UniformSampleCone, y

Percentage Accurate: 57.7% → 98.3%

Time: 6.4s

Alternatives: 23

Speedup: 4.6×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 98.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower--.f32 98.2

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

7. Applied rewrites 98.2%

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

Alternative 2: 98.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower--.f32 98.2

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in ux around inf

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

   1. pow2 N/A

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

   2. lower--.f32 N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. div-add N/A

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

   7. lower-/.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 98.2

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

10. Applied rewrites 98.2%

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

Alternative 3: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

Alternative 4: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 97.6

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

7. Applied rewrites 97.6%

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

Alternative 5: 97.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. pow2 N/A

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

   6. lift-*.f32 N/A

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

   7. lower-+.f32 N/A

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

7. Applied rewrites 54.4%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

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

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

   3. metadata-eval N/A

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

   4. *-lft-identity N/A

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

   5. lower-fma.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. count-2-rev N/A

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

   9. lower-+.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-lft-identity N/A

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

   12. lift-*.f32 N/A

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

   13. lift-*.f32 97.6

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

   14. lift-*.f32 N/A

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

   15. *-lft-identity 97.6

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

10. Applied rewrites 97.6%

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

Alternative 6: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\frac{2 - ux}{ux} \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999949504854e-6)
   (* (sin (* (* uy 2.0) PI)) (sqrt (* (/ (- 2.0 ux) ux) (* ux ux))))
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (sqrt
     (*
      (- (/ (fma -2.0 maxCos 2.0) ux) (* (- 1.0 maxCos) (- 1.0 maxCos)))
      (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999949504854e-6f) {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((((2.0f - ux) / ux) * (ux * ux)));
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf((((fmaf(-2.0f, maxCos, 2.0f) / ux) - ((1.0f - maxCos) * (1.0f - maxCos))) * (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999949504854e-6))
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(Float32(2.0) - ux) / ux) * Float32(ux * ux))));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux) - Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos))) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999999e-6

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in ux around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower--.f32 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in maxCos around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f32 92.3

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

   10. Applied rewrites 92.3%

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

   if 1.99999999e-6 < maxCos

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in ux around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower--.f32 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lower--.f32 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add N/A

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

      7. lower-/.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lift--.f32 N/A

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

      10. lift--.f32 N/A

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

      11. lift-*.f32 98.2

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

   10. Applied rewrites 98.2%

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

   11. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f32 N/A

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

       6. unpow3 N/A

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

       7. lower-*.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lift-PI.f32 N/A

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

       10. lift-PI.f32 N/A

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

       11. lift-PI.f32 N/A

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

       12. lower-*.f32 N/A

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

       13. lift-PI.f32 89.2

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

   13. Applied rewrites 89.2%

       \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 96.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} - 1\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999949504854e-6)
   (* (sqrt (* (* ux ux) (- (/ 2.0 ux) 1.0))) (sin (* PI (+ uy uy))))
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (sqrt
     (*
      (- (/ (fma -2.0 maxCos 2.0) ux) (* (- 1.0 maxCos) (- 1.0 maxCos)))
      (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999949504854e-6f) {
		tmp = sqrtf(((ux * ux) * ((2.0f / ux) - 1.0f))) * sinf((((float) M_PI) * (uy + uy)));
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf((((fmaf(-2.0f, maxCos, 2.0f) / ux) - ((1.0f - maxCos) * (1.0f - maxCos))) * (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999949504854e-6))
		tmp = Float32(sqrt(Float32(Float32(ux * ux) * Float32(Float32(Float32(2.0) / ux) - Float32(1.0)))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux) - Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos))) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999999e-6

   1. Initial program 57.7%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lift-*.f32 20.7

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

   7. Applied rewrites 20.7%

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

   8. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. associate-*r/ N/A

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

      6. metadata-eval N/A

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

      7. lower-/.f32 92.3

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

   10. Applied rewrites 92.3%

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

   if 1.99999999e-6 < maxCos

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in ux around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower--.f32 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lower--.f32 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add N/A

