UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 19.0s

Alternatives: 13

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: 57.4% 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, 0.4× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\left(ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), 1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5} \cdot {\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow
    (*
     ux
     (+
      1.0
      (- (fma ux (* (- 1.0 maxCos) (+ maxCos -1.0)) (- 1.0 maxCos)) maxCos)))
    1.5)
   (pow (sin (* uy (* 2.0 PI))) 3.0))))

C

float code(float ux, float uy, float maxCos) {
	return cbrtf((powf((ux * (1.0f + (fmaf(ux, ((1.0f - maxCos) * (maxCos + -1.0f)), (1.0f - maxCos)) - maxCos))), 1.5f) * powf(sinf((uy * (2.0f * ((float) M_PI)))), 3.0f)));
}

Julia

function code(ux, uy, maxCos)
	return cbrt(Float32((Float32(ux * Float32(Float32(1.0) + Float32(fma(ux, Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))), Float32(Float32(1.0) - maxCos)) - maxCos))) ^ Float32(1.5)) * (sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) ^ Float32(3.0))))
end

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. fma-define 98.2%

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

   4. *-commutative 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. +-commutative 98.2%

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

   8. mul-1-neg 98.2%

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

   9. sub-neg 98.2%

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

   10. metadata-eval 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. sub-neg 98.2%

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

7. Simplified 98.2%

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

   1. *-commutative 98.2%

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

   2. add-cbrt-cube 98.2%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)} \cdot \sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)}\right) \cdot \sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)}}} \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \]

   3. add-cbrt-cube 98.2%

      \[\leadsto \sqrt[3]{\left(\sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)} \cdot \sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)}\right) \cdot \sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)}} \cdot \color{blue}{\sqrt[3]{\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right) \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}} \]

   4. cbrt-unprod 98.1%

      \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)} \cdot \sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)}\right) \cdot \sqrt{ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(-1 + maxCos\right) \cdot \left(1 - maxCos\right), 1 - maxCos\right) - maxCos\right)\right)}\right) \cdot \left(\left(\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right) \cdot \sin \left(uy \cdot \left(2 \cdot \pi\right)\right)\right)}} \]

9. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(-1 + maxCos\right), 1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5} \cdot {\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3}}} \]

10. Final simplification 98.5%

    \[\leadsto \sqrt[3]{{\left(ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(maxCos + -1\right), 1 - maxCos\right) - maxCos\right)\right)\right)}^{1.5} \cdot {\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3}} \]
11. Add Preprocessing

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(\left(\left(2 - maxCos\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - maxCos\right)\right)}^{1.5}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow (sin (* uy (* 2.0 PI))) 3.0)
   (pow
    (*
     ux
     (- (+ (- 2.0 maxCos) (* ux (* (- 1.0 maxCos) (+ maxCos -1.0)))) maxCos))
    1.5))))

C

float code(float ux, float uy, float maxCos) {
	return cbrtf((powf(sinf((uy * (2.0f * ((float) M_PI)))), 3.0f) * powf((ux * (((2.0f - maxCos) + (ux * ((1.0f - maxCos) * (maxCos + -1.0f)))) - maxCos)), 1.5f)));
}

Julia

function code(ux, uy, maxCos)
	return cbrt(Float32((sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) ^ Float32(3.0)) * (Float32(ux * Float32(Float32(Float32(Float32(2.0) - maxCos) + Float32(ux * Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))))) - maxCos)) ^ Float32(1.5))))
end

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in ux around 0 98.2%

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

   1. add-cube-cbrt 97.0%

      \[\leadsto \sin \color{blue}{\left(\left(\sqrt[3]{uy \cdot \left(2 \cdot \pi\right)} \cdot \sqrt[3]{uy \cdot \left(2 \cdot \pi\right)}\right) \cdot \sqrt[3]{uy \cdot \left(2 \cdot \pi\right)}\right)} \cdot \sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)} \]

   2. pow3 96.9%

      \[\leadsto \sin \color{blue}{\left({\left(\sqrt[3]{uy \cdot \left(2 \cdot \pi\right)}\right)}^{3}\right)} \cdot \sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)} \]

