UniformSampleCone, y

Percentage Accurate: 57.2% → 98.4%

Time: 20.5s

Alternatives: 10

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: 57.2% 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: 98.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.7%

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

3. Taylor expanded in ux around 0 98.3%

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

   1. associate--l+ 98.3%

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

   2. associate-*r* 98.3%

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

   3. mul-1-neg 98.3%

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

   4. fmm-def 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

   7. +-commutative 98.3%

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

   8. distribute-lft-neg-in 98.3%

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

   9. metadata-eval 98.3%

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

   10. *-commutative 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around inf 98.3%

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

   1. fma-define 98.3%

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

   2. mul-1-neg 98.3%

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

   3. fmm-undef 98.3%

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

   4. sub-neg 98.3%

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

   5. metadata-eval 98.3%

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

   6. associate-*r* 98.3%

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

   7. *-commutative 98.3%

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

   8. *-commutative 98.3%

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

   9. *-commutative 98.3%

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

8. Simplified 98.3%

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

   1. add-cbrt-cube 98.3%

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

   2. add-cbrt-cube 98.3%

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

   3. pow3 98.3%

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

   4. *-commutative 98.3%

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

   5. *-commutative 98.3%

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

   6. associate-*r* 98.3%

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

   7. cbrt-unprod 98.1%

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

10. Applied egg-rr 98.4%

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

12. Simplified 98.4%

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

13. Final simplification 98.4%

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

3. Taylor expanded in ux around 0 98.3%

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

   1. associate--l+ 98.3%

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

   2. associate-*r* 98.3%

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

   3. mul-1-neg 98.3%

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

   4. fmm-def 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

   7. +-commutative 98.3%

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

   8. distribute-lft-neg-in 98.3%

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

   9. metadata-eval 98.3%

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

   10. *-commutative 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around inf 98.3%

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

   1. fma-define 98.3%

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

   2. mul-1-neg 98.3%

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

   3. fmm-undef 98.3%

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

   4. sub-neg 98.3%

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

   5. metadata-eval 98.3%

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

   6. associate-*r* 98.3%

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

   7. *-commutative 98.3%

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

   8. *-commutative 98.3%

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

   9. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Final simplification 98.3%

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

Alternative 3: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - \left(ux \cdot {\left(maxCos + -1\right)}^{2} + 2 \cdot maxCos\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0010000000474974513)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (* ux (- 2.0 (+ (* ux (pow (+ maxCos -1.0) 2.0)) (* 2.0 maxCos))))))))
   (* ux (* (sin (* PI (* 2.0 uy))) (sqrt (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0010000000474974513f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * (2.0f - ((ux * powf((maxCos + -1.0f), 2.0f)) + (2.0f * maxCos)))))));
	} else {
		tmp = ux * (sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(ux * (Float32(maxCos + Float32(-1.0)) ^ Float32(2.0))) + Float32(Float32(2.0) * maxCos))))))));
	else
		tmp = Float32(ux * Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.0010000000474974513))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * (single(2.0) - ((ux * ((maxCos + single(-1.0)) ^ single(2.0))) + (single(2.0) * maxCos)))))));
	else
		tmp = ux * (sin((single(pi) * (single(2.0) * uy))) * sqrt((single(-1.0) + (single(2.0) / ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00100000005

   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 59.4%

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

   5. Taylor expanded in uy around 0 59.1%

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

   7. Simplified 59.2%

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

   8. Taylor expanded in ux around 0 98.2%

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

      1. associate--l+ 98.2%

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

      2. associate-*r* 98.2%

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

      3. neg-mul-1 98.2%

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

      4. sub-neg 98.2%

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

      5. metadata-eval 98.2%

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

      6. +-commutative 98.2%

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

      7. +-commutative 98.2%

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

   10. Simplified 98.2%

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

   if 0.00100000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.5%

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

   3. Taylor expanded in ux around inf 98.1%

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

   4. Taylor expanded in maxCos around 0 93.8%

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

      1. associate-*l* 93.9%

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

      2. associate-*r* 93.9%

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

      3. *-commutative 93.9%

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

      4. *-commutative 93.9%

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

      5. *-commutative 93.9%

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

      6. sub-neg 93.9%

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

      7. associate-*r/ 93.9%

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

      8. metadata-eval 93.9%

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

      9. metadata-eval 93.9%

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

   6. Simplified 93.9%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(2 - \left(ux \cdot {\left(maxCos + -1\right)}^{2} + 2 \cdot maxCos\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

