UniformSampleCone, y

Percentage Accurate: 57.5% → 98.3%

Time: 6.3s

Alternatives: 19

Speedup: 5.4×

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

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sin(((uy * single(2.0)) * single(pi))) * sqrt(((((-((maxCos - single(1.0)) * (maxCos - single(1.0))) + ((single(2.0) - maxCos) / ux)) * ux) - maxCos) * 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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate--r+ N/A

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

   9. lower--.f32 N/A

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--l+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. div-sub N/A

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

   7. lower-+.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower--.f32 98.2

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

9. Applied rewrites 98.2%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(sin(Float32(Float32(uy + uy) * Float32(pi))) * sqrt(Float32(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - maxCos) - maxCos) * 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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate--r+ N/A

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

   9. lower--.f32 N/A

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

6. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lift-+.f32 98.3

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

8. Applied rewrites 98.3%

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

Alternative 3: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((fmaf(((ux + ux) - 2.0f), maxCos, -ux) + 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(Float32(ux + ux) - Float32(2.0)), maxCos, Float32(-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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

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. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower--.f32 N/A

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

   7. count-2-rev N/A

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

   8. lower-+.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lift-neg.f32 97.6

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

7. Applied rewrites 97.6%

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

Alternative 4: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0036849998869001865:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\left(-ux\right) + 2\right) - maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0036849998869001865)
   (*
    (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)))
    (sqrt
     (*
      (-
       (* (+ (- (* (- maxCos 1.0) (- maxCos 1.0))) (/ (- 2.0 maxCos) ux)) ux)
       maxCos)
      ux)))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (- (+ (- ux) 2.0) maxCos) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0036849998869001865f) {
		tmp = (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * sqrtf(((((-((maxCos - 1.0f) * (maxCos - 1.0f)) + ((2.0f - maxCos) / ux)) * ux) - maxCos) * ux));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((((-ux + 2.0f) - maxCos) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0036849998869001865))
		tmp = Float32(Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) + Float32(pi)))) * sqrt(Float32(Float32(Float32(Float32(Float32(-Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0)))) + Float32(Float32(Float32(2.0) - maxCos) / ux)) * ux) - maxCos) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(Float32(-ux) + Float32(2.0)) - maxCos) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00368499989

   1. Initial program 57.4%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   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. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate--r+ N/A

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

      9. lower--.f32 N/A

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

   6. Applied rewrites 98.5%

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

   7. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--l+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-sub N/A

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

      7. lower-+.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. lift--.f32 N/A

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

      13. lift--.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 98.4

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

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow3 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-*.f32 N/A

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

       10. lift-PI.f32 N/A

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

       11. lift-PI.f32 N/A

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

       12. lift-PI.f32 N/A

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

       13. count-2-rev N/A

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

       14. lower-+.f32 N/A

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

       15. lift-PI.f32 N/A

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

       16. lift-PI.f32 98.4

           \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux} \]

   12. Applied rewrites 98.4%

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

   if 0.00368499989 < uy

   1. Initial program 58.0%

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

   2. Taylor expanded in 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. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 97.8

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

   4. Applied rewrites 97.8%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate--r+ N/A

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

      9. lower--.f32 N/A

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in maxCos around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. lift-+.f32 N/A

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

      4. lift-neg.f32 92.5

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

   9. Applied rewrites 92.5%

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

Alternative 5: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0036849998869001865:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(-ux\right) + 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0036849998869001865)
   (*
    (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)))
    (sqrt
     (*
      (-
       (* (+ (- (* (- maxCos 1.0) (- maxCos 1.0))) (/ (- 2.0 maxCos) ux)) ux)
       maxCos)
      ux)))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (+ (- ux) 2.0) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0036849998869001865f) {
		tmp = (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * sqrtf(((((-((maxCos - 1.0f) * (maxCos - 1.0f)) + ((2.0f - maxCos) / ux)) * ux) - maxCos) * ux));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf(((-ux + 2.0f) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0036849998869001865))
		tmp = Float32(Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) + Float32(pi)))) * sqrt(Float32(Float32(Float32(Float32(Float32(-Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0)))) + Float32(Float32(Float32(2.0) - maxCos) / ux)) * ux) - maxCos) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(-ux) + Float32(2.0)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00368499989