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

      7. lower-/.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lift--.f32 N/A

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

      10. lift--.f32 N/A

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

      11. lift-*.f32 98.2

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

   10. Applied rewrites 98.2%

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

   11. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f32 N/A

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

       6. unpow3 N/A

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

       7. lower-*.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lift-PI.f32 N/A

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

       10. lift-PI.f32 N/A

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

       11. lift-PI.f32 N/A

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

       12. lower-*.f32 N/A

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

       13. lift-PI.f32 89.2

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

   13. Applied rewrites 89.2%

       \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999949504854e-6)
   (* (sqrt (* ux (- 2.0 ux))) (sin (* PI (+ uy uy))))
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (sqrt
     (*
      (- (/ (fma -2.0 maxCos 2.0) ux) (* (- 1.0 maxCos) (- 1.0 maxCos)))
      (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999949504854e-6f) {
		tmp = sqrtf((ux * (2.0f - ux))) * sinf((((float) M_PI) * (uy + uy)));
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf((((fmaf(-2.0f, maxCos, 2.0f) / ux) - ((1.0f - maxCos) * (1.0f - maxCos))) * (ux * ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999949504854e-6))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) / ux) - Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos))) * Float32(ux * ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999999e-6

   1. Initial program 57.7%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lift-*.f32 20.7

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

   7. Applied rewrites 20.7%

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

   8. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 92.4

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

      6. lift-*.f32 N/A

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

      7. *-lft-identity 92.4

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

   10. Applied rewrites 92.4%

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

   if 1.99999999e-6 < maxCos

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in ux around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower--.f32 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lower--.f32 N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add N/A

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

      7. lower-/.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lift--.f32 N/A

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

      10. lift--.f32 N/A

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

      11. lift-*.f32 98.2

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

   10. Applied rewrites 98.2%

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

   11. Taylor expanded in uy around 0

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

       1. lower-*.f32 N/A

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

       2. lower-fma.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. unpow2 N/A

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

       5. lower-*.f32 N/A

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

       6. unpow3 N/A

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

       7. lower-*.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lift-PI.f32 N/A

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

       10. lift-PI.f32 N/A

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

       11. lift-PI.f32 N/A

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

       12. lower-*.f32 N/A

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

       13. lift-PI.f32 89.2

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

   13. Applied rewrites 89.2%

       \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(\frac{\mathsf{fma}\left(-2, maxCos, 2\right)}{ux} - \left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 96.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999949504854e-6)
   (* (sqrt (* ux (- 2.0 ux))) (sin (* PI (+ uy uy))))
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999949504854e-6f) {
		tmp = sqrtf((ux * (2.0f - ux))) * sinf((((float) M_PI) * (uy + uy)));
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf(((fmaf(-ux, ((maxCos - 1.0f) * (maxCos - 1.0f)), 2.0f) - (maxCos + maxCos)) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999949504854e-6))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999999e-6

   1. Initial program 57.7%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lift-*.f32 20.7

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

   7. Applied rewrites 20.7%

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

   8. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 92.4

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

      6. lift-*.f32 N/A

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

      7. *-lft-identity 92.4

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

   10. Applied rewrites 92.4%

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

   if 1.99999999e-6 < maxCos

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow3 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 89.3

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

   7. Applied rewrites 89.3%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in maxCos around 0

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

7. Applied rewrites 96.8%

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

Alternative 11: 96.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

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

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

   2. metadata-eval N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 96.8

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

7. Applied rewrites 96.8%

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

Alternative 12: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999949504854 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(2 + \mathsf{fma}\left(-1, ux, maxCos \cdot \left(2 \cdot ux - 2\right)\right)\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999949504854e-6)
   (* (sqrt (* ux (- 2.0 ux))) (sin (* PI (+ uy uy))))
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (sqrt (* (+ 2.0 (fma -1.0 ux (* maxCos (- (* 2.0 ux) 2.0)))) ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999949504854e-6f) {
		tmp = sqrtf((ux * (2.0f - ux))) * sinf((((float) M_PI) * (uy + uy)));
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf(((2.0f + fmaf(-1.0f, ux, (maxCos * ((2.0f * ux) - 2.0f)))) * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999949504854e-6))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) + fma(Float32(-1.0), ux, Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(2.0))))) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999999e-6

