UniformSampleCone, y

Percentage Accurate: 57.7% → 98.4%

Time: 16.3s

Alternatives: 11

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: 57.7% 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

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.5%

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

   2. associate-*r* 98.5%

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

   3. mul-1-neg 98.5%

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

   4. sub-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

5. Simplified 98.5%

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

   1. *-commutative 98.5%

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

   2. add-cbrt-cube 98.4%

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

   3. add-cbrt-cube 98.4%

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

   4. cbrt-unprod 98.3%

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

7. Applied egg-rr 98.7%

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

8. Final simplification 98.7%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (cbrt
  (*
   (pow (sin (* 2.0 (* uy PI))) 3.0)
   (pow (* ux (- 2.0 (+ ux (* maxCos (+ 2.0 (* ux (+ maxCos -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 - (ux + (maxCos * (2.0f + (ux * (maxCos + -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(2.0) - Float32(ux + Float32(maxCos * Float32(Float32(2.0) + Float32(ux * Float32(maxCos + Float32(-2.0)))))))) ^ Float32(1.5))))
end

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.5%

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

   2. associate-*r* 98.5%

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

   3. mul-1-neg 98.5%

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

   4. sub-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

5. Simplified 98.5%

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

   1. *-commutative 98.5%

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

   2. add-cbrt-cube 98.4%

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

   3. add-cbrt-cube 98.4%

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

   4. cbrt-unprod 98.3%

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

7. Applied egg-rr 98.7%

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

8. Taylor expanded in maxCos around 0 98.7%

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

9. Taylor expanded in ux around -inf 98.7%

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

    1. associate-*r* 98.7%

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

    2. neg-mul-1 98.7%

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

    3. sub-neg 98.7%

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

    4. metadata-eval 98.7%

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

11. Simplified 98.7%

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

12. Final simplification 98.7%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.5%

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

   2. associate-*r* 98.5%

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

   3. mul-1-neg 98.5%

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

   4. sub-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in maxCos around 0 98.5%

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

7. Final simplification 98.5%

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

Alternative 4: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.5%

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

   2. associate-*r* 98.5%

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

   3. mul-1-neg 98.5%

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

   4. sub-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in maxCos around 0 97.5%

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

7. Final simplification 97.5%

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

Alternative 5: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot uy\right)\\ \mathbf{if}\;2 \cdot uy \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(\left(ux \cdot 2 - ux \cdot maxCos\right) - 2\right) - ux\right)\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (* 2.0 uy))))
   (if (<= (* 2.0 uy) 0.00039999998989515007)
     (*
      (sqrt
       (* ux (+ 2.0 (- (* maxCos (- (- (* ux 2.0) (* ux maxCos)) 2.0)) ux))))
      t_0)
     (* (sin t_0) (sqrt (* ux (- 2.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (2.0f * uy);
	float tmp;
	if ((2.0f * uy) <= 0.00039999998989515007f) {
		tmp = sqrtf((ux * (2.0f + ((maxCos * (((ux * 2.0f) - (ux * maxCos)) - 2.0f)) - ux)))) * t_0;
	} else {
		tmp = sinf(t_0) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(Float32(2.0) * uy))
	tmp = Float32(0.0)
	if (Float32(Float32(2.0) * uy) <= Float32(0.00039999998989515007))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(maxCos * Float32(Float32(Float32(ux * Float32(2.0)) - Float32(ux * maxCos)) - Float32(2.0))) - ux)))) * t_0);
	else
		tmp = Float32(sin(t_0) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = single(pi) * (single(2.0) * uy);
	tmp = single(0.0);
	if ((single(2.0) * uy) <= single(0.00039999998989515007))
		tmp = sqrt((ux * (single(2.0) + ((maxCos * (((ux * single(2.0)) - (ux * maxCos)) - single(2.0))) - ux)))) * t_0;
	else
		tmp = sin(t_0) * 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)) < 3.9999999e-4

   1. Initial program 58.7%

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

   3. Taylor expanded in ux around 0 98.5%

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

      1. associate--l+ 98.6%

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

      2. associate-*r* 98.6%

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

      3. mul-1-neg 98.6%

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

      4. sub-neg 98.6%

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

      5. metadata-eval 98.6%

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

      6. +-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in uy around 0 98.4%

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

      1. associate-*r* 98.4%

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

   8. Simplified 98.4%

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

   9. Taylor expanded in maxCos around 0 98.4%

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

   if 3.9999999e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 60.8%

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

   3. Taylor expanded in ux around 0 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. sub-neg 98.3%

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

      5. metadata-eval 98.3%

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

      6. +-commutative 98.3%

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

   5. Simplified 98.3%

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

      1. add-cbrt-cube 98.2%

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

      2. pow1/3 95.8%

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left(\left(\sqrt{ux \cdot \left(2 + \left(\left(-ux\right) \cdot {\left(-1 + maxCos\right)}^{2} - 2 \cdot maxCos\right)\right)} \cdot \sqrt{ux \cdot \left(2 + \left(\left(-ux\right) \cdot {\left(-1 + maxCos\right)}^{2} - 2 \cdot maxCos\right)\right)}\right) \cdot \sqrt{ux \cdot \left(2 + \left(\left(-ux\right) \cdot {\left(-1 + maxCos\right)}^{2} - 2 \cdot maxCos\right)\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 95.9%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \color{blue}{{\left({\left(ux \cdot \left(2 + \mathsf{fma}\left(-ux, {\left(-1 + maxCos\right)}^{2}, -2 \cdot maxCos\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 98.1%