   1. Initial program 57.4%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot {\left(maxCos - 1\right)}^{2}\right)\right) - 2 \cdot maxCos\right)}} \]
   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. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.5

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

   4. Applied rewrites 98.5%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate--r+ N/A

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

      9. lower--.f32 N/A

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

   6. Applied rewrites 98.5%

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

   7. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--l+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-sub N/A

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

      7. lower-+.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. lift--.f32 N/A

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

      13. lift--.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 98.4

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

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow3 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-*.f32 N/A

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

       10. lift-PI.f32 N/A

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

       11. lift-PI.f32 N/A

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

       12. lift-PI.f32 N/A

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

       13. count-2-rev N/A

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

       14. lower-+.f32 N/A

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

       15. lift-PI.f32 N/A

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

       16. lift-PI.f32 98.4

           \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux} \]

   12. Applied rewrites 98.4%

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

   if 0.00368499989 < uy

   1. Initial program 58.0%

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

   2. Taylor expanded in 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. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 97.8

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

   4. Applied rewrites 97.8%

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

   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. +-commutative N/A

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

      2. lower-+.f32 N/A

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

      3. mul-1-neg N/A

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

      4. lift-neg.f32 92.0

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

   7. Applied rewrites 92.0%

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

Alternative 6: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((((-ux + single(2.0)) - (maxCos + maxCos)) * ux)) * sin(((uy + uy) * single(pi)));
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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 96.8

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

7. Applied rewrites 96.8%

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

   1. lift-*.f32 N/A

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

   2. lift-sin.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

9. Applied rewrites 96.8%

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

Alternative 7: 94.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.06849999725818634:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.06849999725818634)
   (*
    (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)))
    (sqrt
     (*
      (-
       (* (+ (- (* (- maxCos 1.0) (- maxCos 1.0))) (/ (- 2.0 maxCos) ux)) ux)
       maxCos)
      ux)))
   (* (sin (* PI (+ uy uy))) (sqrt (* (- 2.0 (+ maxCos maxCos)) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.06849999725818634f) {
		tmp = (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * sqrtf(((((-((maxCos - 1.0f) * (maxCos - 1.0f)) + ((2.0f - maxCos) / ux)) * ux) - maxCos) * ux));
	} else {
		tmp = sinf((((float) M_PI) * (uy + uy))) * sqrtf(((2.0f - (maxCos + maxCos)) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.06849999725818634))
		tmp = Float32(Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) + Float32(pi)))) * sqrt(Float32(Float32(Float32(Float32(Float32(-Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0)))) + Float32(Float32(Float32(2.0) - maxCos) / ux)) * ux) - maxCos) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(uy + uy))) * sqrt(Float32(Float32(Float32(2.0) - Float32(maxCos + maxCos)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0684999973

   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. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.4

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

   4. Applied rewrites 98.4%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift--.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. associate--r+ N/A

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

      9. lower--.f32 N/A

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

   6. Applied rewrites 98.4%

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

   7. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate--l+ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-sub N/A

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

      7. lower-+.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. lift--.f32 N/A

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

      13. lift--.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 98.4

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

   9. Applied rewrites 98.4%

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

   10. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f32 N/A

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

       7. unpow3 N/A

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

       8. lower-*.f32 N/A

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

       9. lower-*.f32 N/A

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

       10. lift-PI.f32 N/A

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

       11. lift-PI.f32 N/A

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

       12. lift-PI.f32 N/A

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

       13. count-2-rev N/A

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

       14. lower-+.f32 N/A

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

       15. lift-PI.f32 N/A

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

       16. lift-PI.f32 96.6

           \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux} \]

