UniformSampleCone, y

Percentage Accurate: 57.5% → 98.3%

Time: 5.9s

Alternatives: 13

Speedup: 2.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((((ux * ux) * ((1.0f - ((fmaf((1.0f - maxCos), (1.0f - maxCos), ((maxCos + maxCos) / ux)) * 0.5f) * 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(ux * ux) * Float32(Float32(Float32(1.0) - Float32(Float32(fma(Float32(Float32(1.0) - maxCos), Float32(Float32(1.0) - maxCos), Float32(Float32(maxCos + maxCos) / ux)) * Float32(0.5)) * ux)) * Float32(2.0))) / ux)))
end

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 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. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.2

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

4. Applied rewrites 98.2%

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

   1. lift--.f32 N/A

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

   2. sub-to-mult N/A

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

   3. lift-*.f32 N/A

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

   4. lift-/.f32 N/A

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

   5. mult-flip-rev N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f32 N/A

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

6. Applied rewrites 98.2%

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

   1. lift-*.f32 N/A

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

   2. lift-/.f32 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f32 N/A

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

8. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

3. Applied rewrites 56.1%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 98.3

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

6. Applied rewrites 98.3%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in 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. lower-*.f32 N/A

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

   2. lower-pow.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-/.f32 N/A

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

   6. lower-fma.f32 N/A

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

   7. lower-/.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower--.f32 98.2

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

4. Applied rewrites 98.2%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.2

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

6. Applied rewrites 98.2%

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

Alternative 4: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - 0.5 \cdot ux\right) \cdot 2\right)}{ux}}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{{ux}^{2} \cdot \left(\frac{1}{ux} + \left(\frac{1}{ux} - \mathsf{fma}\left(1 - maxCos, 1 - maxCos, \frac{maxCos + maxCos}{ux}\right)\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999494757503e-5)
   (*
    (sin (* (* uy 2.0) PI))
    (sqrt (/ (* (* ux ux) (* (- 1.0 (* 0.5 ux)) 2.0)) ux)))
   (*
    (* 2.0 (* uy PI))
    (sqrt
     (*
      (pow ux 2.0)
      (+
       (/ 1.0 ux)
       (-
        (/ 1.0 ux)
        (fma (- 1.0 maxCos) (- 1.0 maxCos) (/ (+ maxCos maxCos) ux)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999494757503e-5f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((((ux * ux) * ((1.0f - (0.5f * ux)) * 2.0f)) / ux));
	} else {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((powf(ux, 2.0f) * ((1.0f / ux) + ((1.0f / ux) - fmaf((1.0f - maxCos), (1.0f - maxCos), ((maxCos + maxCos) / ux))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999494757503e-5))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - Float32(Float32(0.5) * ux)) * Float32(2.0))) / ux)));
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32((ux ^ Float32(2.0)) * Float32(Float32(Float32(1.0) / ux) + Float32(Float32(Float32(1.0) / ux) - fma(Float32(Float32(1.0) - maxCos), Float32(Float32(1.0) - maxCos), Float32(Float32(maxCos + maxCos) / ux)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-5

   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. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift--.f32 N/A

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

      2. sub-to-mult N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f32 N/A

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

   6. Applied rewrites 98.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f32 N/A

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in maxCos around 0

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

   11. Applied rewrites 92.3%

       \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - 0.5 \cdot ux\right) \cdot 2\right)}{ux}} \]

   if 1.99999995e-5 < maxCos

   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. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f32 N/A

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

      6. lower--.f32 98.2

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

      7. lift-fma.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f32 N/A

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

      10. unpow2 N/A

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

      11. lift--.f32 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f32 N/A

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

      14. sub-negate-rev N/A

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

      15. sqr-neg N/A

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

      16. lower-fma.f32 N/A

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

      17. lower--.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lift-/.f32 N/A

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

      20. associate-*r/ N/A

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

      21. lower-/.f32 N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f32 98.2

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

   6. Applied rewrites 98.2%

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

   7. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.4

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

   9. Applied rewrites 81.4%

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

Alternative 5: 95.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-5

   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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.5

