UniformSampleCone, x

Percentage Accurate: 57.3% → 99.1%

Time: 9.6s

Alternatives: 13

Speedup: 11.2×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 99.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* PI (fma uy -2.0 0.5)))
  (sqrt
   (*
    ux
    (- (+ 2.0 (* -1.0 (* ux (pow (- maxCos 1.0) 2.0)))) (* 2.0 maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((((float) M_PI) * fmaf(uy, -2.0f, 0.5f))) * sqrtf((ux * ((2.0f + (-1.0f * (ux * powf((maxCos - 1.0f), 2.0f)))) - (2.0f * maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(pi) * fma(uy, Float32(-2.0), Float32(0.5)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) - Float32(Float32(2.0) * maxCos)))))
end

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]
7. Add Preprocessing

Alternative 2: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - 2 \cdot maxCos\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* PI (fma uy -2.0 0.5)))
  (sqrt
   (*
    ux
    (-
     (+ 1.0 (- 1.0 (* ux (* (- maxCos 1.0) (- maxCos 1.0)))))
     (* 2.0 maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((((float) M_PI) * fmaf(uy, -2.0f, 0.5f))) * sqrtf((ux * ((1.0f + (1.0f - (ux * ((maxCos - 1.0f) * (maxCos - 1.0f))))) - (2.0f * maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(pi) * fma(uy, Float32(-2.0), Float32(0.5)))) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - Float32(ux * Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0)))))) - Float32(Float32(2.0) * maxCos)))))
end

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

8. Applied rewrites 99.1%

   \[\leadsto \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - \color{blue}{2} \cdot maxCos\right)} \]
9. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.0%

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

8. Applied rewrites 99.0%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 2\right) \cdot \mathsf{fma}\left(maxCos - 1, ux, 0\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot uy\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt (* (- (- ux (* maxCos ux)) 2.0) (fma (- maxCos 1.0) ux 0.0)))
  (cos (* (+ PI PI) uy))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((((ux - (maxCos * ux)) - 2.0f) * fmaf((maxCos - 1.0f), ux, 0.0f))) * cosf(((((float) M_PI) + ((float) M_PI)) * uy));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(Float32(ux - Float32(maxCos * ux)) - Float32(2.0)) * fma(Float32(maxCos - Float32(1.0)), ux, Float32(0.0)))) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * uy)))
end

Derivation

1. Initial program 57.3%

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

3. Applied rewrites 57.3%

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

5. Applied rewrites 99.0%

   \[\leadsto \color{blue}{\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 2\right) \cdot \mathsf{fma}\left(maxCos - 1, ux, 0\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot uy\right)} \]
6. Add Preprocessing

Alternative 5: 97.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - ux\right)\right) - 2 \cdot maxCos\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* PI (fma uy -2.0 0.5)))
  (sqrt (* ux (- (+ 1.0 (- 1.0 ux)) (* 2.0 maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	return sinf((((float) M_PI) * fmaf(uy, -2.0f, 0.5f))) * sqrtf((ux * ((1.0f + (1.0f - ux)) - (2.0f * maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(pi) * fma(uy, Float32(-2.0), Float32(0.5)))) * sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(1.0) - ux)) - Float32(Float32(2.0) * maxCos)))))
end

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 99.0%

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

6. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

8. Applied rewrites 99.1%

   \[\leadsto \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - \color{blue}{2} \cdot maxCos\right)} \]

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 97.7%

    \[\leadsto \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - ux\right)\right) - 2 \cdot maxCos\right)} \]
12. Add Preprocessing

Alternative 6: 96.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.4999999621068127 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) + \mathsf{fma}\left(1 - maxCos, ux, -1\right) \cdot \left(maxCos \cdot ux - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.4999999621068127e-5)
   (* (sin (* PI (fma uy -2.0 0.5))) (sqrt (* ux (- 2.0 ux))))
   (sqrt
    (+
     (- (- ux (* maxCos ux)) 0.0)
     (* (fma (- 1.0 maxCos) ux -1.0) (- (* maxCos ux) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.4999999621068127e-5f) {
		tmp = sinf((((float) M_PI) * fmaf(uy, -2.0f, 0.5f))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((((ux - (maxCos * ux)) - 0.0f) + (fmaf((1.0f - maxCos), ux, -1.0f) * ((maxCos * ux) - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.4999999621068127e-5))
		tmp = Float32(sin(Float32(Float32(pi) * fma(uy, Float32(-2.0), Float32(0.5)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(Float32(Float32(ux - Float32(maxCos * ux)) - Float32(0.0)) + Float32(fma(Float32(Float32(1.0) - maxCos), ux, Float32(-1.0)) * Float32(Float32(maxCos * ux) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.49999996e-5

   1. Initial program 57.3%

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

   2. Taylor expanded in ux around 0

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

   4. Applied rewrites 99.0%

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

   6. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right)} \cdot \sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)} \]

   8. Applied rewrites 99.1%

      \[\leadsto \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(\left(1 + \left(1 - ux \cdot \left(\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right)\right) - \color{blue}{2} \cdot maxCos\right)} \]