   12. Applied rewrites 96.6%

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

   if 0.0684999973 < uy

   1. Initial program 58.1%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-sin.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. *-commutative N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. metadata-eval N/A

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

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

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

      15. *-commutative N/A

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

      16. lower-*.f32 N/A

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

   4. Applied rewrites 75.7%

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

Alternative 8: 88.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)))
  (sqrt
   (*
    (-
     (* (+ (- (* (- maxCos 1.0) (- maxCos 1.0))) (/ (- 2.0 maxCos) ux)) ux)
     maxCos)
    ux))))

C

float code(float ux, float uy, float maxCos) {
	return (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * sqrtf(((((-((maxCos - 1.0f) * (maxCos - 1.0f)) + ((2.0f - maxCos) / ux)) * ux) - maxCos) * ux));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(uy * fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) + Float32(pi)))) * sqrt(Float32(Float32(Float32(Float32(Float32(-Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0)))) + Float32(Float32(Float32(2.0) - maxCos) / ux)) * ux) - maxCos) * 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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate--r+ N/A

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

   9. lower--.f32 N/A

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. associate--l+ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. div-sub N/A

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

   7. lower-+.f32 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 N/A

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

   12. lift--.f32 N/A

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

   13. lift--.f32 N/A

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

   14. lower-/.f32 N/A

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

   15. lower--.f32 98.2

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

9. Applied rewrites 98.2%

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

10. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. associate-*r* N/A

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

    3. lower-fma.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. unpow2 N/A

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

    6. lower-*.f32 N/A

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

    7. unpow3 N/A

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

    8. lower-*.f32 N/A

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

    9. lower-*.f32 N/A

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

    10. lift-PI.f32 N/A

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

    11. lift-PI.f32 N/A

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

    12. lift-PI.f32 N/A

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

    13. count-2-rev N/A

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

    14. lower-+.f32 N/A

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

    15. lift-PI.f32 N/A

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

    16. lift-PI.f32 88.8

        \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux} \]

12. Applied rewrites 88.8%

    \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right)} \cdot \sqrt{\left(\left(\left(-\left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right) + \frac{2 - maxCos}{ux}\right) \cdot ux - maxCos\right) \cdot ux} \]
13. Add Preprocessing

Alternative 9: 88.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)) uy)
  (sqrt
   (*
    (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
    ux))))

C

float code(float ux, float uy, float maxCos) {
	return (fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * 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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

7. Applied rewrites 88.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right) \cdot uy\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
8. Add Preprocessing

Alternative 10: 88.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos\right) - maxCos\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)) uy)
  (sqrt
   (*
    (- (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) maxCos) maxCos)
    ux))))

C

float code(float ux, float uy, float maxCos) {
	return (fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf((((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - maxCos) - maxCos) * ux));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - maxCos) - maxCos) * 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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-neg.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift--.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate--r+ N/A

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

   9. lower--.f32 N/A

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

9. Applied rewrites 88.9%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right) \cdot uy\right)} \cdot \sqrt{\left(\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - maxCos\right) - maxCos\right) \cdot ux} \]
10. Add Preprocessing

Alternative 11: 87.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right) \cdot uy\right) \cdot \sqrt{\left(\left(\left(-ux\right) + 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)) uy)
  (sqrt (* (- (+ (- ux) 2.0) (+ maxCos maxCos)) ux))))

C

float code(float ux, float uy, float maxCos) {
	return (fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI))) * uy) * sqrtf((((-ux + 2.0f) - (maxCos + maxCos)) * ux));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(fma(Float32(Float32(-1.3333333333333333) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)), Float32(Float32(pi) + Float32(pi))) * uy) * sqrt(Float32(Float32(Float32(Float32(-ux) + Float32(2.0)) - Float32(maxCos + maxCos)) * 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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. mul-1-neg N/A

