UniformSampleCone, x

Percentage Accurate: 57.2% → 99.1%

Time: 7.5s

Alternatives: 17

Speedup: 3.3×

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

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

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: 57.2% accurate, 1.0× speedup

Math

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

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: 99.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sinf(fmaf(-((float) M_PI), (uy + uy), (((float) M_PI) * 0.5f))) * sqrtf((ux * ((2.0f + (-1.0f * (ux * powf((maxCos - 1.0f), 2.0f)))) - (2.0f * maxCos))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sin(fma(Float32(-Float32(pi)), Float32(uy + uy), Float32(Float32(pi) * Float32(0.5)))) * sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * (Float32(maxCos - Float32(1.0)) ^ Float32(2.0))))) - Float32(Float32(2.0) * maxCos)))))
end

Derivation

1. Initial program 57.2%

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

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. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. count-2 N/A

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

   9. lift-+.f32 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-fma.f32 N/A

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

   13. lower-neg.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. mult-flip N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f32 99.1%

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

6. Applied rewrites 99.1%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

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. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 99.0%

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

6. Applied rewrites 99.0%

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

Alternative 3: 98.4% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

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. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 98.4%

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

7. Applied rewrites 98.4%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.4%

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

9. Applied rewrites 98.4%

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

Alternative 4: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

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. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-*.f32 97.6%

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

7. Applied rewrites 97.6%

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

Alternative 5: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

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. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 99.0%

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

4. Applied rewrites 99.0%

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

5. Taylor expanded in maxCos around 0

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

   1. lower-+.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 98.4%

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 97.6%

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

12. Applied rewrites 97.6%

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

Alternative 6: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.9999995231628418:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot \left(2 + -1 \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(2 + -1 \cdot \left(ux \cdot \left(1 + \mathsf{fma}\left(-2, maxCos, {maxCos}^{2}\right)\right)\right)\right) - 2 \cdot maxCos\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (cos (* (* uy 2.0) PI))))
  (if (<= t_0 0.9999995231628418)
    (* t_0 (sqrt (* ux (+ 2.0 (* -1.0 ux)))))
    (sqrt
     (*
      ux
      (-
       (+
        2.0
        (* -1.0 (* ux (+ 1.0 (fma -2.0 maxCos (pow maxCos 2.0))))))
       (* 2.0 maxCos)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (t_0 <= 0.9999995231628418f) {
		tmp = t_0 * sqrtf((ux * (2.0f + (-1.0f * ux))));
	} else {
		tmp = sqrtf((ux * ((2.0f + (-1.0f * (ux * (1.0f + fmaf(-2.0f, maxCos, powf(maxCos, 2.0f)))))) - (2.0f * maxCos))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999995231628418))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(Float32(2.0) + Float32(Float32(-1.0) * ux)))));
	else
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(2.0) + Float32(Float32(-1.0) * Float32(ux * Float32(Float32(1.0) + fma(Float32(-2.0), maxCos, (maxCos ^ Float32(2.0))))))) - Float32(Float32(2.0) * maxCos))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999999523

   1. Initial program 57.2%

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

   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. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-pow.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 99.0%

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

   4. Applied rewrites 99.0%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 92.8%

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

   7. Applied rewrites 92.8%

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

   if 0.999999523 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-pow.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

      6. lift--.f32 N/A

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

      7. sum-square-pow N/A

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

      8. pow2 N/A

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

      9. lower-+.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-*.f32 N/A

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

      20. lower-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-*.f32 N/A

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

      6. lower-+.f32 N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 80.4%

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

   9. Applied rewrites 80.4%

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

Alternative 7: 80.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-*.f32 49.1%

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

4. Applied rewrites 49.1%

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

   1. lift-pow.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-+r- N/A

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

   6. lift--.f32 N/A

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

   7. sum-square-pow N/A

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

   8. pow2 N/A

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

   9. lower-+.f32 N/A

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

   10. pow2 N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. swap-sqr N/A

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

   16. lower-fma.f32 N/A

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

   17. lower-*.f32 N/A

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

   18. lower-*.f32 N/A

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

   19. lower-*.f32 N/A

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

   20. lower-*.f32 N/A

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

6. Applied rewrites 49.2%

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

7. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-+.f32 N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-pow.f32 N/A

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

   9. lower-*.f32 80.4%

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

9. Applied rewrites 80.4%

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

Alternative 8: 80.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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(((2.0e0 * (ux - (maxcos * ux))) - (((maxcos * ux) - ux) ** 2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-+.f32 N/A

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

   4. lift--.f32 N/A

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

   5. associate-+l- N/A

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

   6. sub-square-pow N/A

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

   7. pow2 N/A

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

   8. lower-+.f32 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 N/A

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

   17. sub-negate-rev N/A

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

   18. sub-negate-rev N/A

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

3. Applied rewrites 60.1%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-pow.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 80.4%