   1. Initial program 57.7%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lift-*.f32 20.7

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

   7. Applied rewrites 20.7%

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

   8. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

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

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

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

      3. lower-*.f32 N/A

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

      4. lift--.f32 N/A

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

      5. lift-*.f32 92.4

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

      6. lift-*.f32 N/A

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

      7. *-lft-identity 92.4

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

   10. Applied rewrites 92.4%

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

   if 1.99999999e-6 < maxCos

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow3 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 88.6

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

   10. Applied rewrites 88.6%

       \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(2 + \mathsf{fma}\left(-1, ux, maxCos \cdot \left(2 \cdot ux - 2\right)\right)\right) \cdot ux} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 93.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.02800000086426735:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(2 + \mathsf{fma}\left(-1, ux, maxCos \cdot \left(2 \cdot ux - 2\right)\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux + ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.02800000086426735)
   (*
    (* uy (fma -1.3333333333333333 (* (* uy uy) (* (* PI PI) PI)) (* 2.0 PI)))
    (sqrt (* (+ 2.0 (fma -1.0 ux (* maxCos (- (* 2.0 ux) 2.0)))) ux)))
   (* (sqrt (+ ux ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.02800000086426735f) {
		tmp = (uy * fmaf(-1.3333333333333333f, ((uy * uy) * ((((float) M_PI) * ((float) M_PI)) * ((float) M_PI))), (2.0f * ((float) M_PI)))) * sqrtf(((2.0f + fmaf(-1.0f, ux, (maxCos * ((2.0f * ux) - 2.0f)))) * ux));
	} else {
		tmp = sqrtf((ux + ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.02800000086426735))
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(uy * uy) * Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi))), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) + fma(Float32(-1.0), ux, Float32(maxCos * Float32(Float32(Float32(2.0) * ux) - Float32(2.0))))) * ux)));
	else
		tmp = Float32(sqrt(Float32(ux + ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.0280000009

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.3

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

   4. Applied rewrites 98.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-+.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-*.f32 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow3 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lift-PI.f32 88.6

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

   10. Applied rewrites 88.6%

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

   if 0.0280000009 < uy

   1. Initial program 57.7%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lift-*.f32 20.7

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

   7. Applied rewrites 20.7%

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

   8. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 72.9

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

   10. Applied rewrites 72.9%

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

Alternative 14: 89.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0020000000949949026:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{\frac{2 + \mathsf{fma}\left(-2, maxCos, -1 \cdot \left(ux \cdot \left(\left(1 - maxCos\right) \cdot \left(1 - maxCos\right)\right)\right)\right)}{ux} \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux + ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0020000000949949026)
   (*
    (* (+ uy uy) PI)
    (sqrt
     (*
      (/
       (+
        2.0
        (fma -2.0 maxCos (* -1.0 (* ux (* (- 1.0 maxCos) (- 1.0 maxCos))))))
       ux)
      (* ux ux))))
   (* (sqrt (+ ux ux)) (sin (* PI (+ uy uy))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0020000000949949026f) {
		tmp = ((uy + uy) * ((float) M_PI)) * sqrtf((((2.0f + fmaf(-2.0f, maxCos, (-1.0f * (ux * ((1.0f - maxCos) * (1.0f - maxCos)))))) / ux) * (ux * ux)));
	} else {
		tmp = sqrtf((ux + ux)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0020000000949949026))
		tmp = Float32(Float32(Float32(uy + uy) * Float32(pi)) * sqrt(Float32(Float32(Float32(Float32(2.0) + fma(Float32(-2.0), maxCos, Float32(Float32(-1.0) * Float32(ux * Float32(Float32(Float32(1.0) - maxCos) * Float32(Float32(1.0) - maxCos)))))) / ux) * Float32(ux * ux))));
	else
		tmp = Float32(sqrt(Float32(ux + ux)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00200000009