3. Taylor expanded in ux around 0 98.3%

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

   1. associate--l+ 98.3%

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

   2. associate-*r* 98.3%

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

   3. mul-1-neg 98.3%

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

   4. fmm-def 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

   7. +-commutative 98.3%

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

   8. distribute-lft-neg-in 98.3%

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

   9. metadata-eval 98.3%

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

   10. *-commutative 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around inf 98.3%

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

   1. fma-define 98.3%

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

   2. mul-1-neg 98.3%

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

   3. fmm-undef 98.3%

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

   4. sub-neg 98.3%

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

   5. metadata-eval 98.3%

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

   6. associate-*r* 98.3%

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

   7. *-commutative 98.3%

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

   8. *-commutative 98.3%

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

   9. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in maxCos around 0 97.7%

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

    1. +-commutative 97.7%

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

    2. mul-1-neg 97.7%

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

    3. unsub-neg 97.7%

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

    4. metadata-eval 97.7%

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

    5. cancel-sign-sub-inv 97.7%

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

    6. cancel-sign-sub-inv 97.7%

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

    7. metadata-eval 97.7%

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

    8. *-commutative 97.7%

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

11. Simplified 97.7%

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

12. Final simplification 97.7%

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

Alternative 5: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) - \left(maxCos + -1\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0010000000474974513)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (*
        ux
        (+
         (- 1.0 maxCos)
         (- (* ux (* (+ maxCos -1.0) (- 1.0 maxCos))) (+ maxCos -1.0))))))))
   (* ux (* (sin (* PI (* 2.0 uy))) (sqrt (+ -1.0 (/ 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0010000000474974513f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * ((1.0f - maxCos) + ((ux * ((maxCos + -1.0f) * (1.0f - maxCos))) - (maxCos + -1.0f)))))));
	} else {
		tmp = ux * (sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((-1.0f + (2.0f / ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))) - Float32(maxCos + Float32(-1.0)))))))));
	else
		tmp = Float32(ux * Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.0010000000474974513))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * ((single(1.0) - maxCos) + ((ux * ((maxCos + single(-1.0)) * (single(1.0) - maxCos))) - (maxCos + single(-1.0))))))));
	else
		tmp = ux * (sin((single(pi) * (single(2.0) * uy))) * sqrt((single(-1.0) + (single(2.0) / ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00100000005

