UniformSampleCone, x

Percentage Accurate: 58.0% → 99.1%

Time: 5.0s

Alternatives: 13

Speedup: 5.9×

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
  :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)))
  (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: 58.0% accurate, 1.0× speedup

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

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
  :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)))
  (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, 0.9× speedup

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

Math

\[\begin{array}{l} t_0 := ux - maxCos \cdot ux\\ \sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right) \cdot \sqrt{\left(t\_0 - 0\right) \cdot \left(-\left(t\_0 - 2\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (let* ((t_0 (- ux (* maxCos ux))))
  (*
   (sin (fma (- PI) (+ uy uy) (* PI 0.5)))
   (sqrt (* (- t_0 0.0) (- (- t_0 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux - (maxCos * ux);
	return sinf(fmaf(-((float) M_PI), (uy + uy), (((float) M_PI) * 0.5f))) * sqrtf(((t_0 - 0.0f) * -(t_0 - 2.0f)));
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux - Float32(maxCos * ux))
	return Float32(sin(fma(Float32(-Float32(pi)), Float32(uy + uy), Float32(Float32(pi) * Float32(0.5)))) * sqrt(Float32(Float32(t_0 - Float32(0.0)) * Float32(-Float32(t_0 - Float32(2.0))))))
end

Derivation

1. Initial program 58.0%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-sqr-1 N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]
4. Step-by-step derivation

   1. lift-cos.f32 N/A

      \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

   2. cos-neg-rev N/A

      \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

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

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

   4. lower-sin.f32 N/A

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

5. Applied rewrites 99.1%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]
6. Add Preprocessing

Alternative 2: 98.9% accurate, 1.1× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow1/3 N/A

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

   9. lift-PI.f32 N/A

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

   10. pow1/3 N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f32 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lower-cbrt.f32 58.0%

       \[\leadsto \cos \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. count-2-rev N/A

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

   20. lower-+.f32 58.0%

       \[\leadsto \cos \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(uy + uy\right)}\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

3. Applied rewrites 58.0%

   \[\leadsto \cos \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(uy + uy\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 98.9%

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

   1. lift--.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 98.9%

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

7. Applied rewrites 98.9%

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

Alternative 3: 98.9% accurate, 1.1× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f32 N/A

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

   4. add-cube-cbrt N/A

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

   5. associate-*l* N/A

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

   6. lower-*.f32 N/A

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

   7. lift-PI.f32 N/A

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

   8. pow1/3 N/A

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

   9. lift-PI.f32 N/A

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

   10. pow1/3 N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f32 N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lower-cbrt.f32 58.0%

       \[\leadsto \cos \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{\sqrt[3]{\pi}} \cdot \left(uy \cdot 2\right)\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   17. lift-*.f32 N/A

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

   18. *-commutative N/A

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

   19. count-2-rev N/A

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

   20. lower-+.f32 58.0%

       \[\leadsto \cos \left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \color{blue}{\left(uy + uy\right)}\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

3. Applied rewrites 58.0%

   \[\leadsto \cos \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\sqrt[3]{\pi} \cdot \left(uy + uy\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 98.9%

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

Alternative 4: 97.5% accurate, 1.1× speedup

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

Math

\[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(2 - ux\right)} \]

FPCore

(FPCore (ux uy maxCos)
  :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 (* (- (- ux (* maxCos ux)) 0.0) (- 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	return cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((((ux - (maxCos * ux)) - 0.0f) * (2.0f - ux)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(Float32(ux - Float32(maxCos * ux)) - Float32(0.0)) * Float32(Float32(2.0) - ux))))
end

MATLAB

function tmp = code(ux, uy, maxCos)
	tmp = cos(((uy * single(2.0)) * single(pi))) * sqrt((((ux - (maxCos * ux)) - single(0.0)) * (single(2.0) - ux)));
end

Derivation

1. Initial program 58.0%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

   5. difference-of-sqr-1 N/A

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

   6. distribute-rgt-neg-in N/A

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.9%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

4. Taylor expanded in maxCos around 0

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \color{blue}{\left(2 - ux\right)}} \]
5. Step-by-step derivation

   1. lower--.f32 97.5%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(2 - \color{blue}{ux}\right)} \]

6. Applied rewrites 97.5%

   \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \color{blue}{\left(2 - ux\right)}} \]
7. Add Preprocessing