   1. Initial program 57.7%

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

   2. Taylor expanded in ux around -inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   4. Applied rewrites 98.2%

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

   5. Taylor expanded in ux around 0

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

      1. lower-/.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower--.f32 98.2

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in uy around 0

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

      1. associate-*r* N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lift-PI.f32 81.5

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

   10. Applied rewrites 81.5%

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

   if 0.00200000009 < uy

   1. Initial program 57.7%

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

   2. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower--.f32 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-*r* N/A

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

      10. *-commutative N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. *-commutative N/A

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

      16. count-2-rev N/A

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

      17. lower-+.f32 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in ux around inf

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

      1. pow2 N/A

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

      2. lift-*.f32 20.7

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

   7. Applied rewrites 20.7%

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

   8. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. lower-+.f32 72.9

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

   10. Applied rewrites 72.9%

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

Alternative 15: 81.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower--.f32 98.2

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in uy around 0

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

   1. associate-*r* N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lift-PI.f32 81.5

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

10. Applied rewrites 81.5%

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

Alternative 16: 81.5% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around -inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

4. Applied rewrites 98.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-/.f32 N/A

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

   2. lower-+.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower--.f32 98.2

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

7. Applied rewrites 98.2%

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

8. Taylor expanded in ux around inf

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

   1. pow2 N/A

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

   2. lower--.f32 N/A

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

   3. associate-*r/ N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. div-add N/A

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

   7. lower-/.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 98.2

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

10. Applied rewrites 98.2%

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

11. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lift-PI.f32 81.5

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

13. Applied rewrites 81.5%

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

Alternative 17: 81.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 81.6

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

7. Applied rewrites 81.6%

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

Alternative 18: 81.1% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.7%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.3

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

4. Applied rewrites 98.3%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 97.6

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

7. Applied rewrites 97.6%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lift-PI.f32 81.1

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

10. Applied rewrites 81.1%

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

Alternative 19: 77.2% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sqrt (* (* ux ux) (- (* 2.0 (/ 1.0 ux)) 1.0))) (* (+ uy uy) PI)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf(((ux * ux) * ((2.0f * (1.0f / ux)) - 1.0f))) * ((uy + uy) * ((float) M_PI));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(ux * ux) * Float32(Float32(Float32(2.0) * Float32(Float32(1.0) / ux)) - Float32(1.0)))) * Float32(Float32(uy + uy) * Float32(pi)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt(((ux * ux) * ((single(2.0) * (single(1.0) / ux)) - single(1.0)))) * ((uy + uy) * single(pi));
end

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \left(\color{blue}{2} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   11. lower-sin.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

   14. lift-PI.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \]

   16. count-2-rev N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   17. lower-+.f32 55.7

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

4. Applied rewrites 55.7%

   \[\leadsto \color{blue}{\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

5. Taylor expanded in ux around inf

   \[\leadsto \sqrt{1 - {ux}^{2}} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
6. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   2. lift-*.f32 20.7

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

7. Applied rewrites 20.7%

   \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

8. Taylor expanded in uy around 0

   \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   2. count-2-rev N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   3. lift-+.f32 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   5. lift-PI.f32 20.0

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

10. Applied rewrites 20.0%

    \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \color{blue}{\pi}\right) \]

11. Taylor expanded in ux around inf

    \[\leadsto \sqrt{{ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - 1\right)} \cdot \left(\left(\color{blue}{uy} + uy\right) \cdot \pi\right) \]
12. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \sqrt{{ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - 1\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    2. pow2 N/A

       \[\leadsto \sqrt{\left(ux \cdot ux\right) \cdot \left(2 \cdot \frac{1}{ux} - 1\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    3. lift-*.f32 N/A

       \[\leadsto \sqrt{\left(ux \cdot ux\right) \cdot \left(2 \cdot \frac{1}{ux} - 1\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    4. lower--.f32 N/A

       \[\leadsto \sqrt{\left(ux \cdot ux\right) \cdot \left(2 \cdot \frac{1}{ux} - 1\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    5. lower-*.f32 N/A

       \[\leadsto \sqrt{\left(ux \cdot ux\right) \cdot \left(2 \cdot \frac{1}{ux} - 1\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    6. lower-/.f32 77.1

       \[\leadsto \sqrt{\left(ux \cdot ux\right) \cdot \left(2 \cdot \frac{1}{ux} - 1\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

