UniformSampleCone, y

Percentage Accurate: 57.8% → 98.3%

Time: 19.0s

Alternatives: 12

Speedup: 3.3×

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: 57.8% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

8. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

Alternative 4: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 97.6%

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

9. Applied rewrites 97.6%

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

Alternative 5: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 97.6%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 96.9%

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

Alternative 6: 95.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \mathbf{if}\;uy \leq 0.00011999999696854502:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot t\_0 - {t\_0}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- ux (* maxCos ux))))
   (if (<= uy 0.00011999999696854502)
     (* 2.0 (* uy (* PI (sqrt (- (* 2.0 t_0) (pow t_0 2.0))))))
     (* (sin (* (+ PI PI) uy)) (sqrt (* (- 2.0 ux) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (maxCos * ux);
	float tmp;
	if (uy <= 0.00011999999696854502f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((2.0f * t_0) - powf(t_0, 2.0f)))));
	} else {
		tmp = sinf(((((float) M_PI) + ((float) M_PI)) * uy)) * sqrtf(((2.0f - ux) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(maxCos * ux))
	tmp = Float32(0.0)
	if (uy <= Float32(0.00011999999696854502))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(Float32(2.0) * t_0) - (t_0 ^ Float32(2.0)))))));
	else
		tmp = Float32(sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux - (maxCos * ux);
	tmp = single(0.0);
	if (uy <= single(0.00011999999696854502))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((single(2.0) * t_0) - (t_0 ^ single(2.0))))));
	else
		tmp = sin(((single(pi) + single(pi)) * uy)) * sqrt(((single(2.0) - ux) * ux));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.19999997e-4

   1. Initial program 57.8%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 51.0%

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

   6. Applied rewrites 52.0%

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

   7. Taylor expanded in uy around 0

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

   9. Applied rewrites 81.4%

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

   if 1.19999997e-4 < uy

   1. Initial program 57.8%

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

   2. Taylor expanded in ux around 0

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

   4. Applied rewrites 98.3%

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in maxCos around 0

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

   9. Applied rewrites 92.5%

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

Alternative 7: 81.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 51.0%

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

6. Applied rewrites 52.0%

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

7. Taylor expanded in uy around 0

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

9. Applied rewrites 81.4%

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

Alternative 8: 81.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 98.3%

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in uy around 0

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

9. Applied rewrites 81.4%

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

Alternative 9: 80.9% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 97.6%

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

8. Taylor expanded in uy around 0

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

10. Applied rewrites 80.9%

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

Alternative 10: 75.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.999779999256134:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi + \pi\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.999779999256134)
     (* (* 2.0 (* uy PI)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))
     (* (* (+ PI PI) uy) (sqrt (* ux (- 2.0 (* 2.0 maxCos))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.999779999256134f) {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	} else {
		tmp = ((((float) M_PI) + ((float) M_PI)) * uy) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.999779999256134))
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(Float32(Float32(Float32(pi) + Float32(pi)) * uy) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = single(0.0);
	if ((t_0 * t_0) <= single(0.999779999256134))
		tmp = (single(2.0) * (uy * single(pi))) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	else
		tmp = ((single(pi) + single(pi)) * uy) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999779999

   1. Initial program 57.8%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in maxCos around 0

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

   7. Applied rewrites 49.6%

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

   8. Taylor expanded in maxCos around 0

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

   10. Applied rewrites 49.5%

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

   if 0.999779999 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 57.8%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 7.1%

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

   9. Applied rewrites 7.1%

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

   10. Taylor expanded in ux around 0

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

   12. Applied rewrites 65.9%

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

Alternative 11: 65.9% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

9. Applied rewrites 7.1%

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

10. Taylor expanded in ux around 0

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

12. Applied rewrites 65.9%

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

Alternative 12: 7.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.8%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

9. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025153 
(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)))))))