UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 11.2s

Alternatives: 14

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: 57.4% 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(\left(ux - 1\right) - ux \cdot maxCos\right) - 1\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (* (- (- (- ux 1.0) (* ux maxCos)) 1.0) (fma maxCos ux (+ (- ux) 0.0))))
  (sin (* (+ PI PI) uy))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((((ux - 1.0f) - (ux * maxCos)) - 1.0f) * fmaf(maxCos, ux, (-ux + 0.0f)))) * sinf(((((float) M_PI) + ((float) M_PI)) * uy));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(Float32(Float32(ux - Float32(1.0)) - Float32(ux * maxCos)) - Float32(1.0)) * fma(maxCos, ux, Float32(Float32(-ux) + Float32(0.0))))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)))
end

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 57.4%

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

7. Applied rewrites 98.3%

   \[\leadsto \sqrt{\color{blue}{\left(\left(\left(ux - 1\right) - ux \cdot maxCos\right) - 1\right)} \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \]
8. Add Preprocessing

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* (- (- ux (* ux maxCos)) 2.0) (fma maxCos ux (+ (- ux) 0.0))))
  (sin (* (+ PI PI) uy))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((((ux - (ux * maxCos)) - 2.0f) * fmaf(maxCos, ux, (-ux + 0.0f)))) * sinf(((((float) M_PI) + ((float) M_PI)) * uy));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(Float32(ux - Float32(ux * maxCos)) - Float32(2.0)) * fma(maxCos, ux, Float32(Float32(-ux) + Float32(0.0))))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)))
end

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 57.4%

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]
6. Add Preprocessing

Alternative 3: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 57.4%

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

7. Applied rewrites 98.3%

   \[\leadsto \sqrt{\color{blue}{\left(\left(\left(ux - 1\right) - ux \cdot maxCos\right) - 1\right)} \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \]

9. Applied rewrites 98.3%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 57.4%

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

7. Applied rewrites 98.3%

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

Alternative 5: 97.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(ux - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* (- ux 2.0) (fma maxCos ux (+ (- ux) 0.0))))
  (sin (* (+ PI PI) uy))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((ux - 2.0f) * fmaf(maxCos, ux, (-ux + 0.0f)))) * sinf(((((float) M_PI) + ((float) M_PI)) * uy));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(ux - Float32(2.0)) * fma(maxCos, ux, Float32(Float32(-ux) + Float32(0.0))))) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)))
end

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 57.4%

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

6. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{\color{blue}{\left(ux - 2\right)} \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \]

8. Applied rewrites 97.1%

   \[\leadsto \sqrt{\color{blue}{\left(ux - 2\right)} \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \]
9. Add Preprocessing

Alternative 6: 96.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. 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%

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

7. Taylor expanded in maxCos around 0

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

9. Applied rewrites 96.9%

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

Alternative 7: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.4999999621068127 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(2 - ux\right) \cdot ux} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(\left(ux - 1\right) - ux \cdot maxCos\right) - 1\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.4999999621068127e-5)
   (* (sqrt (* (- 2.0 ux) ux)) (sin (* (+ PI PI) uy)))
   (*
    (sqrt
     (* (- (- (- ux 1.0) (* ux maxCos)) 1.0) (fma maxCos ux (+ (- ux) 0.0))))
    (* 2.0 (* uy PI)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.4999999621068127e-5f) {
		tmp = sqrtf(((2.0f - ux) * ux)) * sinf(((((float) M_PI) + ((float) M_PI)) * uy));
	} else {
		tmp = sqrtf(((((ux - 1.0f) - (ux * maxCos)) - 1.0f) * fmaf(maxCos, ux, (-ux + 0.0f)))) * (2.0f * (uy * ((float) M_PI)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.4999999621068127e-5))
		tmp = Float32(sqrt(Float32(Float32(Float32(2.0) - ux) * ux)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)));
	else
		tmp = Float32(sqrt(Float32(Float32(Float32(Float32(ux - Float32(1.0)) - Float32(ux * maxCos)) - Float32(1.0)) * fma(maxCos, ux, Float32(Float32(-ux) + Float32(0.0))))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.49999996e-5

   1. Initial program 57.4%

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

   3. Applied rewrites 57.4%

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

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 92.4%

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

   10. Applied rewrites 92.4%

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

   if 1.49999996e-5 < maxCos

   1. Initial program 57.4%

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

   3. Applied rewrites 57.4%

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

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

   7. Applied rewrites 98.3%

      \[\leadsto \sqrt{\color{blue}{\left(\left(\left(ux - 1\right) - ux \cdot maxCos\right) - 1\right)} \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right) \]