13. Applied rewrites 77.1%

    \[\leadsto \sqrt{\left(ux \cdot ux\right) \cdot \left(2 \cdot \frac{1}{ux} - 1\right)} \cdot \left(\left(\color{blue}{uy} + uy\right) \cdot \pi\right) \]
14. Add Preprocessing

Alternative 20: 77.1% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \left(2 - 1 \cdot ux\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sqrt (* ux (- 2.0 (* 1.0 ux)))) (* (+ uy uy) PI)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * (2.0f - (1.0f * ux)))) * ((uy + uy) * ((float) M_PI));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(1.0) * ux)))) * Float32(Float32(uy + uy) * Float32(pi)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((ux * (single(2.0) - (single(1.0) * ux)))) * ((uy + uy) * single(pi));
end

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \left(\color{blue}{2} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   11. lower-sin.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

   14. lift-PI.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \]

   16. count-2-rev N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   17. lower-+.f32 55.7

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

4. Applied rewrites 55.7%

   \[\leadsto \color{blue}{\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

5. Taylor expanded in ux around inf

   \[\leadsto \sqrt{1 - {ux}^{2}} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
6. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   2. lift-*.f32 20.7

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

7. Applied rewrites 20.7%

   \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

8. Taylor expanded in uy around 0

   \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   2. count-2-rev N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   3. lift-+.f32 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   5. lift-PI.f32 20.0

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

10. Applied rewrites 20.0%

    \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \color{blue}{\pi}\right) \]

11. Taylor expanded in ux around 0

    \[\leadsto \sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \left(\left(\color{blue}{uy} + uy\right) \cdot \pi\right) \]
12. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(2 + -1 \cdot ux\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    2. fp-cancel-sign-sub-inv N/A

       \[\leadsto \sqrt{ux \cdot \left(2 - \left(\mathsf{neg}\left(-1\right)\right) \cdot ux\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    3. metadata-eval N/A

       \[\leadsto \sqrt{ux \cdot \left(2 - 1 \cdot ux\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    4. lower--.f32 N/A

       \[\leadsto \sqrt{ux \cdot \left(2 - 1 \cdot ux\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

    5. lower-*.f32 77.2

       \[\leadsto \sqrt{ux \cdot \left(2 - 1 \cdot ux\right)} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

13. Applied rewrites 77.2%

    \[\leadsto \sqrt{ux \cdot \left(2 - 1 \cdot ux\right)} \cdot \left(\left(\color{blue}{uy} + uy\right) \cdot \pi\right) \]
14. Add Preprocessing

Alternative 21: 65.9% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - \left(maxCos + maxCos\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* PI (+ uy uy)) (sqrt (* ux (- 2.0 (+ maxCos maxCos))))))

C

float code(float ux, float uy, float maxCos) {
	return (((float) M_PI) * (uy + uy)) * sqrtf((ux * (2.0f - (maxCos + maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(pi) * Float32(uy + uy)) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(maxCos + maxCos)))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = (single(pi) * (uy + uy)) * sqrt((ux * (single(2.0) - (maxCos + maxCos))));
end

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   8. *-commutative N/A

      \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   9. count-2-rev N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   10. lower-+.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   11. lower-sqrt.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   12. lower--.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   13. unpow2 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \]
6. Step-by-step derivation

7. Applied rewrites 7.1%

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \]

8. Taylor expanded in ux around 0

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]
9. Step-by-step derivation

   1. fp-cancel-sub-sign-inv N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)} \]

   2. metadata-eval N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)} \]