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

   3. Applied rewrites 57.7%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower--.f32 55.5

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 92.3

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

   9. Applied rewrites 92.3%

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

   if 1.99999995e-5 < maxCos

   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. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. associate--l+ N/A

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

      5. lower-+.f32 N/A

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

      6. lower--.f32 98.2

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

      7. lift-fma.f32 N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f32 N/A

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

      10. unpow2 N/A

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

      11. lift--.f32 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f32 N/A

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

      14. sub-negate-rev N/A

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

      15. sqr-neg N/A

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

      16. lower-fma.f32 N/A

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

      17. lower--.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lift-/.f32 N/A

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

      20. associate-*r/ N/A

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

      21. lower-/.f32 N/A

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

      22. count-2-rev N/A

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

      23. lower-+.f32 98.2

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

   6. Applied rewrites 98.2%

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

   7. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.4

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

   9. Applied rewrites 81.4%

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

Alternative 6: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - \left(\mathsf{fma}\left(1 - maxCos, 1 - maxCos, \frac{maxCos + maxCos}{ux}\right) \cdot 0.5\right) \cdot ux\right) \cdot 2\right)}{ux}}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999494757503e-5)
   (* (sqrt (* ux (+ 2.0 (* -1.0 ux)))) (sin (* PI (+ uy uy))))
   (*
    (* 2.0 (* uy PI))
    (sqrt
     (/
      (*
       (* ux ux)
       (*
        (-
         1.0
         (*
          (* (fma (- 1.0 maxCos) (- 1.0 maxCos) (/ (+ maxCos maxCos) ux)) 0.5)
          ux))
        2.0))
      ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999494757503e-5f) {
		tmp = sqrtf((ux * (2.0f + (-1.0f * ux)))) * sinf((((float) M_PI) * (uy + uy)));
	} else {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((((ux * ux) * ((1.0f - ((fmaf((1.0f - maxCos), (1.0f - maxCos), ((maxCos + maxCos) / ux)) * 0.5f) * ux)) * 2.0f)) / ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999494757503e-5))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(-1.0) * ux)))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - Float32(Float32(fma(Float32(Float32(1.0) - maxCos), Float32(Float32(1.0) - maxCos), Float32(Float32(maxCos + maxCos) / ux)) * Float32(0.5)) * ux)) * Float32(2.0))) / ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-5

   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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.5

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

   3. Applied rewrites 57.7%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower--.f32 55.5

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 92.3

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

   9. Applied rewrites 92.3%

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

   if 1.99999995e-5 < maxCos

   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. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift--.f32 N/A

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

      2. sub-to-mult N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f32 N/A

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

   6. Applied rewrites 98.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f32 N/A

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-PI.f32 81.4

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

   11. Applied rewrites 81.4%

       \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \cdot \sqrt{\frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - \left(\mathsf{fma}\left(1 - maxCos, 1 - maxCos, \frac{maxCos + maxCos}{ux}\right) \cdot 0.5\right) \cdot ux\right) \cdot 2\right)}{ux}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 89.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.001449999981559813:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{\frac{\left(ux \cdot ux\right) \cdot \left(\left(1 - \left(\mathsf{fma}\left(1 - maxCos, 1 - maxCos, \frac{maxCos + maxCos}{ux}\right) \cdot 0.5\right) \cdot ux\right) \cdot 2\right)}{ux}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.001449999981559813)
   (*
    (* 2.0 (* uy PI))
    (sqrt
     (/
      (*
       (* ux ux)
       (*
        (-
         1.0
         (*
          (* (fma (- 1.0 maxCos) (- 1.0 maxCos) (/ (+ maxCos maxCos) ux)) 0.5)
          ux))
        2.0))
      ux)))
   (* (sqrt (* 2.0 ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.001449999981559813f) {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((((ux * ux) * ((1.0f - ((fmaf((1.0f - maxCos), (1.0f - maxCos), ((maxCos + maxCos) / ux)) * 0.5f) * ux)) * 2.0f)) / ux));
	} else {
		tmp = sqrtf((2.0f * ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.001449999981559813))
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(Float32(ux * ux) * Float32(Float32(Float32(1.0) - Float32(Float32(fma(Float32(Float32(1.0) - maxCos), Float32(Float32(1.0) - maxCos), Float32(Float32(maxCos + maxCos) / ux)) * Float32(0.5)) * ux)) * Float32(2.0))) / ux)));
	else
		tmp = Float32(sqrt(Float32(Float32(2.0) * ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00144999998