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

6. Applied rewrites 80.4%

   \[\leadsto \color{blue}{\sqrt{2 \cdot \left(ux - maxCos \cdot ux\right) - {\left(maxCos \cdot ux - ux\right)}^{2}}} \]
7. Add Preprocessing

Alternative 9: 80.3% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.2%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-*.f32 49.1%

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

4. Applied rewrites 49.1%

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

   1. lift--.f32 N/A

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

   2. sub-flip N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lower-pow.f32 N/A

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

   14. pow2 N/A

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

   15. lift-*.f32 N/A

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

   16. lower-+.f32 N/A

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

   17. lift-*.f32 N/A

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

6. Applied rewrites 49.2%

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

7. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-+.f32 N/A

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

   4. lower-fma.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower-*.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower--.f32 80.3%

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

9. Applied rewrites 80.3%

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

Alternative 10: 76.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \mathbf{if}\;ux \leq 8.800000068731606 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - t\_0 \cdot \left(2 - t\_0\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (- ux (* maxCos ux))))
  (if (<= ux 8.800000068731606e-5)
    (sqrt (* ux (- (+ 1.0 (* -1.0 (- maxCos 1.0))) maxCos)))
    (sqrt (- 1.0 (- 1.0 (* t_0 (- 2.0 t_0))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (maxCos * ux);
	float tmp;
	if (ux <= 8.800000068731606e-5f) {
		tmp = sqrtf((ux * ((1.0f + (-1.0f * (maxCos - 1.0f))) - maxCos)));
	} else {
		tmp = sqrtf((1.0f - (1.0f - (t_0 * (2.0f - t_0)))));
	}
	return tmp;
}

Fortran

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = ux - (maxcos * ux)
    if (ux <= 8.800000068731606e-5) then
        tmp = sqrt((ux * ((1.0e0 + ((-1.0e0) * (maxcos - 1.0e0))) - maxcos)))
    else
        tmp = sqrt((1.0e0 - (1.0e0 - (t_0 * (2.0e0 - t_0)))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(maxCos * ux))
	tmp = Float32(0.0)
	if (ux <= Float32(8.800000068731606e-5))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(-1.0) * Float32(maxCos - Float32(1.0)))) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(Float32(1.0) - Float32(t_0 * Float32(Float32(2.0) - t_0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = ux - (maxCos * ux);
	tmp = single(0.0);
	if (ux <= single(8.800000068731606e-5))
		tmp = sqrt((ux * ((single(1.0) + (single(-1.0) * (maxCos - single(1.0)))) - maxCos)));
	else
		tmp = sqrt((single(1.0) - (single(1.0) - (t_0 * (single(2.0) - t_0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 8.80000007e-5

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 64.9%

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

   9. Applied rewrites 64.9%

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

   if 8.80000007e-5 < ux

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-pow.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

      6. lift--.f32 N/A

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

      7. +-commutative N/A

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

      8. +-commutative N/A

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

      9. lift--.f32 N/A

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

      10. associate-+r- N/A

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

      11. +-commutative N/A

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

      12. associate--l+ N/A

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

      13. sub-negate-rev N/A

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

      14. lift--.f32 N/A

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

      15. sub-flip-reverse N/A

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

      16. sub-square-pow N/A

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

      17. metadata-eval N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. pow2 N/A

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

   6. Applied rewrites 51.7%

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

Alternative 11: 75.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ \mathbf{if}\;ux \leq 0.0002099999983329326:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (fma maxCos ux (- 1.0 ux))))
  (if (<= ux 0.0002099999983329326)
    (sqrt (* ux (- (+ 1.0 (* -1.0 (- maxCos 1.0))) maxCos)))
    (sqrt (- 1.0 (* t_0 t_0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, (1.0f - ux));
	float tmp;
	if (ux <= 0.0002099999983329326f) {
		tmp = sqrtf((ux * ((1.0f + (-1.0f * (maxCos - 1.0f))) - maxCos)));
	} else {
		tmp = sqrtf((1.0f - (t_0 * t_0)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = fma(maxCos, ux, Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (ux <= Float32(0.0002099999983329326))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(-1.0) * Float32(maxCos - Float32(1.0)))) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.09999998e-4

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 64.9%

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

   9. Applied rewrites 64.9%

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

   if 2.09999998e-4 < ux

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-pow.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. +-commutative N/A