   8. Taylor expanded in uy around 0

      \[\leadsto \sqrt{\left(\left(\left(ux - 1\right) - ux \cdot maxCos\right) - 1\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   10. Applied rewrites 81.4%

       \[\leadsto \sqrt{\left(\left(\left(ux - 1\right) - ux \cdot maxCos\right) - 1\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \mathbf{if}\;uy \leq 0.003800000064074993:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{2 \cdot t\_0 - {t\_0}^{2}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- ux (* maxCos ux))))
   (if (<= uy 0.003800000064074993)
     (* 2.0 (* uy (* PI (sqrt (- (* 2.0 t_0) (pow t_0 2.0))))))
     (* (sqrt (* 2.0 ux)) (sin (* (+ PI PI) uy))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (maxCos * ux);
	float tmp;
	if (uy <= 0.003800000064074993f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((2.0f * t_0) - powf(t_0, 2.0f)))));
	} else {
		tmp = sqrtf((2.0f * ux)) * sinf(((((float) M_PI) + ((float) M_PI)) * uy));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(maxCos * ux))
	tmp = Float32(0.0)
	if (uy <= Float32(0.003800000064074993))
		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(sqrt(Float32(Float32(2.0) * ux)) * sin(Float32(Float32(Float32(pi) + Float32(pi)) * uy)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux - (maxCos * ux);
	tmp = single(0.0);
	if (uy <= single(0.003800000064074993))
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((single(2.0) * t_0) - (t_0 ^ single(2.0))))));
	else
		tmp = sqrt((single(2.0) * ux)) * sin(((single(pi) + single(pi)) * uy));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00380000006

   1. Initial program 57.4%

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

   2. 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 50.6%

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

   6. Applied rewrites 52.8%

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

   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 0.00380000006 < uy

   1. Initial program 57.4%

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

   3. Applied rewrites 57.4%

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

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in ux around 0

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

   11. Applied rewrites 73.2%

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

Alternative 9: 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 57.4%

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

2. 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 50.6%

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

6. Applied rewrites 52.8%

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

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 10: 81.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

3. Applied rewrites 57.4%

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

5. Applied rewrites 98.3%

   \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

6. Taylor expanded in uy around 0

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

8. Applied rewrites 81.4%

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

Alternative 11: 81.4% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. 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 50.6%

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

6. Applied rewrites 52.8%

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

8. Applied rewrites 52.8%

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

10. Applied rewrites 81.4%

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

Alternative 12: 79.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\pi + \pi\right) \cdot uy\right) \cdot \sqrt{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999494757503e-5)
   (* (sqrt (* -1.0 (* ux (- ux 2.0)))) (* 2.0 (* uy PI)))
   (* (* (+ PI PI) uy) (sqrt (* 2.0 (* ux (- 1.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999494757503e-5f) {
		tmp = sqrtf((-1.0f * (ux * (ux - 2.0f)))) * (2.0f * (uy * ((float) M_PI)));
	} else {
		tmp = ((((float) M_PI) + ((float) M_PI)) * uy) * sqrtf((2.0f * (ux * (1.0f - maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999494757503e-5))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(ux * Float32(ux - Float32(2.0))))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(Float32(Float32(Float32(pi) + Float32(pi)) * uy) * sqrt(Float32(Float32(2.0) * Float32(ux * Float32(Float32(1.0) - maxCos)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.9999999494757503e-5))
		tmp = sqrt((single(-1.0) * (ux * (ux - single(2.0))))) * (single(2.0) * (uy * single(pi)));
	else
		tmp = ((single(pi) + single(pi)) * uy) * sqrt((single(2.0) * (ux * (single(1.0) - maxCos))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-5

   1. Initial program 57.4%

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

   3. Applied rewrites 57.4%

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

   5. Applied rewrites 98.3%

      \[\leadsto \color{blue}{\sqrt{\left(\left(ux - ux \cdot maxCos\right) - 2\right) \cdot \mathsf{fma}\left(maxCos, ux, \left(-ux\right) + 0\right)} \cdot \sin \left(\left(\pi + \pi\right) \cdot uy\right)} \]

   6. Taylor expanded in maxCos around 0

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

   8. Applied rewrites 92.4%

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

   9. Taylor expanded in uy around 0

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

   11. Applied rewrites 77.3%

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

   if 1.99999995e-5 < maxCos

   1. Initial program 57.4%

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

   2. 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 50.6%

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

   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

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

   12. Applied rewrites 66.0%

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

Alternative 13: 66.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. 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 50.6%

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

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

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

12. Applied rewrites 66.0%

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

Alternative 14: 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 57.4%

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

2. 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 50.6%

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

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 2025161 
(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)))))))