UniformSampleCone, x

Percentage Accurate: 57.5% → 99.0%

Time: 6.3s

Alternatives: 14

Speedup: 11.2×

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

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Initial Program: 57.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (cos (* (* 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 cosf(((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(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((single(1.0) - (t_0 * t_0)));
end

Alternative 1: 99.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-cos.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. sin-+PI/2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 98.8

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

6. Applied rewrites 98.8%

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

8. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

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

   4. lower-sin.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 99.0

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

10. Applied rewrites 99.0%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-cos.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. sin-+PI/2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 98.8

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

6. Applied rewrites 98.8%

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

8. Applied rewrites 99.0%

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

Alternative 3: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-cos.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. sin-+PI/2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 98.8

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

6. Applied rewrites 98.8%

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in maxCos around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. +-commutative N/A

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

    5. distribute-rgt-out N/A

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

    6. lower-fma.f32 N/A

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

    7. lower-+.f32 98.9

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

11. Applied rewrites 98.9%

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

Alternative 4: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-cos.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. sin-+PI/2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. lower-fma.f32 N/A

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

   9. lift-PI.f32 N/A

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

   10. lower-/.f32 N/A

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

   11. lift-PI.f32 98.8

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

6. Applied rewrites 98.8%

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in maxCos around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. +-commutative N/A

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

    5. lower-fma.f32 98.2

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

11. Applied rewrites 98.2%

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

Alternative 5: 97.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 6.000000212225132 \cdot 10^{-7}:\\ \;\;\;\;\cos \left(uy \cdot \left(\pi + \pi\right)\right) \cdot \sqrt{\left(2 - ux\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - 2 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(-ux, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right), 2\right) - \left(maxCos + maxCos\right)\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 6.000000212225132e-7)
   (* (cos (* uy (+ PI PI))) (sqrt (* (- 2.0 ux) ux)))
   (*
    (- 1.0 (* 2.0 (* (* uy uy) (* PI PI))))
    (sqrt
     (*
      (- (fma (- ux) (* (- maxCos 1.0) (- maxCos 1.0)) 2.0) (+ maxCos maxCos))
      ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 6.000000212225132e-7f) {
		tmp = cosf((uy * (((float) M_PI) + ((float) M_PI)))) * sqrtf(((2.0f - ux) * ux));
	} else {
		tmp = (1.0f - (2.0f * ((uy * uy) * (((float) M_PI) * ((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(6.000000212225132e-7))
		tmp = Float32(cos(Float32(uy * Float32(Float32(pi) + Float32(pi)))) * sqrt(Float32(Float32(Float32(2.0) - ux) * ux)));
	else
		tmp = Float32(Float32(Float32(1.0) - Float32(Float32(2.0) * Float32(Float32(uy * uy) * Float32(Float32(pi) * Float32(pi))))) * sqrt(Float32(Float32(fma(Float32(-ux), Float32(Float32(maxCos - Float32(1.0)) * Float32(maxCos - Float32(1.0))), Float32(2.0)) - Float32(maxCos + maxCos)) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 6.0000002e-7

   1. Initial program 57.5%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 99.0

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

   4. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. sin-+PI/2-rev N/A

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

      6. lower-sin.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. lower-fma.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lower-/.f32 N/A

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

      11. lift-PI.f32 98.8

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

   6. Applied rewrites 98.8%

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

   8. Applied rewrites 99.0%

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

   9. Taylor expanded in maxCos around 0

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

   11. Applied rewrites 98.9%

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

   if 6.0000002e-7 < maxCos

   1. Initial program 57.5%

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

   2. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-neg.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. count-2-rev N/A

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

      14. lower-+.f32 98.8

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

   4. Applied rewrites 98.8%

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

   5. Taylor expanded in uy around 0

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

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

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

      2. lower--.f32 N/A

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

      3. metadata-eval N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f32 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lift-PI.f32 87.7

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

   7. Applied rewrites 87.7%

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

Alternative 6: 97.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

function code(ux, uy, maxCos)
	return Float32(cos(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 = cos(((uy * single(2.0)) * single(pi))) * sqrt((((single(2.0) - (single(1.0) * ux)) - (maxCos + maxCos)) * ux));
end

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in maxCos around 0

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

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

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

   2. metadata-eval N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 97.4

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

7. Applied rewrites 97.4%

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

Alternative 7: 88.2% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. count-2-rev N/A

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

   14. lower-+.f32 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in uy around 0

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

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

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

   2. lower--.f32 N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lift-PI.f32 88.2

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

7. Applied rewrites 88.2%

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

Alternative 8: 79.9% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

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

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

   4. lower--.f32 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 79.8

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

7. Applied rewrites 79.8%

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

   1. lift-*.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate--l- N/A

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

   10. *-lft-identity N/A

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

   11. pow2 N/A

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

   12. +-commutative N/A

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

9. Applied rewrites 79.9%

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

Alternative 9: 79.3% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

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

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

   4. lower--.f32 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 79.8

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in maxCos around 0

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

   1. metadata-eval N/A

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

   2. pow2 N/A

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

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

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

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

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

10. Applied rewrites 75.8%

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

11. Taylor expanded in maxCos around 0

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

    1. +-commutative N/A

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

    2. lower-+.f32 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. lower--.f32 N/A

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

    7. count-2-rev N/A

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

    8. lower-+.f32 N/A

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

    9. mul-1-neg N/A

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

    10. lower-neg.f32 79.3

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

13. Applied rewrites 79.3%

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

Alternative 10: 79.3% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

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

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

   4. lower--.f32 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 79.8

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in maxCos around 0

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

   1. metadata-eval N/A

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

   2. pow2 N/A

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

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

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

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

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

10. Applied rewrites 79.3%

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

Alternative 11: 78.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * ((2.0e0 - ux) - (2.0e0 * maxcos))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

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

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

   4. lower--.f32 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 79.8

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 78.8%

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

Alternative 12: 75.8% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * (2.0e0 - ux)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

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

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

   4. lower--.f32 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 79.8

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in maxCos around 0

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

   1. metadata-eval N/A

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

   2. pow2 N/A

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

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

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

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

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

10. Applied rewrites 75.8%

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

Alternative 13: 62.2% accurate, 11.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot 2} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((ux * 2.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

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

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

   4. lower--.f32 N/A

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

   5. metadata-eval N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. pow2 N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. lift-*.f32 N/A

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

   12. lower-*.f32 79.8

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

7. Applied rewrites 79.8%

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

8. Taylor expanded in maxCos around 0

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

   1. metadata-eval N/A

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

   2. pow2 N/A

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

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

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

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

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

   5. metadata-eval N/A

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

   6. pow2 N/A

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

10. Applied rewrites 75.8%

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

11. Taylor expanded in ux around 0

    \[\leadsto \sqrt{ux \cdot 2} \]
12. Step-by-step derivation

13. Applied rewrites 62.2%

    \[\leadsto \sqrt{ux \cdot 2} \]
14. Add Preprocessing

Alternative 14: 6.6% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{1 - 1} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((1.0e0 - 1.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f32 49.2

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

4. Applied rewrites 49.2%

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

5. Taylor expanded in ux around 0

   \[\leadsto \sqrt{1 - 1} \]
6. Step-by-step derivation

7. Applied rewrites 6.6%

   \[\leadsto \sqrt{1 - 1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, x"
  :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)))
  (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))