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

      5. associate-+r- N/A

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

      6. lift--.f32 N/A

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

      7. sum-square-pow N/A

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

      8. pow2 N/A

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

      9. lower-+.f32 N/A

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

      10. pow2 N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-*.f32 N/A

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

      14. *-commutative N/A

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

      15. swap-sqr N/A

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

      16. lower-fma.f32 N/A

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

      17. lower-*.f32 N/A

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

      18. lower-*.f32 N/A

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

      19. lower-*.f32 N/A

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

      20. lower-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

      1. lift-+.f32 N/A

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

      2. lift-fma.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. pow2 N/A

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

      5. lift-*.f32 N/A

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

      6. pow2 N/A

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

      7. pow-prod-down N/A

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

      8. *-commutative N/A

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

      9. lift-*.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. lift-*.f32 N/A

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

      12. lift-*.f32 N/A

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

      13. sqr-neg-rev N/A

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

      14. lift--.f32 N/A

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

      15. sub-negate-rev N/A

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

      16. lift--.f32 N/A

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

      17. lift--.f32 N/A

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

      18. sub-negate-rev N/A

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

      19. lift--.f32 N/A

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

      20. pow2 N/A

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

   8. Applied rewrites 49.2%

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

Alternative 12: 75.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.0002099999983329326:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(ux - \mathsf{fma}\left(maxCos, ux, 1\right)\right) \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= ux 0.0002099999983329326)
  (sqrt (* ux (- (+ 1.0 (* -1.0 (- maxCos 1.0))) maxCos)))
  (sqrt
   (+ 1.0 (* (- ux (fma maxCos ux 1.0)) (fma maxCos ux (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0002099999983329326f) {
		tmp = sqrtf((ux * ((1.0f + (-1.0f * (maxCos - 1.0f))) - maxCos)));
	} else {
		tmp = sqrtf((1.0f + ((ux - fmaf(maxCos, ux, 1.0f)) * fmaf(maxCos, ux, (1.0f - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0002099999983329326))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(-1.0) * Float32(maxCos - Float32(1.0)))) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(ux - fma(maxCos, ux, Float32(1.0))) * fma(maxCos, ux, Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 2.09999998e-4

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 64.9%

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

   9. Applied rewrites 64.9%

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

   if 2.09999998e-4 < ux

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

Alternative 13: 75.6% accurate, 2.5× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= ux 0.00011000000085914508)
  (sqrt (* ux (- (+ 1.0 (* -1.0 (- maxCos 1.0))) maxCos)))
  (sqrt
   (fma (fma maxCos ux (- 1.0 ux)) (- ux (fma maxCos ux 1.0)) 1.0))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = sqrtf((ux * ((1.0f + (-1.0f * (maxCos - 1.0f))) - maxCos)));
	} else {
		tmp = sqrtf(fmaf(fmaf(maxCos, ux, (1.0f - ux)), (ux - fmaf(maxCos, ux, 1.0f)), 1.0f));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(-1.0) * Float32(maxCos - Float32(1.0)))) - maxCos)));
	else
		tmp = sqrt(fma(fma(maxCos, ux, Float32(Float32(1.0) - ux)), Float32(ux - fma(maxCos, ux, Float32(1.0))), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 64.9%

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

   9. Applied rewrites 64.9%

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

   if 1.10000001e-4 < ux

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lift-fma.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. +-commutative N/A

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

      8. lift--.f32 N/A

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. lift--.f32 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. lift-fma.f32 49.2%

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

   8. Applied rewrites 49.2%

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

Alternative 14: 74.8% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00013000000035390258:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(ux \cdot \left(2 + -1 \cdot ux\right) - 1\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= ux 0.00013000000035390258)
  (sqrt (* ux (- (+ 1.0 (* -1.0 (- maxCos 1.0))) maxCos)))
  (sqrt (+ 1.0 (- (* ux (+ 2.0 (* -1.0 ux))) 1.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00013000000035390258f) {
		tmp = sqrtf((ux * ((1.0f + (-1.0f * (maxCos - 1.0f))) - maxCos)));
	} else {
		tmp = sqrtf((1.0f + ((ux * (2.0f + (-1.0f * ux))) - 1.0f)));
	}
	return tmp;
}

Fortran

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
    real(4) :: tmp
    if (ux <= 0.00013000000035390258e0) then
        tmp = sqrt((ux * ((1.0e0 + ((-1.0e0) * (maxcos - 1.0e0))) - maxcos)))
    else
        tmp = sqrt((1.0e0 + ((ux * (2.0e0 + ((-1.0e0) * ux))) - 1.0e0)))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00013000000035390258))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(-1.0) * Float32(maxCos - Float32(1.0)))) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(ux * Float32(Float32(2.0) + Float32(Float32(-1.0) * ux))) - Float32(1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.00013000000035390258))
		tmp = sqrt((ux * ((single(1.0) + (single(-1.0) * (maxCos - single(1.0)))) - maxCos)));
	else
		tmp = sqrt((single(1.0) + ((ux * (single(2.0) + (single(-1.0) * ux))) - single(1.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.3e-4