   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 59.4%

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

   5. Taylor expanded in uy around 0 59.1%

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

   7. Simplified 59.2%

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

   8. Taylor expanded in ux around -inf 97.9%

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

   9. Taylor expanded in ux around 0 98.1%

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

       1. associate-+r+ 98.1%

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

       2. mul-1-neg 98.1%

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

       3. unsub-neg 98.1%

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

       4. +-commutative 98.1%

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

       5. mul-1-neg 98.1%

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

       6. unsub-neg 98.1%

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

       7. *-commutative 98.1%

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

       8. sub-neg 98.1%

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

       9. metadata-eval 98.1%

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

       10. mul-1-neg 98.1%

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

       11. unsub-neg 98.1%

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

       12. sub-neg 98.1%

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

       13. metadata-eval 98.1%

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

   11. Simplified 98.1%

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

   if 0.00100000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.5%

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

   3. Taylor expanded in ux around inf 98.1%

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

   4. Taylor expanded in maxCos around 0 93.8%

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

      1. associate-*l* 93.9%

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

      2. associate-*r* 93.9%

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

      3. *-commutative 93.9%

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

      4. *-commutative 93.9%

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

      5. *-commutative 93.9%

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

      6. sub-neg 93.9%

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

      7. associate-*r/ 93.9%

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

      8. metadata-eval 93.9%

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

      9. metadata-eval 93.9%

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

   6. Simplified 93.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) - \left(maxCos + -1\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;ux \cdot \left(\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) - \left(maxCos + -1\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0010000000474974513)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (*
        ux
        (+
         (- 1.0 maxCos)
         (- (* ux (* (+ maxCos -1.0) (- 1.0 maxCos))) (+ maxCos -1.0))))))))
   (* (sin (* PI (* 2.0 uy))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.0010000000474974513f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * ((1.0f - maxCos) + ((ux * ((maxCos + -1.0f) * (1.0f - maxCos))) - (maxCos + -1.0f)))))));
	} else {
		tmp = sinf((((float) M_PI) * (2.0f * uy))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.0010000000474974513))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))) - Float32(maxCos + Float32(-1.0)))))))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(Float32(2.0) * uy))) * 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 ((single(2.0) * uy) <= single(0.0010000000474974513))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * ((single(1.0) - maxCos) + ((ux * ((maxCos + single(-1.0)) * (single(1.0) - maxCos))) - (maxCos + single(-1.0))))))));
	else
		tmp = sin((single(pi) * (single(2.0) * uy))) * 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)) < 0.00100000005

   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 59.4%

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

   5. Taylor expanded in uy around 0 59.1%

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

   7. Simplified 59.2%

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

   8. Taylor expanded in ux around -inf 97.9%

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

   9. Taylor expanded in ux around 0 98.1%

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

       1. associate-+r+ 98.1%

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

       2. mul-1-neg 98.1%

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

       3. unsub-neg 98.1%

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

       4. +-commutative 98.1%

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

       5. mul-1-neg 98.1%

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

       6. unsub-neg 98.1%

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

       7. *-commutative 98.1%

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

       8. sub-neg 98.1%

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

       9. metadata-eval 98.1%

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

       10. mul-1-neg 98.1%

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

       11. unsub-neg 98.1%

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

       12. sub-neg 98.1%

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

       13. metadata-eval 98.1%

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

   11. Simplified 98.1%

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

   if 0.00100000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.5%

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

   3. Taylor expanded in ux around 0 98.1%

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

      1. associate--l+ 98.1%

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

      2. associate-*r* 98.1%

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

      3. mul-1-neg 98.1%

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

      4. fmm-def 98.1%

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

      5. sub-neg 98.1%

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

      6. metadata-eval 98.1%

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

      7. +-commutative 98.1%

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

      8. distribute-lft-neg-in 98.1%

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

      9. metadata-eval 98.1%

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

      10. *-commutative 98.1%

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

   5. Simplified 98.1%

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

   6. Taylor expanded in uy around inf 98.1%

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

      1. fma-define 98.1%

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

      2. mul-1-neg 98.1%

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

      3. fmm-undef 98.1%

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

      4. sub-neg 98.1%

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

      5. metadata-eval 98.1%

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

      6. associate-*r* 98.1%

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

      7. *-commutative 98.1%

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

      8. *-commutative 98.1%

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

      9. *-commutative 98.1%

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

   8. Simplified 98.1%

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

   9. Taylor expanded in maxCos around 0 93.9%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0010000000474974513:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) - \left(maxCos + -1\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.002099999925121665:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) - \left(maxCos + -1\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.002099999925121665)
   (*
    2.0
    (*
     uy
     (*
      PI
      (sqrt
       (*
        ux
        (+
         (- 1.0 maxCos)
         (- (* ux (* (+ maxCos -1.0) (- 1.0 maxCos))) (+ maxCos -1.0))))))))
   (* (sin (* uy (* 2.0 PI))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((2.0f * uy) <= 0.002099999925121665f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((ux * ((1.0f - maxCos) + ((ux * ((maxCos + -1.0f) * (1.0f - maxCos))) - (maxCos + -1.0f)))))));
	} else {
		tmp = sinf((uy * (2.0f * ((float) M_PI)))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.002099999925121665))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) - maxCos) + Float32(Float32(ux * Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos))) - Float32(maxCos + Float32(-1.0)))))))));
	else
		tmp = Float32(sin(Float32(uy * Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.002099999925121665))
		tmp = single(2.0) * (uy * (single(pi) * sqrt((ux * ((single(1.0) - maxCos) + ((ux * ((maxCos + single(-1.0)) * (single(1.0) - maxCos))) - (maxCos + single(-1.0))))))));
	else
		tmp = sin((uy * (single(2.0) * single(pi)))) * sqrt((single(2.0) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00209999993