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

      2. fma-define 98.1%

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

      3. *-commutative 98.1%

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

      4. fma-undefine 98.1%

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

      5. +-commutative 98.1%

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

      6. *-commutative 98.1%

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

   9. Simplified 98.1%

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

   10. Taylor expanded in maxCos around 0 93.1%

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

       1. mul-1-neg 93.1%

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

       2. unsub-neg 93.1%

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

   12. Simplified 93.1%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.00039999998989515007:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + \left(maxCos \cdot \left(\left(ux \cdot 2 - ux \cdot maxCos\right) - 2\right) - ux\right)\right)} \cdot \left(\pi \cdot \left(2 \cdot uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.5%

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

   2. associate-*r* 98.5%

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

   3. mul-1-neg 98.5%

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

   4. sub-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in uy around 0 83.3%

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

   1. associate-*r* 83.3%

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

8. Simplified 83.3%

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

9. Taylor expanded in maxCos around 0 83.3%

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

10. Final simplification 83.3%

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

Alternative 7: 80.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.5%

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

   2. associate-*r* 98.5%

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

   3. mul-1-neg 98.5%

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

   4. sub-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in uy around 0 83.3%

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

   1. associate-*r* 83.3%

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

8. Simplified 83.3%

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

9. Taylor expanded in maxCos around 0 82.6%

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

10. Final simplification 82.6%

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

Alternative 8: 76.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

3. Taylor expanded in ux around 0 98.5%

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

   1. associate--l+ 98.5%

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

   2. associate-*r* 98.5%

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

   3. mul-1-neg 98.5%

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

   4. sub-neg 98.5%

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

   5. metadata-eval 98.5%

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

   6. +-commutative 98.5%

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

5. Simplified 98.5%

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

6. Taylor expanded in uy around 0 83.3%

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

   1. associate-*r* 83.3%

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

8. Simplified 83.3%

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

9. Taylor expanded in maxCos around 0 78.3%

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

    1. *-commutative 78.3%

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

    2. mul-1-neg 78.3%

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

    3. unsub-neg 78.3%

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

11. Simplified 78.3%

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

Alternative 9: 19.2% accurate, 17.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. associate-*l* 59.4%

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

   2. sub-neg 59.4%

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

   3. +-commutative 59.4%

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

   4. distribute-rgt-neg-in 59.4%

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

   5. fma-define 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 52.3%

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

7. Simplified 52.5%

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

8. Taylor expanded in maxCos around -inf -0.0%

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

   1. associate-*r* -0.0%

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

   2. *-commutative -0.0%

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

   3. associate-*r* -0.0%

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

10. Simplified -0.0%

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

12. Applied egg-rr 6.2%

    \[\leadsto \left(maxCos \cdot -2\right) \cdot \color{blue}{{\left(\sqrt[3]{ux \cdot \left(uy \cdot \pi\right)}\right)}^{3}} \]
13. Step-by-step derivation

    1. rem-cube-cbrt 6.2%

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

14. Simplified 6.2%

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

16. Applied egg-rr 19.1%

    \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(maxCos \cdot -2\right)} - -1\right)} \cdot \left(ux \cdot \left(uy \cdot \pi\right)\right) \]
17. Step-by-step derivation

    1. sub-neg 19.1%

       \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(maxCos \cdot -2\right)} + \left(--1\right)\right)} \cdot \left(ux \cdot \left(uy \cdot \pi\right)\right) \]

    2. log1p-undefine 19.1%

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

    3. rem-exp-log 19.1%

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

    4. *-commutative 19.1%

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

    5. metadata-eval 19.1%

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

18. Simplified 19.1%

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

19. Final simplification 19.1%

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

Alternative 10: 9.8% accurate, 22.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. associate-*l* 59.4%

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

   2. sub-neg 59.4%

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

   3. +-commutative 59.4%

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

   4. distribute-rgt-neg-in 59.4%

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

   5. fma-define 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 52.3%

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

7. Simplified 52.5%

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

8. Taylor expanded in maxCos around -inf -0.0%

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

   1. associate-*r* -0.0%

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

   2. *-commutative -0.0%

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

   3. associate-*r* -0.0%

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

10. Simplified -0.0%

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

12. Applied egg-rr 9.7%

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

13. Final simplification 9.7%

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

Alternative 11: 6.0% accurate, 24.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.4%

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

   1. associate-*l* 59.4%

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

   2. sub-neg 59.4%

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

   3. +-commutative 59.4%

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

   4. distribute-rgt-neg-in 59.4%

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

   5. fma-define 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 52.3%

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

7. Simplified 52.5%

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

8. Taylor expanded in maxCos around -inf -0.0%

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

   1. associate-*r* -0.0%

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

   2. *-commutative -0.0%

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

   3. associate-*r* -0.0%

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

10. Simplified -0.0%

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

12. Applied egg-rr 6.2%

    \[\leadsto \left(maxCos \cdot -2\right) \cdot \color{blue}{{\left(\sqrt[3]{ux \cdot \left(uy \cdot \pi\right)}\right)}^{3}} \]
13. Step-by-step derivation

    1. rem-cube-cbrt 6.2%

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

14. Simplified 6.2%

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

15. Taylor expanded in maxCos around 0 6.2%

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

Reproduce

herbie shell --seed 2024137 
(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)))))))