UniformSampleCone, y

Percentage Accurate: 57.6% → 98.3%

Time: 6.2s

Alternatives: 22

Speedup: 4.6×

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: 57.6% 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

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. +-commutative N/A

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

   8. unpow2 N/A

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

   9. lower-fma.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower--.f32 N/A

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

   12. associate-*r/ N/A

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

   13. lower-/.f32 N/A

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

   14. count-2-rev N/A

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

   15. lower-+.f32 N/A

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

   16. unpow2 N/A

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

   17. lower-*.f32 98.2

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-/.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. pow2 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 98.2

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

7. Applied rewrites 98.2%

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

Alternative 2: 98.2% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sin (* (* uy 2.0) 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 * 2.0f) * ((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 * Float32(2.0)) * 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 57.6%

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

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. Add Preprocessing

Alternative 3: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.012000000104308128:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.012000000104308128)
   (*
    (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (- (/ 2.0 ux) 1.0) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.012000000104308128f) {
		tmp = (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((((2.0f / ux) - 1.0f) * (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.012000000104308128))
		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(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(Float32(2.0) / ux) - Float32(1.0)) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0120000001

   1. Initial program 57.6%

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

   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. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      3. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(\frac{-4}{3}, {\mathsf{PI}\left(\right)}^{3}, {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.025396825396825397, \left(uy \cdot uy\right) \cdot {\pi}^{7}, 0.26666666666666666 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   8. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow3 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 98.5

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

   10. Applied rewrites 98.5%

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

   if 0.0120000001 < uy

   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 inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f32 N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. lower-fma.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower--.f32 N/A

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

      12. associate-*r/ N/A

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

      13. lower-/.f32 N/A

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

      14. count-2-rev N/A

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

      15. lower-+.f32 N/A

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

      16. unpow2 N/A

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

      17. lower-*.f32 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in maxCos around 0

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

   7. Applied rewrites 92.1%

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

Alternative 4: 97.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.012000000104308128:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 + \left(-ux\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.012000000104308128)
   (*
    (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos 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 <= 0.012000000104308128f) {
		tmp = (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + 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(0.012000000104308128))
		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(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(-ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0120000001

   1. Initial program 57.6%

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

   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. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      3. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(\frac{-4}{3}, {\mathsf{PI}\left(\right)}^{3}, {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.025396825396825397, \left(uy \cdot uy\right) \cdot {\pi}^{7}, 0.26666666666666666 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   8. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow3 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 98.5

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

   10. Applied rewrites 98.5%

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

   if 0.0120000001 < uy

   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 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-fma.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-*.f32 96.9

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

   7. Applied rewrites 96.9%

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

   8. Taylor expanded in maxCos around 0

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

      1. lift-+.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 92.2

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

      4. lift-*.f32 N/A

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

      5. mul-1-neg N/A

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

      6. lift-neg.f32 92.2

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

   10. Applied rewrites 92.2%

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

Alternative 5: 97.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 97.6

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

7. Applied rewrites 97.6%

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

Alternative 6: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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

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

   1. lower-fma.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-*.f32 97.6

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

7. Applied rewrites 97.6%

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

8. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. count-2-rev N/A

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

   7. lift-+.f32 97.6

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

10. Applied rewrites 97.6%

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

Alternative 7: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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. lower-+.f32 N/A

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

   2. lower-*.f32 96.8

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

7. Applied rewrites 96.8%

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

Alternative 8: 94.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.026000000536441803:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux} \cdot \left(\sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.026000000536441803)
   (*
    (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (sqrt ux) (* (sin (* (+ uy uy) PI)) (sqrt 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.026000000536441803f) {
		tmp = (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sqrtf(ux) * (sinf(((uy + uy) * ((float) M_PI))) * sqrtf(2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.026000000536441803))
		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(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	else
		tmp = Float32(sqrt(ux) * Float32(sin(Float32(Float32(uy + uy) * Float32(pi))) * sqrt(Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0260000005

   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 98.5

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

   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. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      3. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(\frac{-4}{3}, {\mathsf{PI}\left(\right)}^{3}, {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.025396825396825397, \left(uy \cdot uy\right) \cdot {\pi}^{7}, 0.26666666666666666 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   8. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow3 N/A

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

      7. lift-*.f32 N/A

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

      8. lift-PI.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. count-2-rev N/A