   4. metadata-eval N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 + \left(\mathsf{neg}\left(2\right)\right) \cdot maxCos\right)} \]

   5. fp-cancel-sub-sign-inv N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

   6. lower--.f32 N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

   7. count-2-rev N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - \left(maxCos + maxCos\right)\right)} \]

   8. lower-+.f32 65.9

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - \left(maxCos + maxCos\right)\right)} \]

10. Applied rewrites 65.9%

    \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{ux \cdot \left(2 - \left(maxCos + maxCos\right)\right)} \]
11. Add Preprocessing

Alternative 22: 63.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (sqrt (* 2.0 ux)) (* (+ uy uy) PI)))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((2.0f * ux)) * ((uy + uy) * ((float) M_PI));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(2.0) * ux)) * Float32(Float32(uy + uy) * Float32(pi)))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = sqrt((single(2.0) * ux)) * ((uy + uy) * single(pi));
end

Derivation

1. Initial program 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in maxCos around 0

   \[\leadsto \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{1 - {\left(1 - ux\right)}^{2}}} \]
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   3. lower-sqrt.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \sqrt{1 - {\left(1 - ux\right)}^{2}} \cdot \sin \left(\color{blue}{2} \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   6. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   7. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   8. lift--.f32 N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \]

   9. associate-*r* N/A

      \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   11. lower-sin.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \]

   12. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \]

   14. lift-PI.f32 N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(2 \cdot uy\right)\right) \]

   16. count-2-rev N/A

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   17. lower-+.f32 55.7

       \[\leadsto \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

4. Applied rewrites 55.7%

   \[\leadsto \color{blue}{\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)} \]

5. Taylor expanded in ux around inf

   \[\leadsto \sqrt{1 - {ux}^{2}} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
6. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

   2. lift-*.f32 20.7

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

7. Applied rewrites 20.7%

   \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]

8. Taylor expanded in uy around 0

   \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   2. count-2-rev N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   3. lift-+.f32 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   4. lower-*.f32 N/A

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \]

   5. lift-PI.f32 20.0

      \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

10. Applied rewrites 20.0%

    \[\leadsto \sqrt{1 - ux \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \color{blue}{\pi}\right) \]

11. Taylor expanded in ux around 0

    \[\leadsto \sqrt{2 \cdot ux} \cdot \left(\left(\color{blue}{uy} + uy\right) \cdot \pi\right) \]
12. Step-by-step derivation

    1. lift-*.f32 63.2

       \[\leadsto \sqrt{2 \cdot ux} \cdot \left(\left(uy + uy\right) \cdot \pi\right) \]

13. Applied rewrites 63.2%

    \[\leadsto \sqrt{2 \cdot ux} \cdot \left(\left(\color{blue}{uy} + uy\right) \cdot \pi\right) \]
14. Add Preprocessing

Alternative 23: 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 57.7%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

2. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{2 \cdot \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}\right)} \]
3. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   2. associate-*r* N/A

      \[\leadsto \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   3. *-commutative N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \color{blue}{\sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   5. *-commutative N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   6. lower-*.f32 N/A

      \[\leadsto \left(\mathsf{PI}\left(\right) \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   7. lift-PI.f32 N/A

      \[\leadsto \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{\color{blue}{1} - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   8. *-commutative N/A

      \[\leadsto \left(\pi \cdot \left(2 \cdot uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   9. count-2-rev N/A

      \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   10. lower-+.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \color{blue}{{\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}}} \]

   11. lower-sqrt.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   12. lower--.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - {\left(\left(1 + maxCos \cdot ux\right) - ux\right)}^{2}} \]

   13. unpow2 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

   14. lower-*.f32 N/A

       \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\left(1 + maxCos \cdot ux\right) - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)} \]

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{\left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux\right)}} \]

5. Taylor expanded in ux around 0

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \]
6. Step-by-step derivation

7. Applied rewrites 7.1%

   \[\leadsto \left(\pi \cdot \left(uy + uy\right)\right) \cdot \sqrt{1 - 1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025123 
(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)))))))