   1. Initial program 59.3%

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

      1. associate-*l* 59.3%

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

      2. sub-neg 59.3%

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

      3. +-commutative 59.3%

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

      4. distribute-rgt-neg-in 59.3%

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

      5. fma-define 59.4%

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

   3. Simplified 59.6%

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

   5. Taylor expanded in uy around 0 58.9%

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

   7. Simplified 58.9%

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

   8. Taylor expanded in ux around -inf 96.6%

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

   9. Taylor expanded in ux around 0 96.8%

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

       1. associate-+r+ 96.8%

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

       2. mul-1-neg 96.8%

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

       3. unsub-neg 96.8%

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

       4. +-commutative 96.8%

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

       5. mul-1-neg 96.8%

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

       6. unsub-neg 96.8%

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

       7. *-commutative 96.8%

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

       8. sub-neg 96.8%

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

       9. metadata-eval 96.8%

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

       10. mul-1-neg 96.8%

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

       11. unsub-neg 96.8%

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

       12. sub-neg 96.8%

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

       13. metadata-eval 96.8%

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

   11. Simplified 96.8%

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

   if 0.00209999993 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.6%

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

      1. associate-*l* 60.6%

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

      2. sub-neg 60.6%

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

      3. +-commutative 60.6%

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

      4. distribute-rgt-neg-in 60.6%

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

      5. fma-define 60.4%

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

   3. Simplified 60.5%

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

   5. Taylor expanded in maxCos around 0 57.9%

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

   6. Taylor expanded in ux around 0 73.9%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.002099999925121665:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{ux \cdot \left(\left(1 - maxCos\right) + \left(ux \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) - \left(maxCos + -1\right)\right)\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(uy \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 81.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. associate-*l* 59.7%

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

   2. sub-neg 59.7%

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

   3. +-commutative 59.7%

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

   4. distribute-rgt-neg-in 59.7%

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

   5. fma-define 59.7%

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

3. Simplified 59.8%

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

5. Taylor expanded in uy around 0 53.3%

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

7. Simplified 53.4%

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

8. Taylor expanded in ux around -inf 83.2%

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

9. Taylor expanded in ux around 0 83.3%

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

    1. associate-+r+ 83.3%

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

    2. mul-1-neg 83.3%

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

    3. unsub-neg 83.3%

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

    4. +-commutative 83.3%

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

    5. mul-1-neg 83.3%

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

    6. unsub-neg 83.3%

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

    7. *-commutative 83.3%

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

    8. sub-neg 83.3%

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

    9. metadata-eval 83.3%

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

    10. mul-1-neg 83.3%

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

    11. unsub-neg 83.3%

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

    12. sub-neg 83.3%

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

    13. metadata-eval 83.3%

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

11. Simplified 83.3%

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

12. Final simplification 83.3%

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

Alternative 9: 77.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

3. Taylor expanded in ux around 0 98.3%

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

   1. associate--l+ 98.3%

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

   2. associate-*r* 98.3%

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

   3. mul-1-neg 98.3%

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

   4. fmm-def 98.3%

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

   5. sub-neg 98.3%

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

   6. metadata-eval 98.3%

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

   7. +-commutative 98.3%

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

   8. distribute-lft-neg-in 98.3%

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

   9. metadata-eval 98.3%

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

   10. *-commutative 98.3%

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

5. Simplified 98.3%

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

6. Taylor expanded in uy around inf 98.3%

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

   1. fma-define 98.3%

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

   2. mul-1-neg 98.3%

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

   3. fmm-undef 98.3%

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

   4. sub-neg 98.3%

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

   5. metadata-eval 98.3%

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

   6. associate-*r* 98.3%

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

   7. *-commutative 98.3%

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

   8. *-commutative 98.3%

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

   9. *-commutative 98.3%

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

8. Simplified 98.3%

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

9. Taylor expanded in maxCos around 0 93.5%

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

10. Taylor expanded in uy around 0 79.9%

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

    1. associate-*r* 79.9%

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

12. Simplified 79.9%

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

13. Final simplification 79.9%

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

Alternative 10: 63.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.7%

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

   1. associate-*l* 59.7%

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

   2. sub-neg 59.7%

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

   3. +-commutative 59.7%

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

   4. distribute-rgt-neg-in 59.7%

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

   5. fma-define 59.7%

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

3. Simplified 59.8%

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

5. Taylor expanded in uy around 0 53.3%

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

7. Simplified 53.4%

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

8. Taylor expanded in ux around 0 66.9%

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

9. Taylor expanded in maxCos around 0 64.8%

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

10. Final simplification 64.8%

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

Reproduce

herbie shell --seed 2024131 
(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)))))))