UniformSampleCone, y

Percentage Accurate: 57.5% → 98.2%

Time: 13.0s

Alternatives: 8

Speedup: 1.0×

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: 57.5% 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: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in ux around inf 98.2%

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

4. Final simplification 98.2%

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

Alternative 2: 98.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in ux around inf 98.2%

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

4. Taylor expanded in maxCos around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

   4. mul-1-neg 98.2%

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

   5. associate-*r/ 98.2%

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

   6. metadata-eval 98.2%

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

6. Simplified 98.2%

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

7. Final simplification 98.2%

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

Alternative 3: 98.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in ux around 0 98.2%

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

4. Final simplification 98.2%

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

Alternative 4: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in ux around inf 98.2%

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

4. Taylor expanded in maxCos around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

   4. mul-1-neg 98.2%

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

   5. associate-*r/ 98.2%

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

   6. metadata-eval 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in ux around 0 98.2%

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

8. Final simplification 98.2%

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

Alternative 5: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in ux around inf 98.2%

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

4. Taylor expanded in maxCos around 0 98.2%

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

   1. associate--l+ 98.2%

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

   2. associate-*r/ 98.2%

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

   3. metadata-eval 98.2%

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

   4. mul-1-neg 98.2%

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

   5. associate-*r/ 98.2%

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

   6. metadata-eval 98.2%

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

6. Simplified 98.2%

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

7. Taylor expanded in uy around inf 98.1%

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

   1. associate-*l* 98.1%

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

   2. sub-neg 98.1%

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

   3. associate-*r/ 98.1%

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

   4. metadata-eval 98.1%

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

   5. associate--r+ 98.1%

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

   6. associate-*r/ 98.1%

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

   7. metadata-eval 98.1%

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

   8. sub-neg 98.1%

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

   9. distribute-neg-frac 98.1%

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

   10. metadata-eval 98.1%

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

   11. metadata-eval 98.1%

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

9. Simplified 98.1%

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

10. Final simplification 98.1%

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

Alternative 6: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in ux around 0 60.1%

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

4. Taylor expanded in maxCos around 0 97.4%

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

   1. associate-*r* 97.4%

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

   2. mul-1-neg 97.4%

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

   3. sub-neg 97.4%

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

   4. metadata-eval 97.4%

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

6. Simplified 97.4%

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

7. Final simplification 97.4%

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

Alternative 7: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in ux around 0 46.6%

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

4. Taylor expanded in ux around inf 77.6%

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

5. Final simplification 77.6%

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

Alternative 8: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.7%

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

3. Taylor expanded in ux around 0 46.6%

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

4. Taylor expanded in maxCos around 0 77.5%

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

   1. distribute-lft-out-- 77.5%

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

6. Simplified 77.5%

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

7. Final simplification 77.5%

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

Reproduce

herbie shell --seed 2024179 
(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)))))))