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

      13. lower-+.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 98.0

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

   10. Applied rewrites 98.0%

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

   if 0.0260000005 < uy

   1. Initial program 57.9%

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

   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. *-commutative N/A

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

      14. lower-*.f32 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 75.6

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

   4. Applied rewrites 75.6%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

   7. Applied rewrites 72.6%

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

Alternative 9: 89.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00039999998989515007:\\ \;\;\;\;\left(uy \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux} \cdot \left(\sin \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00039999998989515007)
   (*
    (* uy (+ PI PI))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (sqrt ux) (* (sin (* (+ uy uy) PI)) (sqrt 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00039999998989515007f) {
		tmp = (uy * (((float) M_PI) + ((float) M_PI))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sqrtf(ux) * (sinf(((uy + uy) * ((float) M_PI))) * sqrtf(2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00039999998989515007))
		tmp = Float32(Float32(uy * Float32(Float32(pi) + 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)));
	else
		tmp = Float32(sqrt(ux) * Float32(sin(Float32(Float32(uy + uy) * Float32(pi))) * sqrt(Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.9999999e-4

   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 98.5

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

   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. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      3. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(\frac{-4}{3}, {\mathsf{PI}\left(\right)}^{3}, {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.025396825396825397, \left(uy \cdot uy\right) \cdot {\pi}^{7}, 0.26666666666666666 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   8. Taylor expanded in uy around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-PI.f32 98.0

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

   10. Applied rewrites 98.0%

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

   if 3.9999999e-4 < uy

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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. *-commutative N/A

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

      14. lower-*.f32 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

   7. Applied rewrites 73.3%

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

Alternative 10: 89.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00039999998989515007:\\ \;\;\;\;\left(uy \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux + ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00039999998989515007)
   (*
    (* uy (+ PI PI))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (* (sin (* PI (+ uy uy))) (sqrt (+ ux ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00039999998989515007f) {
		tmp = (uy * (((float) M_PI) + ((float) M_PI))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sinf((((float) M_PI) * (uy + uy))) * sqrtf((ux + ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00039999998989515007))
		tmp = Float32(Float32(uy * Float32(Float32(pi) + 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)));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(uy + uy))) * sqrt(Float32(ux + ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.9999999e-4

   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 98.5

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

   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. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      3. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(\frac{-4}{3}, {\mathsf{PI}\left(\right)}^{3}, {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.025396825396825397, \left(uy \cdot uy\right) \cdot {\pi}^{7}, 0.26666666666666666 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   8. Taylor expanded in uy around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-PI.f32 98.0

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

   10. Applied rewrites 98.0%

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

   if 3.9999999e-4 < uy

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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. *-commutative N/A

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

      14. lower-*.f32 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in maxCos around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 73.2

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

   7. Applied rewrites 73.2%

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

Alternative 11: 85.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00039999998989515007:\\ \;\;\;\;\left(uy \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00039999998989515007)
   (*
    (* uy (+ PI PI))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (*
    (* uy (fma (* -1.3333333333333333 (* uy uy)) (* (* PI PI) PI) (+ PI PI)))
    (sqrt (* (fma -2.0 maxCos 2.0) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00039999998989515007f) {
		tmp = (uy * (((float) M_PI) + ((float) M_PI))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = (uy * fmaf((-1.3333333333333333f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00039999998989515007))
		tmp = Float32(Float32(uy * Float32(Float32(pi) + 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)));
	else
		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(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.9999999e-4

   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 98.5

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

   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. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      3. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(\frac{-4}{3}, {\mathsf{PI}\left(\right)}^{3}, {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.025396825396825397, \left(uy \cdot uy\right) \cdot {\pi}^{7}, 0.26666666666666666 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   8. Taylor expanded in uy around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-PI.f32 98.0

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

   10. Applied rewrites 98.0%

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

   if 3.9999999e-4 < uy

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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. *-commutative N/A

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

      14. lower-*.f32 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. associate-*r* N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. pow2 N/A

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

      6. lift-*.f32 N/A

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

      7. pow3 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lift-PI.f32 60.1

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

   7. Applied rewrites 60.1%

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \pi, \pi + \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 84.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.00039999998989515007:\\ \;\;\;\;\left(uy \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux} \cdot \left(\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.00039999998989515007)
   (*
    (* uy (+ PI PI))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))
   (*
    (sqrt ux)
    (*
     (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
     (sqrt 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00039999998989515007f) {
		tmp = (uy * (((float) M_PI) + ((float) M_PI))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	} else {
		tmp = sqrtf(ux) * ((uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf(2.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00039999998989515007))
		tmp = Float32(Float32(uy * Float32(Float32(pi) + 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)));
	else
		tmp = Float32(sqrt(ux) * Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(2.0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 3.9999999e-4

