UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 16.1s

Alternatives: 7

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in ux around 0 98.2%

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

4. Taylor expanded in uy around inf 98.2%

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

6. Simplified 98.2%

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

   1. add-cbrt-cube 98.2%

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

   2. pow1/3 96.2%

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

8. Applied egg-rr 96.2%

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

   1. unpow1/3 98.3%

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

10. Simplified 98.3%

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

Alternative 2: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in ux around 0 98.2%

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

4. Taylor expanded in uy around inf 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 3: 97.6% 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 59.2%

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

3. Taylor expanded in ux around 0 98.2%

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

4. Taylor expanded in uy around inf 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in maxCos around 0 97.5%

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

   1. +-commutative 97.5%

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

   2. mul-1-neg 97.5%

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

   3. unsub-neg 97.5%

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

   4. metadata-eval 97.5%

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

   5. cancel-sign-sub-inv 97.5%

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

   6. *-commutative 97.5%

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

9. Simplified 97.5%

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

10. Final simplification 97.5%

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

Alternative 4: 92.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in ux around 0 98.2%

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

4. Taylor expanded in maxCos around 0 90.8%

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

   1. neg-mul-1 90.8%

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

   2. unsub-neg 90.8%

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

6. Simplified 90.8%

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

7. Final simplification 90.8%

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

Alternative 5: 80.3% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in ux around 0 98.2%

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

4. Taylor expanded in uy around 0 82.3%

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

6. Simplified 82.3%

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

7. Taylor expanded in maxCos around 0 82.0%

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

   1. +-commutative 97.5%

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

   2. mul-1-neg 97.5%

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

   3. unsub-neg 97.5%

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

   4. metadata-eval 97.5%

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

   5. cancel-sign-sub-inv 97.5%

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

   6. *-commutative 97.5%

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

9. Simplified 82.0%

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

10. Final simplification 82.0%

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

Alternative 6: 76.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in ux around 0 98.2%

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

4. Taylor expanded in uy around 0 82.3%

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

6. Simplified 82.3%

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

7. Taylor expanded in maxCos around 0 77.2%

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

   1. *-commutative 77.2%

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

9. Simplified 77.2%

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

10. Final simplification 77.2%

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

Alternative 7: 62.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.2%

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

3. Taylor expanded in ux around 0 98.2%

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

4. Taylor expanded in uy around 0 82.3%

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

6. Simplified 82.3%

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

7. Taylor expanded in maxCos around 0 77.2%

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

   1. *-commutative 77.2%

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

9. Simplified 77.2%

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

10. Taylor expanded in ux around 0 63.2%

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

    1. *-commutative 63.2%

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

12. Simplified 63.2%

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

13. Final simplification 63.2%

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

Reproduce

herbie shell --seed 2024096 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))