UniformSampleCone, x

Percentage Accurate: 56.9% → 99.0%

Time: 14.0s

Alternatives: 9

Speedup: 2.2×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 56.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.4%

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

5. Taylor expanded in ux around 0 99.0%

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

   1. distribute-rgt-in 99.1%

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

   2. associate-*r* 99.1%

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

   3. neg-mul-1 99.1%

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

   4. sub-neg 99.1%

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

   5. metadata-eval 99.1%

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

   6. +-commutative 99.1%

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

   7. mul-1-neg 99.1%

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

   8. *-commutative 99.1%

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

   9. fma-neg 99.1%

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

   10. metadata-eval 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.4%

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

5. Taylor expanded in ux around 0 99.0%

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

6. Final simplification 99.0%

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

Alternative 3: 99.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

3. Taylor expanded in ux around 0 99.0%

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

   1. cancel-sign-sub-inv 99.0%

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

   2. associate-*r* 99.0%

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

   3. mul-1-neg 99.0%

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

   4. sub-neg 99.0%

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

   5. metadata-eval 99.0%

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

   6. +-commutative 99.0%

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

   7. metadata-eval 99.0%

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

   8. *-commutative 99.0%

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.4%

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

5. Taylor expanded in ux around 0 99.0%

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

6. Taylor expanded in maxCos around 0 98.4%

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

7. Final simplification 98.4%

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

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.4%

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

5. Taylor expanded in maxCos around 0 57.6%

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

6. Taylor expanded in ux around 0 98.4%

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

7. Final simplification 98.4%

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

Alternative 6: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0007999999797903001)
   (sqrt (* ux (- 2.0 ux)))
   (* (cos (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0007999999797903001f) {
		tmp = sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = cosf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0007999999797903001))
		tmp = sqrt(Float32(ux * Float32(Float32(2.0) - ux)));
	else
		tmp = Float32(cos(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.0007999999797903001))
		tmp = sqrt((ux * (single(2.0) - ux)));
	else
		tmp = cos((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 7.9999998e-4

   1. Initial program 59.9%

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

      1. associate-*l* 59.9%

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

      2. sub-neg 59.9%

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

      3. +-commutative 59.9%

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

      4. distribute-rgt-neg-in 59.9%

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

      5. fma-define 60.2%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in ux around 0 99.3%

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

      1. distribute-rgt-in 99.5%

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

      2. associate-*r* 99.5%

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

      3. neg-mul-1 99.5%

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

      4. sub-neg 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. mul-1-neg 99.5%

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

      8. *-commutative 99.5%

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

      9. fma-neg 99.5%

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

      10. metadata-eval 99.5%

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

   7. Applied egg-rr 99.5%

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

   8. Taylor expanded in maxCos around 0 94.9%

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

      1. associate-*r* 94.9%

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

      2. +-commutative 94.9%

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

      3. mul-1-neg 94.9%

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

      4. unpow2 94.9%

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

      5. distribute-lft-neg-in 94.9%

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

      6. distribute-rgt-in 94.8%

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

      7. unsub-neg 94.8%

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

   10. Simplified 94.8%

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

   11. Taylor expanded in uy around 0 93.3%

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

   if 7.9999998e-4 < uy

   1. Initial program 53.2%

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

      1. associate-*l* 53.2%

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

      2. sub-neg 53.2%

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

      3. +-commutative 53.2%

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

      4. distribute-rgt-neg-in 53.2%

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

      5. fma-define 53.1%

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

   3. Simplified 53.2%

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

   5. Taylor expanded in maxCos around 0 51.5%

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

   6. Taylor expanded in ux around 0 74.1%

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

      1. *-commutative 74.1%

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

   8. Simplified 74.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0007999999797903001:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 93.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.4%

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

5. Taylor expanded in ux around 0 99.0%

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

6. Taylor expanded in maxCos around 0 94.1%

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

   1. neg-mul-1 94.1%

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

   2. unsub-neg 94.1%

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

8. Simplified 94.1%

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

Alternative 8: 75.6% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - ux)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.4%

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

5. Taylor expanded in ux around 0 99.0%

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

   1. distribute-rgt-in 99.1%

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

   2. associate-*r* 99.1%

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

   3. neg-mul-1 99.1%

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

   4. sub-neg 99.1%

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

   5. metadata-eval 99.1%

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

   6. +-commutative 99.1%

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

   7. mul-1-neg 99.1%

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

   8. *-commutative 99.1%

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

   9. fma-neg 99.1%

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

   10. metadata-eval 99.1%

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in maxCos around 0 94.1%

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

   1. associate-*r* 94.1%

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

   2. +-commutative 94.1%

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

   3. mul-1-neg 94.1%

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

   4. unpow2 94.1%

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

   5. distribute-lft-neg-in 94.1%

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

   6. distribute-rgt-in 94.1%

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

   7. unsub-neg 94.1%

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

10. Simplified 94.1%

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

11. Taylor expanded in uy around 0 78.3%

    \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - ux\right)}} \]
12. Add Preprocessing

Alternative 9: 62.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux} \end{array} \]

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((2.0e0 * ux))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

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

   1. associate-*l* 58.0%

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

   2. sub-neg 58.0%

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

   3. +-commutative 58.0%

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

   4. distribute-rgt-neg-in 58.0%

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

   5. fma-define 58.1%

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

3. Simplified 58.4%

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

5. Taylor expanded in uy around 0 51.1%

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

7. Simplified 51.2%

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

8. Taylor expanded in ux around 0 63.9%

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

9. Taylor expanded in maxCos around 0 62.3%

   \[\leadsto \sqrt{\color{blue}{2 \cdot ux}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024108 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))