   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 98.5

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

   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. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
   6. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      2. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      3. lift-PI.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      5. unpow2 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      6. lower-*.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

      7. lower-fma.f32 N/A

         \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(\frac{-4}{3}, {\mathsf{PI}\left(\right)}^{3}, {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   7. Applied rewrites 98.5%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.025396825396825397, \left(uy \cdot uy\right) \cdot {\pi}^{7}, 0.26666666666666666 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   8. Taylor expanded in uy around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 N/A

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

      3. lift-PI.f32 N/A

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

      4. lift-PI.f32 98.0

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

   10. Applied rewrites 98.0%

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

   if 3.9999999e-4 < uy

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   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. *-commutative N/A

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

      14. lower-*.f32 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 76.1

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

   4. Applied rewrites 76.1%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

   7. Applied rewrites 73.3%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow3 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 58.4

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

   10. Applied rewrites 58.4%

       \[\leadsto \sqrt{ux} \cdot \left(\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 82.1% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* uy (+ PI 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 * (((float) M_PI) + ((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(uy * Float32(Float32(pi) + 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 57.6%

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

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

   \[\leadsto \color{blue}{\left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
6. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right) + {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)}\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   2. lower-fma.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \color{blue}{\mathsf{PI}\left(\right)}, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   3. lift-PI.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, {uy}^{2} \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   5. unpow2 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   6. lower-*.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \left(\frac{-4}{3} \cdot {\mathsf{PI}\left(\right)}^{3} + {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

   7. lower-fma.f32 N/A

      \[\leadsto \left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(\frac{-4}{3}, {\mathsf{PI}\left(\right)}^{3}, {uy}^{2} \cdot \left(\frac{-8}{315} \cdot \left({uy}^{2} \cdot {\mathsf{PI}\left(\right)}^{7}\right) + \frac{4}{15} \cdot {\mathsf{PI}\left(\right)}^{5}\right)\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

7. Applied rewrites 94.1%

   \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(2, \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \pi\right) \cdot \pi, \left(uy \cdot uy\right) \cdot \mathsf{fma}\left(-0.025396825396825397, \left(uy \cdot uy\right) \cdot {\pi}^{7}, 0.26666666666666666 \cdot {\pi}^{5}\right)\right)\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]

8. Taylor expanded in uy around 0

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

   1. count-2-rev N/A

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

   2. lower-+.f32 N/A

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

   3. lift-PI.f32 N/A

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

   4. lift-PI.f32 82.1

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

10. Applied rewrites 82.1%

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

Alternative 14: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0.999779999256134:\\ \;\;\;\;\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \pi\right) \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot \sqrt{ux}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (* t_0 t_0)))
   (if (<= t_1 0.999779999256134)
     (* (* PI (+ uy uy)) (sqrt (- 1.0 t_1)))
     (* (* (+ uy uy) PI) (* (sqrt (fma -2.0 maxCos 2.0)) (sqrt ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float t_1 = t_0 * t_0;
	float tmp;
	if (t_1 <= 0.999779999256134f) {
		tmp = (((float) M_PI) * (uy + uy)) * sqrtf((1.0f - t_1));
	} else {
		tmp = ((uy + uy) * ((float) M_PI)) * (sqrtf(fmaf(-2.0f, maxCos, 2.0f)) * sqrtf(ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	t_1 = Float32(t_0 * t_0)
	tmp = Float32(0.0)
	if (t_1 <= Float32(0.999779999256134))
		tmp = Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(Float32(1.0) - t_1)));
	else
		tmp = Float32(Float32(Float32(uy + uy) * Float32(pi)) * Float32(sqrt(fma(Float32(-2.0), maxCos, Float32(2.0))) * sqrt(ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.5%

