ux

   1. Initial program 88.1%

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

      1. associate-*l* 88.1%

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

      2. sub-neg 88.1%

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

      3. +-commutative 88.1%

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

      4. distribute-rgt-neg-in 88.1%

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

      5. fma-define 88.7%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in uy around 0 73.9%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in maxCos around 0 70.6%

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

      1. neg-mul-1 70.6%

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

      2. sub-neg 70.6%

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

   10. Simplified 70.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00015300000086426735:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 75.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00015300000086426735:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00015300000086426735)
   (* 2.0 (* (* uy PI) (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))
   (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (- 1.0 ux) (+ ux -1.0)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00015300000086426735f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * (2.0f - (2.0f * maxCos)))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((1.0f - ux) * (ux + -1.0f))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00015300000086426735))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos))))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux + Float32(-1.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00015300000086426735))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt((single(1.0) + ((single(1.0) - ux) * (ux + single(-1.0)))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.53e-4

   1. Initial program 36.2%

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

      1. associate-*l* 36.2%

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

      2. sub-neg 36.2%

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

      3. +-commutative 36.2%

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

      4. distribute-rgt-neg-in 36.2%

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

      5. fma-define 36.4%

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

   3. Simplified 36.5%

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

   5. Taylor expanded in uy around 0 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in ux around 0 78.4%

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

   if 1.53e-4 < ux

   1. Initial program 88.1%

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

      1. associate-*l* 88.1%

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

      2. sub-neg 88.1%

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

      3. +-commutative 88.1%

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

      4. distribute-rgt-neg-in 88.1%

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

      5. fma-define 88.7%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in uy around 0 73.9%

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

   7. Simplified 74.1%

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

   8. Taylor expanded in maxCos around 0 70.6%

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

      1. associate-*l* 70.5%

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

      2. neg-mul-1 70.5%

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

      3. sub-neg 70.5%

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

   10. Simplified 70.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00015300000086426735:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 53.5%

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

   1. associate-*l* 53.5%

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

   2. sub-neg 53.5%

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

   3. +-commutative 53.5%

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

   4. distribute-rgt-neg-in 53.5%

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

   5. fma-define 53.8%

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

3. Simplified 53.9%

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

5. Taylor expanded in uy around 0 47.4%

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

7. Simplified 47.5%

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

8. Taylor expanded in ux around 0 68.3%

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

9. Final simplification 68.3%

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

Alternative 15: 7.1% 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 53.5%

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

   1. associate-*l* 53.5%

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

   2. sub-neg 53.5%

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

   3. +-commutative 53.5%

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

   4. distribute-rgt-neg-in 53.5%

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

   5. fma-define 53.8%

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

3. Simplified 53.9%

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

5. Taylor expanded in uy around 0 47.4%

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

7. Simplified 47.5%

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

8. Taylor expanded in ux around 0 7.2%

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

9. Taylor expanded in uy around 0 7.2%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))