   9. Taylor expanded in maxCos around 0

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

   11. Applied rewrites 93.1%

       \[\leadsto \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{ux}\right)} \]

   if 1.49999996e-5 < maxCos

   1. Initial program 57.3%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 80.0%

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

Alternative 7: 80.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.3%

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

6. Applied rewrites 55.9%

   \[\leadsto \sqrt{\left(1 - \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) - \mathsf{fma}\left(ux, maxCos, 1 - ux\right) \cdot \left(maxCos \cdot ux - ux\right)} \]

8. Applied rewrites 80.0%

   \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(ux, maxCos, 1 - ux\right), ux - maxCos \cdot ux, 0 - \left(maxCos \cdot ux - ux\right)\right)} \]
9. Add Preprocessing

Alternative 8: 80.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(maxCos - 1, ux, 2\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 0\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (* (fma (- maxCos 1.0) ux 2.0) (- (- ux (* maxCos ux)) 0.0))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((fmaf((maxCos - 1.0f), ux, 2.0f) * ((ux - (maxCos * ux)) - 0.0f)));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(fma(Float32(maxCos - Float32(1.0)), ux, Float32(2.0)) * Float32(Float32(ux - Float32(maxCos * ux)) - Float32(0.0))))
end

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.3%

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

6. Applied rewrites 80.0%

   \[\leadsto \sqrt{\mathsf{fma}\left(maxCos - 1, ux, 2\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 0\right)} \]
7. Add Preprocessing

Alternative 9: 80.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 2\right) \cdot \mathsf{fma}\left(maxCos - 1, ux, 0\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (sqrt (* (- (- ux (* maxCos ux)) 2.0) (fma (- maxCos 1.0) ux 0.0))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((((ux - (maxCos * ux)) - 2.0f) * fmaf((maxCos - 1.0f), ux, 0.0f)));
}

Julia

function code(ux, uy, maxCos)
	return sqrt(Float32(Float32(Float32(ux - Float32(maxCos * ux)) - Float32(2.0)) * fma(Float32(maxCos - Float32(1.0)), ux, Float32(0.0))))
end

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.3%

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

6. Applied rewrites 80.0%

   \[\leadsto \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 2\right) \cdot \mathsf{fma}\left(maxCos - 1, ux, 0\right)} \]
7. Add Preprocessing

Alternative 10: 77.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-5

   1. Initial program 57.3%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 55.9%

      \[\leadsto \sqrt{\left(1 - \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) - \mathsf{fma}\left(ux, maxCos, 1 - ux\right) \cdot \left(maxCos \cdot ux - ux\right)} \]

   7. Taylor expanded in maxCos around 0

      \[\leadsto \sqrt{ux - -1 \cdot \left(ux \cdot \left(1 - ux\right)\right)} \]

   9. Applied rewrites 75.9%

      \[\leadsto \sqrt{ux - -1 \cdot \left(ux \cdot \left(1 - ux\right)\right)} \]

   11. Applied rewrites 75.9%

       \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(1 - ux, ux, ux\right)}} \]

   if 1.99999995e-5 < maxCos

   1. Initial program 57.3%

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

   2. Taylor expanded in uy around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 55.9%

      \[\leadsto \sqrt{\left(1 - \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) - \mathsf{fma}\left(ux, maxCos, 1 - ux\right) \cdot \left(maxCos \cdot ux - ux\right)} \]

   7. Taylor expanded in maxCos around 0

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

   9. Applied rewrites 79.5%

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

   10. Taylor expanded in ux around 0

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

   12. Applied rewrites 64.8%

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

Alternative 11: 75.9% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.3%

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

6. Applied rewrites 55.9%

   \[\leadsto \sqrt{\left(1 - \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) - \mathsf{fma}\left(ux, maxCos, 1 - ux\right) \cdot \left(maxCos \cdot ux - ux\right)} \]

7. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{ux - -1 \cdot \left(ux \cdot \left(1 - ux\right)\right)} \]

9. Applied rewrites 75.9%

   \[\leadsto \sqrt{ux - -1 \cdot \left(ux \cdot \left(1 - ux\right)\right)} \]

11. Applied rewrites 75.9%

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(1 - ux, ux, ux\right)}} \]
12. Add Preprocessing

Alternative 12: 62.3% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((2.0e0 * ux))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.3%

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

6. Applied rewrites 55.9%

   \[\leadsto \sqrt{\left(1 - \mathsf{fma}\left(ux, maxCos, 1 - ux\right)\right) - \mathsf{fma}\left(ux, maxCos, 1 - ux\right) \cdot \left(maxCos \cdot ux - ux\right)} \]

7. Taylor expanded in maxCos around 0

   \[\leadsto \sqrt{ux - -1 \cdot \left(ux \cdot \left(1 - ux\right)\right)} \]

9. Applied rewrites 75.9%

   \[\leadsto \sqrt{ux - -1 \cdot \left(ux \cdot \left(1 - ux\right)\right)} \]

10. Taylor expanded in ux around 0

    \[\leadsto \sqrt{2 \cdot ux} \]

12. Applied rewrites 62.3%

    \[\leadsto \sqrt{2 \cdot ux} \]
13. Add Preprocessing

Alternative 13: 6.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - 1} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((1.0e0 - 1.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.3%

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

2. Taylor expanded in uy around 0

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

4. Applied rewrites 49.3%

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

5. Taylor expanded in ux around 0

   \[\leadsto \sqrt{1 - 1} \]

7. Applied rewrites 6.6%

   \[\leadsto \sqrt{1 - 1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025161 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))