7. Applied egg-rr 96.9%

   \[\leadsto \sin \color{blue}{\left({\left(\sqrt[3]{uy \cdot \left(2 \cdot \pi\right)}\right)}^{3}\right)} \cdot \sqrt{ux \cdot \left(\left(1 + \left(-1 \cdot \left(maxCos - 1\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - maxCos\right)} \]

9. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(ux \cdot \left(1 + \left(\mathsf{fma}\left(ux, \left(1 - maxCos\right) \cdot \left(-1 + maxCos\right), -\left(-1 + maxCos\right)\right) - maxCos\right)\right)\right)}^{1.5} \cdot {\sin \left(\left(uy \cdot 2\right) \cdot \pi\right)}^{3}}} \]

11. Simplified 98.4%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(ux \cdot \left(\left(\left(2 - maxCos\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(-1 + maxCos\right)\right)\right) - maxCos\right)\right)}^{1.5} \cdot {\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3}}} \]

12. Final simplification 98.4%

    \[\leadsto \sqrt[3]{{\sin \left(uy \cdot \left(2 \cdot \pi\right)\right)}^{3} \cdot {\left(ux \cdot \left(\left(\left(2 - maxCos\right) + ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right)\right)\right) - maxCos\right)\right)}^{1.5}} \]
13. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in ux around 0 98.2%

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

6. Taylor expanded in ux around -inf 98.2%

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

   1. distribute-rgt-in 98.3%

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

   2. mul-1-neg 98.3%

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

   3. sub-neg 98.3%

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

   4. metadata-eval 98.3%

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

   5. +-commutative 98.3%

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

   6. mul-1-neg 98.3%

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

   7. sub-neg 98.3%

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

   8. metadata-eval 98.3%

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

   9. mul-1-neg 98.3%

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

   10. sub-neg 98.3%

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

   11. associate--r+ 98.3%

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

8. Applied egg-rr 98.3%

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

9. Final simplification 98.3%

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

Alternative 4: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.000750000006519258:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right) + \frac{1 + \left(1 - maxCos\right)}{ux}\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(-1 + \frac{2}{ux}\right) - maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.000750000006519258)
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (*
       ux
       (-
        (*
         ux
         (+ (* (- 1.0 maxCos) (+ maxCos -1.0)) (/ (+ 1.0 (- 1.0 maxCos)) ux)))
        maxCos)))))
   (*
    (sin (* uy (* 2.0 PI)))
    (sqrt (* ux (- (* ux (+ -1.0 (/ 2.0 ux))) maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.000750000006519258f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * ((ux * (((1.0f - maxCos) * (maxCos + -1.0f)) + ((1.0f + (1.0f - maxCos)) / ux))) - maxCos))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * ((ux * (-1.0f + (2.0f / ux))) - maxCos)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.000750000006519258))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))) + Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) / ux))) - maxCos)))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))) - maxCos))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.000750000006519258))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * ((ux * (((single(1.0) - maxCos) * (maxCos + single(-1.0))) + ((single(1.0) + (single(1.0) - maxCos)) / ux))) - maxCos))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * ((ux * (single(-1.0) + (single(2.0) / ux))) - maxCos)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 7.50000007e-4

   1. Initial program 60.1%

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

      1. associate-*l* 60.1%

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

      2. sub-neg 60.1%

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

      3. +-commutative 60.1%

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

      4. distribute-rgt-neg-in 60.1%

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

      5. fma-define 60.1%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in ux around 0 98.5%

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

   6. Taylor expanded in ux around -inf 98.6%

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

   7. Taylor expanded in uy around 0 98.0%

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

   if 7.50000007e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 57.4%

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

      1. associate-*l* 57.4%

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

      2. sub-neg 57.4%

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

      3. +-commutative 57.4%

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

      4. distribute-rgt-neg-in 57.4%

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

      5. fma-define 57.5%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in ux around 0 97.6%

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

   6. Taylor expanded in ux around -inf 97.5%

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

   7. Taylor expanded in maxCos around 0 90.3%

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

      1. associate-*r/ 90.3%

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

      2. metadata-eval 90.3%

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

   9. Simplified 90.3%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.000750000006519258:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right) + \frac{1 + \left(1 - maxCos\right)}{ux}\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(-1 + \frac{2}{ux}\right) - maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. fma-define 98.2%

