UniformSampleCone, y

Percentage Accurate: 57.5% → 98.3%

Time: 4.8s

Alternatives: 12

Speedup: 1.8×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lift-pow.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 98.2

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

7. Applied rewrites 98.2%

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

Alternative 2: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 97.6

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

7. Applied rewrites 97.6%

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

Alternative 4: 96.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, 1, 2\right) - maxCos \cdot 2\right) \cdot ux} \]
6. Step-by-step derivation

7. Applied rewrites 96.8%

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

Alternative 5: 95.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;maxCos \leq 0.00019999999494757503:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(2 + -1 \cdot ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (* uy 2.0) PI))))
   (if (<= maxCos 0.00019999999494757503)
     (* t_0 (sqrt (* (+ 2.0 (* -1.0 ux)) ux)))
     (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (maxCos <= 0.00019999999494757503f) {
		tmp = t_0 * sqrtf(((2.0f + (-1.0f * ux)) * ux));
	} else {
		tmp = t_0 * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (maxCos <= Float32(0.00019999999494757503))
		tmp = Float32(t_0 * sqrt(Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * ux)) * ux)));
	else
		tmp = Float32(t_0 * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-4

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower--.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 97.0

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

   7. Applied rewrites 97.0%

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

   if 1.99999995e-4 < maxCos

   1. Initial program 56.4%

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

   2. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 76.8

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

   4. Applied rewrites 76.8%

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

Alternative 6: 91.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9995999932289124:\\ \;\;\;\;t\_1 \cdot \sin \cos^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (sin (* (* uy 2.0) PI))))
   (if (<= (* t_0 t_0) 0.9995999932289124)
     (* t_1 (sin (acos (fma maxCos ux (- 1.0 ux)))))
     (* t_1 (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float t_1 = sinf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if ((t_0 * t_0) <= 0.9995999932289124f) {
		tmp = t_1 * sinf(acosf(fmaf(maxCos, ux, (1.0f - ux))));
	} else {
		tmp = t_1 * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	t_1 = sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (Float32(t_0 * t_0) <= Float32(0.9995999932289124))
		tmp = Float32(t_1 * sin(acos(fma(maxCos, ux, Float32(Float32(1.0) - ux)))));
	else
		tmp = Float32(t_1 * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)));
	end
	return 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))) < 0.999599993

   1. Initial program 89.4%

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

      1. lift-sqrt.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. sin-acos-rev N/A

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

      11. lower-sin.f32 N/A

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

      12. lower-acos.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

      18. lift--.f32 89.6

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

   3. Applied rewrites 89.6%

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

   if 0.999599993 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 37.3%

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

   2. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 92.0

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

   4. Applied rewrites 92.0%

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

Alternative 7: 91.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \cos \sin^{-1} \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (* uy 2.0) PI))))
   (if (<= ux 0.00019999999494757503)
     (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (* t_0 (cos (asin (fma maxCos ux (- 1.0 ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = sinf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (ux <= 0.00019999999494757503f) {
		tmp = t_0 * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	} else {
		tmp = t_0 * cosf(asinf(fmaf(maxCos, ux, (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = Float32(t_0 * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)));
	else
		tmp = Float32(t_0 * cos(asin(fma(maxCos, ux, Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.99999995e-4

   1. Initial program 37.2%

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

   2. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 92.1

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

   4. Applied rewrites 92.1%

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

   if 1.99999995e-4 < ux

   1. Initial program 89.3%

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

      1. lift-sqrt.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. cos-asin-rev N/A

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

      11. lower-cos.f32 N/A

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

      12. lower-asin.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

      18. lift--.f32 89.5

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

   3. Applied rewrites 89.5%

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

Alternative 8: 84.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9983900189399719:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9983900189399719)
     (* (* PI (* 2.0 uy)) (sqrt (- 1.0 (pow (- 1.0 ux) 2.0))))
     (* (sin (* (* uy 2.0) PI)) (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9983900189399719f) {
		tmp = (((float) M_PI) * (2.0f * uy)) * sqrtf((1.0f - powf((1.0f - ux), 2.0f)));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	}
	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(0.9983900189399719))
		tmp = Float32(Float32(Float32(pi) * Float32(Float32(2.0) * uy)) * sqrt(Float32(Float32(1.0) - (Float32(Float32(1.0) - ux) ^ Float32(2.0)))));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)));
	end
	return 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))) < 0.998390019