   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. lower-*.f32 N/A

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

      2. lower-pow.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-/.f32 N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower--.f32 98.2

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

   4. Applied rewrites 98.2%

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

      1. lift--.f32 N/A

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

      2. sub-to-mult N/A

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

      3. lift-*.f32 N/A

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

      4. lift-/.f32 N/A

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

      5. mult-flip-rev N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f32 N/A

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

   6. Applied rewrites 98.2%

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

      1. lift-*.f32 N/A

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

      2. lift-/.f32 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f32 N/A

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

   8. Applied rewrites 98.3%

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

   9. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-PI.f32 81.4

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

   11. Applied rewrites 81.4%

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

   if 0.00144999998 < 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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.5

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

   3. Applied rewrites 57.7%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower--.f32 55.5

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 73.1

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

   9. Applied rewrites 73.1%

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

Alternative 8: 82.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathbf{if}\;ux \leq 0.0003000000142492354:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \mathsf{fma}\left(maxCos, ux, 1 - ux\right), 1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (if (<= ux 0.0003000000142492354)
     (* (sqrt (* 2.0 ux)) (sin t_0))
     (*
      (sqrt (fma (- ux (fma maxCos ux 1.0)) (fma maxCos ux (- 1.0 ux)) 1.0))
      t_0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float tmp;
	if (ux <= 0.0003000000142492354f) {
		tmp = sqrtf((2.0f * ux)) * sinf(t_0);
	} else {
		tmp = sqrtf(fmaf((ux - fmaf(maxCos, ux, 1.0f)), fmaf(maxCos, ux, (1.0f - ux)), 1.0f)) * t_0;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	tmp = Float32(0.0)
	if (ux <= Float32(0.0003000000142492354))
		tmp = Float32(sqrt(Float32(Float32(2.0) * ux)) * sin(t_0));
	else
		tmp = Float32(sqrt(fma(Float32(ux - fma(maxCos, ux, Float32(1.0))), fma(maxCos, ux, Float32(Float32(1.0) - ux)), Float32(1.0))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.00000014e-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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.5

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

   3. Applied rewrites 57.7%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower--.f32 55.5

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

   6. Applied rewrites 55.5%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 73.1

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

   9. Applied rewrites 73.1%

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

   if 3.00000014e-4 < ux

   1. Initial program 57.5%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 50.6

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 50.7%

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

Alternative 9: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(uy + uy\right)\\ \mathbf{if}\;ux \leq 0.0001500000071246177:\\ \;\;\;\;\sqrt{-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 - 2 \cdot \frac{1}{maxCos}\right)\right)\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \mathsf{fma}\left(maxCos, ux, 1 - ux\right), 1\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* PI (+ uy uy))))
   (if (<= ux 0.0001500000071246177)
     (* (sqrt (* -1.0 (* maxCos (* ux (- 2.0 (* 2.0 (/ 1.0 maxCos))))))) t_0)
     (*
      (sqrt (fma (- ux (fma maxCos ux 1.0)) (fma maxCos ux (- 1.0 ux)) 1.0))
      t_0))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ((float) M_PI) * (uy + uy);
	float tmp;
	if (ux <= 0.0001500000071246177f) {
		tmp = sqrtf((-1.0f * (maxCos * (ux * (2.0f - (2.0f * (1.0f / maxCos))))))) * t_0;
	} else {
		tmp = sqrtf(fmaf((ux - fmaf(maxCos, ux, 1.0f)), fmaf(maxCos, ux, (1.0f - ux)), 1.0f)) * t_0;
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(pi) * Float32(uy + uy))
	tmp = Float32(0.0)
	if (ux <= Float32(0.0001500000071246177))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(maxCos * Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * Float32(Float32(1.0) / maxCos))))))) * t_0);
	else
		tmp = Float32(sqrt(fma(Float32(ux - fma(maxCos, ux, Float32(1.0))), fma(maxCos, ux, Float32(Float32(1.0) - ux)), Float32(1.0))) * t_0);
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.50000007e-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 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. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 50.6