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

   4. *-commutative 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. +-commutative 98.2%

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

   8. mul-1-neg 98.2%

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

   9. sub-neg 98.2%

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

   10. metadata-eval 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. sub-neg 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in uy around inf 98.2%

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

9. Final simplification 98.2%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. fma-define 98.2%

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

   4. *-commutative 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. +-commutative 98.2%

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

   8. mul-1-neg 98.2%

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

   9. sub-neg 98.2%

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

   10. metadata-eval 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. sub-neg 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in maxCos around 0 97.5%

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

9. Final simplification 97.5%

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

Alternative 7: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. fma-define 98.2%

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

   4. *-commutative 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. +-commutative 98.2%

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

   8. mul-1-neg 98.2%

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

   9. sub-neg 98.2%

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

   10. metadata-eval 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. sub-neg 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in maxCos around 0 97.5%

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

9. Final simplification 97.5%

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

Alternative 8: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.000750000006519258:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right) + \frac{1 + \left(1 - maxCos\right)}{ux}\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.000750000006519258)
   (*
    2.0
    (*
     (* uy PI)
     (sqrt
      (*
       ux
       (-
        (*
         ux
         (+ (* (- 1.0 maxCos) (+ maxCos -1.0)) (/ (+ 1.0 (- 1.0 maxCos)) ux)))
        maxCos)))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.000750000006519258f) {
		tmp = 2.0f * ((uy * ((float) M_PI)) * sqrtf((ux * ((ux * (((1.0f - maxCos) * (maxCos + -1.0f)) + ((1.0f + (1.0f - maxCos)) / ux))) - maxCos))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.000750000006519258))
		tmp = Float32(Float32(2.0) * Float32(Float32(uy * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(ux * Float32(Float32(Float32(Float32(1.0) - maxCos) * Float32(maxCos + Float32(-1.0))) + Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - maxCos)) / ux))) - maxCos)))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((uy * single(2.0)) <= single(0.000750000006519258))
		tmp = single(2.0) * ((uy * single(pi)) * sqrt((ux * ((ux * (((single(1.0) - maxCos) * (maxCos + single(-1.0))) + ((single(1.0) + (single(1.0) - maxCos)) / ux))) - maxCos))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((ux * (single(2.0) - ux)));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 7.50000007e-4

   1. Initial program 60.1%

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

      1. associate-*l* 60.1%

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

      2. sub-neg 60.1%

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

      3. +-commutative 60.1%

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

      4. distribute-rgt-neg-in 60.1%

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

      5. fma-define 60.1%

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

   3. Simplified 60.2%

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

   5. Taylor expanded in ux around 0 98.5%

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

   6. Taylor expanded in ux around -inf 98.6%

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

   7. Taylor expanded in uy around 0 98.0%

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

   if 7.50000007e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 57.4%

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

      1. associate-*l* 57.4%

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

      2. sub-neg 57.4%

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

      3. +-commutative 57.4%

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

      4. distribute-rgt-neg-in 57.4%

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

      5. fma-define 57.5%

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

   3. Simplified 57.5%

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

   5. Taylor expanded in ux around 0 97.6%

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

      1. associate--l+ 97.7%

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

      2. +-commutative 97.7%

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

      3. fma-define 97.7%

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

      4. *-commutative 97.7%

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

      5. sub-neg 97.7%

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

      6. metadata-eval 97.7%

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

      7. +-commutative 97.7%

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

      8. mul-1-neg 97.7%

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

      9. sub-neg 97.7%

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

      10. metadata-eval 97.7%

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

      11. +-commutative 97.7%

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

      12. distribute-neg-in 97.7%

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

      13. metadata-eval 97.7%

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

      14. sub-neg 97.7%

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

   7. Simplified 97.7%

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

   8. Taylor expanded in maxCos around 0 89.6%

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

      1. *-commutative 89.6%

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

      2. associate-*r* 89.6%

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

      3. *-commutative 89.6%

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

      4. associate-*r* 89.6%

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

      5. mul-1-neg 89.6%

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

      6. unsub-neg 89.6%

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

   10. Simplified 89.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.000750000006519258:\\ \;\;\;\;2 \cdot \left(\left(uy \cdot \pi\right) \cdot \sqrt{ux \cdot \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(maxCos + -1\right) + \frac{1 + \left(1 - maxCos\right)}{ux}\right) - maxCos\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in ux around 0 98.2%