   1. Initial program 92.0%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 74.0%

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

   if 0.998390019 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 41.2%

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

   2. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 89.7

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

   4. Applied rewrites 89.7%

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

Alternative 9: 75.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.019999999552965164:\\ \;\;\;\;\left(2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \pi\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.019999999552965164)
     (* (* 2.0 (* (sqrt (* ux (- 2.0 (* 2.0 maxCos)))) PI)) uy)
     (* (* PI (* 2.0 uy)) (sqrt (- 1.0 (pow (- 1.0 ux) 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if (sqrtf((1.0f - (t_0 * t_0))) <= 0.019999999552965164f) {
		tmp = (2.0f * (sqrtf((ux * (2.0f - (2.0f * maxCos)))) * ((float) M_PI))) * uy;
	} else {
		tmp = (((float) M_PI) * (2.0f * uy)) * sqrtf((1.0f - powf((1.0f - ux), 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) <= Float32(0.019999999552965164))
		tmp = Float32(Float32(Float32(2.0) * Float32(sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))) * Float32(pi))) * uy);
	else
		tmp = Float32(Float32(Float32(pi) * Float32(Float32(2.0) * uy)) * sqrt(Float32(Float32(1.0) - (Float32(Float32(1.0) - ux) ^ Float32(2.0)))));
	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 (sqrt((single(1.0) - (t_0 * t_0))) <= single(0.019999999552965164))
		tmp = (single(2.0) * (sqrt((ux * (single(2.0) - (single(2.0) * maxCos)))) * single(pi))) * uy;
	else
		tmp = (single(pi) * (single(2.0) * uy)) * sqrt((single(1.0) - ((single(1.0) - ux) ^ single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0199999996

   1. Initial program 37.3%

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

   2. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

   4. Applied rewrites 35.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\pi}^{3} \cdot \left(uy \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}, -1.3333333333333333, \left(\pi \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}\right) \cdot uy} \]

   5. Taylor expanded in ux around 0

      \[\leadsto \left(\frac{-4}{3} \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot uy \]
   6. Step-by-step derivation

      1. lower-fma.f32 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-4}{3}, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right), 2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot uy \]

   7. Applied rewrites 83.9%

      \[\leadsto \mathsf{fma}\left(-1.3333333333333333, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(\left(uy \cdot uy\right) \cdot {\pi}^{3}\right), 2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \pi\right)\right) \cdot uy \]

   8. Taylor expanded in uy around 0

      \[\leadsto \left(2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot uy \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-sqrt.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. lift-PI.f32 N/A

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

      7. lift-*.f32 77.2

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

   10. Applied rewrites 77.2%

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

   if 0.0199999996 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 89.4%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. lower--.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 75.6

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 72.4%

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

Alternative 10: 65.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right)} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\frac{-4}{3} \cdot \left(\left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right) + 2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)\right) \cdot \color{blue}{uy} \]

4. Applied rewrites 53.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left({\pi}^{3} \cdot \left(uy \cdot uy\right)\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}, -1.3333333333333333, \left(\pi \cdot 2\right) \cdot \sqrt{1 - {\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}^{2}}\right) \cdot uy} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\frac{-4}{3} \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) + 2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot uy \]
6. Step-by-step derivation

   1. lower-fma.f32 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-4}{3}, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right), 2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot uy \]

7. Applied rewrites 70.7%

   \[\leadsto \mathsf{fma}\left(-1.3333333333333333, \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \left(\left(uy \cdot uy\right) \cdot {\pi}^{3}\right), 2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \pi\right)\right) \cdot uy \]

8. Taylor expanded in uy around 0

   \[\leadsto \left(2 \cdot \left(\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot uy \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-PI.f32 N/A

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

   7. lift-*.f32 65.9

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

10. Applied rewrites 65.9%

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

Alternative 11: 20.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 7.1%

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

5. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - 1} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 7.1

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

7. Applied rewrites 7.1%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-pow.f32 N/A

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

   2. lower--.f32 48.9

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

10. Applied rewrites 48.9%

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

11. Taylor expanded in ux around inf

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

    1. unpow2 N/A

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

    2. lower-*.f32 20.0

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

13. Applied rewrites 20.0%

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

Alternative 12: 7.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

4. Applied rewrites 7.1%

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

5. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - 1} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 7.1

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

7. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025101 
(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)))))))