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

   4. lift-neg.f32 96.8

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

7. Applied rewrites 96.8%

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

8. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 87.6%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right) \cdot uy\right)} \cdot \sqrt{\left(\left(\left(-ux\right) + 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
11. Add Preprocessing

Alternative 12: 81.3% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(uy + uy) * Float32(pi)) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * 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. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. lower-*.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. lift-PI.f32 81.3

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

7. Applied rewrites 81.3%

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

Alternative 13: 76.7% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	t_0 = fma(maxCos, ux, Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(Float32(Float32(2.0) * sqrt(Float32(Float32(Float32(2.0) - Float32(maxCos + maxCos)) * ux))) * Float32(pi)) * uy);
	else
		tmp = Float32(Float32(pi) * Float32(Float32(uy + uy) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))));
	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 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. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 34.2%

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

   5. Taylor expanded in ux around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 77.3

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

   7. Applied rewrites 77.3%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

   9. Applied rewrites 77.3%

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

   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. 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. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 75.6%

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

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-sqrt.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lift--.f32 N/A

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

      9. lift-fma.f32 N/A

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

      10. lift--.f32 N/A

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

      11. lift-fma.f32 N/A

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

      12. associate-*l* N/A

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

      13. lower-*.f32 N/A

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

   6. Applied rewrites 75.7%

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

Alternative 14: 75.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00019999999494757503))
		tmp = Float32(Float32(Float32(Float32(2.0) * sqrt(Float32(Float32(Float32(2.0) - Float32(maxCos + maxCos)) * ux))) * Float32(pi)) * uy);
	else
		tmp = Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(Float32(1.0) - Float32(Float32(fma(maxCos, ux, Float32(1.0)) - 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 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. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 34.2%

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

   5. Taylor expanded in ux around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 77.3

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

   7. Applied rewrites 77.3%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

   9. Applied rewrites 77.3%

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

   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. 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. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 72.6%

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

Alternative 15: 75.3% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00019999999494757503:\\ \;\;\;\;\left(\left(2 \cdot \sqrt{\left(2 - \left(maxCos + maxCos\right)\right) \cdot ux}\right) \cdot \pi\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00019999999494757503))
		tmp = ((single(2.0) * sqrt(((single(2.0) - (maxCos + maxCos)) * ux))) * single(pi)) * uy;
	else
		tmp = (single(pi) * (uy + uy)) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	end
	tmp_2 = 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 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. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 34.2%

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

   5. Taylor expanded in ux around 0

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

      1. associate-*r* N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. count-2-rev N/A

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

      9. lift-+.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lift-PI.f32 77.3

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

   7. Applied rewrites 77.3%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f32 N/A

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

   9. Applied rewrites 77.3%

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

   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. 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. count-2-rev N/A

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

      10. lower-+.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 N/A

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in maxCos around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift--.f32 72.2

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

   7. Applied rewrites 72.2%

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

Alternative 16: 65.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.4%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. count-2-rev N/A

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

   9. lift-+.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 65.9

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

7. Applied rewrites 65.9%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-+.f32 N/A

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

   6. lift--.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. associate-*r* N/A

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

   10. lower-*.f32 N/A

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

9. Applied rewrites 65.9%

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

Alternative 17: 65.9% accurate, 4.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.4%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. count-2-rev N/A

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

   9. lift-+.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 65.9

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

7. Applied rewrites 65.9%

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

Alternative 18: 63.2% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

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

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.4%

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

5. Taylor expanded in ux around 0

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

   1. associate-*r* N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-sqrt.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. count-2-rev N/A

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

   9. lift-+.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lift-PI.f32 65.9

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

7. Applied rewrites 65.9%

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

8. Taylor expanded in maxCos around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f32 63.2

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

10. Applied rewrites 63.2%

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

Alternative 19: 7.1% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(pi) * (uy + uy)) * 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 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. count-2-rev N/A

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

   10. lower-+.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 N/A

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

4. Applied rewrites 50.4%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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