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

   4. Applied rewrites 50.6%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. lift-+.f32 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-fma.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower-*.f32 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 45.8

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-/.f32 65.9

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

   11. Applied rewrites 65.9%

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

   if 1.50000007e-4 < ux

   1. Initial program 57.5%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 50.6

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

   4. Applied rewrites 50.6%

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

   6. Applied rewrites 50.7%

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

Alternative 10: 75.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0003150000120513141:\\ \;\;\;\;\sqrt{-1 \cdot \left(maxCos \cdot \left(ux \cdot \left(2 - 2 \cdot \frac{1}{maxCos}\right)\right)\right)} \cdot \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0003150000120513141)
   (*
    (sqrt (* -1.0 (* maxCos (* ux (- 2.0 (* 2.0 (/ 1.0 maxCos)))))))
    (* PI (+ uy uy)))
   (* (* 2.0 (* uy PI)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0003150000120513141f) {
		tmp = sqrtf((-1.0f * (maxCos * (ux * (2.0f - (2.0f * (1.0f / maxCos))))))) * (((float) M_PI) * (uy + uy));
	} else {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0003150000120513141))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(maxCos * Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * Float32(Float32(1.0) / maxCos))))))) * Float32(Float32(pi) * Float32(uy + uy)));
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0003150000120513141))
		tmp = sqrt((single(-1.0) * (maxCos * (ux * (single(2.0) - (single(2.0) * (single(1.0) / maxCos))))))) * (single(pi) * (uy + uy));
	else
		tmp = (single(2.0) * (uy * single(pi))) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.15000012e-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 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. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 50.6

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

   4. Applied rewrites 50.6%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. +-commutative N/A

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

      4. sum-to-mult N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. lift-+.f32 N/A

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

      9. +-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-fma.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lower-*.f32 N/A

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

   6. Applied rewrites 45.8%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 45.8

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

   8. Applied rewrites 45.8%

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

   9. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower--.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. lower-/.f32 65.9

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

   11. Applied rewrites 65.9%

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

   if 3.15000012e-4 < ux

   1. Initial program 57.5%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 49.1

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

   10. Applied rewrites 49.1%

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

Alternative 11: 49.9% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 50.6

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

4. Applied rewrites 50.6%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. +-commutative N/A

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

   4. sum-to-mult N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lift-+.f32 N/A

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

   9. +-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-fma.f32 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. lower-*.f32 N/A

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

6. Applied rewrites 45.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 45.8

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

8. Applied rewrites 45.8%

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

9. Taylor expanded in maxCos around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower--.f32 N/A

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

    4. lower--.f32 N/A

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

    5. lower-/.f32 49.9

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

11. Applied rewrites 49.9%

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

Alternative 12: 49.1% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in 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. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 50.6

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in maxCos around 0

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

   1. lower--.f32 49.2

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

7. Applied rewrites 49.2%

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

8. Taylor expanded in maxCos around 0

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

   1. lower--.f32 49.1

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

10. Applied rewrites 49.1%

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

Alternative 13: 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 57.5%

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

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. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 50.6

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. distribute-rgt-in N/A

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

   6. lift-+.f32 N/A

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

   7. lift-*.f32 7.1

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

9. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025156 
(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)))))))