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

   2. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. *-commutative N/A

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

      7. count-2-rev N/A

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

      8. lower-+.f32 75.3

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

   4. Applied rewrites 75.3%

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

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

   1. Initial program 35.9%

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

   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. *-commutative N/A

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

      14. lower-*.f32 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 78.7

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

   7. Applied rewrites 78.7%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-fma.f32 N/A

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

      8. lift-sqrt.f32 78.7

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

   9. Applied rewrites 78.7%

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

Alternative 15: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1\right) - ux\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_1 \cdot t\_1 \leq 0.999779999256134:\\ \;\;\;\;\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \pi\right) \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot \sqrt{ux}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (fma maxCos ux 1.0) ux)) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_1 t_1) 0.999779999256134)
     (* (* PI (+ uy uy)) (sqrt (- 1.0 (* t_0 t_0))))
     (* (* (+ uy uy) PI) (* (sqrt (fma -2.0 maxCos 2.0)) (sqrt ux))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, 1.0f) - ux;
	float t_1 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_1 * t_1) <= 0.999779999256134f) {
		tmp = (((float) M_PI) * (uy + uy)) * sqrtf((1.0f - (t_0 * t_0)));
	} else {
		tmp = ((uy + uy) * ((float) M_PI)) * (sqrtf(fmaf(-2.0f, maxCos, 2.0f)) * sqrtf(ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(fma(maxCos, ux, Float32(1.0)) - ux)
	t_1 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_1 * t_1) <= Float32(0.999779999256134))
		tmp = Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))));
	else
		tmp = Float32(Float32(Float32(uy + uy) * Float32(pi)) * Float32(sqrt(fma(Float32(-2.0), maxCos, Float32(2.0))) * sqrt(ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.5%

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

   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 75.2%

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

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

   1. Initial program 35.9%

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

   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. *-commutative N/A

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

      14. lower-*.f32 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 92.8

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

   4. Applied rewrites 92.8%

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

   5. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 78.7

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

   7. Applied rewrites 78.7%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-fma.f32 N/A

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

      8. lift-sqrt.f32 78.7

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

   9. Applied rewrites 78.7%

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

Alternative 16: 75.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \pi\right) \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot \sqrt{ux}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (* (* (+ uy uy) PI) (* (sqrt (fma -2.0 maxCos 2.0)) (sqrt ux)))
   (* (* 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 <= 0.00011000000085914508f) {
		tmp = ((uy + uy) * ((float) M_PI)) * (sqrtf(fmaf(-2.0f, maxCos, 2.0f)) * sqrtf(ux));
	} 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(0.00011000000085914508))
		tmp = Float32(Float32(Float32(uy + uy) * Float32(pi)) * Float32(sqrt(fma(Float32(-2.0), maxCos, Float32(2.0))) * sqrt(ux)));
	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

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 35.9%

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

   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. *-commutative N/A

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

      14. lower-*.f32 N/A

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

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

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

      16. metadata-eval N/A

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

      17. +-commutative N/A

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

      18. lower-fma.f32 92.9

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

   4. Applied rewrites 92.9%

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

   5. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 78.7

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

   7. Applied rewrites 78.7%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-fma.f32 N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift-fma.f32 N/A

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

      8. lift-sqrt.f32 78.7

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

   9. Applied rewrites 78.7%

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

   if 1.10000001e-4 < ux

   1. Initial program 88.5%

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

   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 75.2%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 72.3%

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

   8. Taylor expanded in ux around 0

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

   10. Applied rewrites 71.9%

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

Alternative 17: 66.4% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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. *-commutative N/A

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

   14. lower-*.f32 N/A

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f32 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 66.4

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

7. Applied rewrites 66.4%

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

   1. lift-sqrt.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-fma.f32 N/A

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

   4. sqrt-prod N/A

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

   5. lower-*.f32 N/A

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

   6. lower-sqrt.f32 N/A

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

   7. lift-fma.f32 N/A

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

   8. lift-sqrt.f32 66.4

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

9. Applied rewrites 66.4%

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

Alternative 18: 66.4% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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. *-commutative N/A

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

   14. lower-*.f32 N/A

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f32 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 66.4

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

7. Applied rewrites 66.4%

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

Alternative 19: 63.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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. *-commutative N/A

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

   14. lower-*.f32 N/A

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f32 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-*.f32 N/A

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

7. Applied rewrites 73.3%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 63.9

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

10. Applied rewrites 63.9%

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

Alternative 20: 63.9% accurate, 3.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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. *-commutative N/A

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

   14. lower-*.f32 N/A

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f32 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-sqrt.f32 N/A

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

   3. lower-*.f32 N/A

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

7. Applied rewrites 73.3%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-sqrt.f32 63.9

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

10. Applied rewrites 63.9%

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

Alternative 21: 63.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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. *-commutative N/A

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

   14. lower-*.f32 N/A

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

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

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

   16. metadata-eval N/A

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

   17. +-commutative N/A

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

   18. lower-fma.f32 76.4

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

4. Applied rewrites 76.4%

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

5. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 66.4

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

7. Applied rewrites 66.4%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 63.9%

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

Alternative 22: 7.1% accurate, 4.7× 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 57.6%

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

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 50.7%

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

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 2025122 
(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)))))))