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

6. Taylor expanded in ux around -inf 98.2%

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

7. Taylor expanded in uy around 0 82.4%

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

8. Final simplification 82.4%

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

Alternative 10: 80.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in ux around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. +-commutative 98.2%

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

   3. fma-define 98.2%

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

   4. *-commutative 98.2%

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

   5. sub-neg 98.2%

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

   6. metadata-eval 98.2%

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

   7. +-commutative 98.2%

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

   8. mul-1-neg 98.2%

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

   9. sub-neg 98.2%

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

   10. metadata-eval 98.2%

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

   11. +-commutative 98.2%

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

   12. distribute-neg-in 98.2%

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

   13. metadata-eval 98.2%

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

   14. sub-neg 98.2%

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

7. Simplified 98.2%

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

8. Taylor expanded in uy around 0 82.3%

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

9. Final simplification 82.3%

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

Alternative 11: 74.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0002099999983329326:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0002099999983329326)
   (* 2.0 (* uy (* PI (sqrt (* ux (- 2.0 (* maxCos 2.0)))))))
   (* 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 <= 0.0002099999983329326f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (2.0f - (maxCos * 2.0f))))));
	} 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(0.0002099999983329326))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos * Float32(2.0))))))));
	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(0.0002099999983329326))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(2.0) - (maxCos * single(2.0)))))));
	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 < 2.09999998e-4

   1. Initial program 40.2%

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

      1. associate-*l* 40.2%

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

      2. sub-neg 40.2%

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

      3. +-commutative 40.2%

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

      4. distribute-rgt-neg-in 40.2%

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

      5. fma-define 40.2%

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

   3. Simplified 40.3%

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

   5. Taylor expanded in uy around 0 36.9%

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

   7. Simplified 36.9%

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

   8. Taylor expanded in ux around 0 76.0%

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

   if 2.09999998e-4 < ux

   1. Initial program 88.4%

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

      1. associate-*l* 88.4%

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

      2. sub-neg 88.4%

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

      3. +-commutative 88.4%

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

      4. distribute-rgt-neg-in 88.4%

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

      5. fma-define 88.5%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in uy around 0 77.7%

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

      1. associate-*l* 77.7%

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

      2. +-commutative 77.7%

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

      3. sub-neg 77.7%

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

      4. metadata-eval 77.7%

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

      5. sub-neg 77.7%

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

      6. metadata-eval 77.7%

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

      7. +-commutative 77.7%

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

      8. fma-undefine 77.9%

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

   7. Applied egg-rr 77.9%

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

      1. +-commutative 77.9%

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

   9. Simplified 77.9%

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

   10. Taylor expanded in maxCos around 0 73.6%

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

       1. mul-1-neg 73.6%

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

       2. unsub-neg 73.6%

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

       3. sub-neg 73.6%

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

       4. metadata-eval 73.6%

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

   12. Simplified 73.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.0002099999983329326:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - maxCos \cdot 2\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux + -1\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in uy around 0 53.0%

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

7. Simplified 53.0%

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

8. Taylor expanded in ux around 0 66.2%

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

9. Final simplification 66.2%

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

Alternative 13: 62.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

   1. associate-*l* 59.2%

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

   2. sub-neg 59.2%

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

   3. +-commutative 59.2%

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

   4. distribute-rgt-neg-in 59.2%

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

   5. fma-define 59.3%

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

3. Simplified 59.3%

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

5. Taylor expanded in uy around 0 53.0%

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

6. Taylor expanded in ux around 0 66.2%

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

   1. mul-1-neg 66.2%

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

   2. sub-neg 66.2%

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

   3. sub-neg 66.2%

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

   4. metadata-eval 66.2%

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

8. Simplified 66.2%

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

9. Taylor expanded in maxCos around 0 63.2%

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

Reproduce

herbie shell --seed 2024096 
(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)))))))