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 64.9%

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

   9. Applied rewrites 64.9%

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

   if 1.3e-4 < ux

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower--.f32 47.8%

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

   9. Applied rewrites 47.8%

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

   10. Taylor expanded in ux around 0

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

       1. lower--.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower-+.f32 N/A

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

       4. lower-*.f32 50.1%

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

   12. Applied rewrites 50.1%

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

Alternative 15: 74.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.030249999836087227:\\ \;\;\;\;\sqrt{ux \cdot \left(\left(1 + -1 \cdot \left(maxCos - 1\right)\right) - maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \left(1 - ux\right) \cdot \left(ux - 1\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
  (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.030249999836087227)
    (sqrt (* ux (- (+ 1.0 (* -1.0 (- maxCos 1.0))) maxCos)))
    (sqrt (+ 1.0 (* (- 1.0 ux) (- ux 1.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if (sqrtf((1.0f - (t_0 * t_0))) <= 0.030249999836087227f) {
		tmp = sqrtf((ux * ((1.0f + (-1.0f * (maxCos - 1.0f))) - maxCos)));
	} else {
		tmp = sqrtf((1.0f + ((1.0f - ux) * (ux - 1.0f))));
	}
	return tmp;
}

Fortran

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
    real(4) :: t_0
    real(4) :: tmp
    t_0 = (1.0e0 - ux) + (ux * maxcos)
    if (sqrt((1.0e0 - (t_0 * t_0))) <= 0.030249999836087227e0) then
        tmp = sqrt((ux * ((1.0e0 + ((-1.0e0) * (maxcos - 1.0e0))) - maxcos)))
    else
        tmp = sqrt((1.0e0 + ((1.0e0 - ux) * (ux - 1.0e0))))
    end if
    code = tmp
end function

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) <= Float32(0.030249999836087227))
		tmp = sqrt(Float32(ux * Float32(Float32(Float32(1.0) + Float32(Float32(-1.0) * Float32(maxCos - Float32(1.0)))) - maxCos)));
	else
		tmp = sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux - Float32(1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = single(0.0);
	if (sqrt((single(1.0) - (t_0 * t_0))) <= single(0.030249999836087227))
		tmp = sqrt((ux * ((single(1.0) + (single(-1.0) * (maxCos - single(1.0)))) - maxCos)));
	else
		tmp = sqrt((single(1.0) + ((single(1.0) - ux) * (ux - single(1.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0302499998

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-+.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 64.9%

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

   9. Applied rewrites 64.9%

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

   if 0.0302499998 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 57.2%

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

   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. lower-pow.f32 N/A

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

      4. lower--.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. lift--.f32 N/A

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

      4. lift-+.f32 N/A

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

      5. +-commutative N/A

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

      6. associate-+r- N/A

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

      7. lift--.f32 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f32 N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. lower-pow.f32 N/A

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

      14. pow2 N/A

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

      15. lift-*.f32 N/A

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

      16. lower-+.f32 N/A

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

      17. lift-*.f32 N/A

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

   6. Applied rewrites 49.2%

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

   7. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower--.f32 47.8%

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

   9. Applied rewrites 47.8%

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

Alternative 16: 47.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Fortran

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 - ux) * (ux - 1.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-*.f32 49.1%

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

4. Applied rewrites 49.1%

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

   1. lift--.f32 N/A

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

   2. sub-flip N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lower-pow.f32 N/A

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

   14. pow2 N/A

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

   15. lift-*.f32 N/A

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

   16. lower-+.f32 N/A

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

   17. lift-*.f32 N/A

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

6. Applied rewrites 49.2%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower--.f32 47.8%

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

9. Applied rewrites 47.8%

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

Alternative 17: 39.9% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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 + ((2.0e0 * ux) - 1.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.2%

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

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. lower-pow.f32 N/A

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

   4. lower--.f32 N/A

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

   5. lower-+.f32 N/A

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

   6. lower-*.f32 49.1%

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

4. Applied rewrites 49.1%

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

   1. lift--.f32 N/A

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

   2. sub-flip N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f32 N/A

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

   10. lift-*.f32 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. lower-pow.f32 N/A

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

   14. pow2 N/A

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

   15. lift-*.f32 N/A

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

   16. lower-+.f32 N/A

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

   17. lift-*.f32 N/A

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

6. Applied rewrites 49.2%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower--.f32 47.8%

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

9. Applied rewrites 47.8%

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

10. Taylor expanded in ux around 0

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

    1. lower--.f32 N/A

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

    2. lower-*.f32 39.9%

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

12. Applied rewrites 39.9%

    \[\leadsto \sqrt{1 + \left(2 \cdot ux - 1\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025325 
(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)))))))