Alternative 5: 96.4% accurate, 0.7× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999989867210388:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{ux - \mathsf{fma}\left(maxCos, ux, \left(maxCos \cdot ux - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (if (<= (cos (* (* uy 2.0) PI)) 0.9999989867210388)
  (* (sin (fma (- PI) (+ uy uy) (* PI 0.5))) (sqrt (* ux (- 2.0 ux))))
  (sqrt
   (-
    ux
    (fma
     maxCos
     ux
     (* (- (* maxCos ux) ux) (- (+ 1.0 (* maxCos ux)) ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.9999989867210388f) {
		tmp = sinf(fmaf(-((float) M_PI), (uy + uy), (((float) M_PI) * 0.5f))) * sqrtf((ux * (2.0f - ux)));
	} else {
		tmp = sqrtf((ux - fmaf(maxCos, ux, (((maxCos * ux) - ux) * ((1.0f + (maxCos * ux)) - ux)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.9999989867210388))
		tmp = Float32(sin(fma(Float32(-Float32(pi)), Float32(uy + uy), Float32(Float32(pi) * Float32(0.5)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	else
		tmp = sqrt(Float32(ux - fma(maxCos, ux, Float32(Float32(Float32(maxCos * ux) - ux) * Float32(Float32(Float32(1.0) + Float32(maxCos * ux)) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999998987

   1. Initial program 58.0%

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

      1. lift--.f32 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-sqr-1 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f32 N/A

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

   3. Applied rewrites 98.9%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

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

         \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower--.f32 92.8%

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

   8. Applied rewrites 92.8%

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

   if 0.999998987 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 58.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. remove-double-neg N/A

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

      5. remove-double-neg N/A

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

      6. lift-+.f32 N/A

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

      7. lift--.f32 N/A

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

      8. sub-flip N/A

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

      9. associate-+l+ N/A

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

      10. add-flip-rev N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-+.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f32 N/A

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

      19. lower-*.f32 N/A

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

   3. Applied rewrites 58.8%

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

   4. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 79.7%

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

   6. Applied rewrites 79.7%

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

Alternative 6: 96.3% accurate, 1.1× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{ux - \mathsf{fma}\left(maxCos, ux, \left(maxCos \cdot ux - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(-2, uy \cdot \pi, 0.5 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (if (<= uy 0.00022000000171829015)
  (sqrt
   (-
    ux
    (fma
     maxCos
     ux
     (* (- (* maxCos ux) ux) (- (+ 1.0 (* maxCos ux)) ux)))))
  (* (sin (fma -2.0 (* uy PI) (* 0.5 PI))) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00022000000171829015f) {
		tmp = sqrtf((ux - fmaf(maxCos, ux, (((maxCos * ux) - ux) * ((1.0f + (maxCos * ux)) - ux)))));
	} else {
		tmp = sinf(fmaf(-2.0f, (uy * ((float) M_PI)), (0.5f * ((float) M_PI)))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00022000000171829015))
		tmp = sqrt(Float32(ux - fma(maxCos, ux, Float32(Float32(Float32(maxCos * ux) - ux) * Float32(Float32(Float32(1.0) + Float32(maxCos * ux)) - ux)))));
	else
		tmp = Float32(sin(fma(Float32(-2.0), Float32(uy * Float32(pi)), Float32(Float32(0.5) * Float32(pi)))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 2.20000002e-4

   1. Initial program 58.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. remove-double-neg N/A

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

      5. remove-double-neg N/A

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

      6. lift-+.f32 N/A

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

      7. lift--.f32 N/A

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

      8. sub-flip N/A

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

      9. associate-+l+ N/A

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

      10. add-flip-rev N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-+.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f32 N/A

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

      19. lower-*.f32 N/A

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

   3. Applied rewrites 58.8%

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

   4. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 79.7%

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

   6. Applied rewrites 79.7%

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

   if 2.20000002e-4 < uy

   1. Initial program 58.0%

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

      1. lift--.f32 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-sqr-1 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f32 N/A

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

   3. Applied rewrites 98.9%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-cos.f32 N/A

         \[\leadsto \color{blue}{\cos \left(\left(uy \cdot 2\right) \cdot \pi\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

      2. cos-neg-rev N/A

         \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

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

         \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(uy \cdot 2\right) \cdot \pi\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

   5. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(-\pi, uy + uy, \pi \cdot 0.5\right)\right)} \cdot \sqrt{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)} \]

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sin.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-PI.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-PI.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 92.6%

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

   8. Applied rewrites 92.6%

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

Alternative 7: 96.3% accurate, 1.2× speedup

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

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.00022000000171829015:\\ \;\;\;\;\sqrt{ux - \mathsf{fma}\left(maxCos, ux, \left(maxCos \cdot ux - ux\right) \cdot \left(\left(1 + maxCos \cdot ux\right) - ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (if (<= uy 0.00022000000171829015)
  (sqrt
   (-
    ux
    (fma
     maxCos
     ux
     (* (- (* maxCos ux) ux) (- (+ 1.0 (* maxCos ux)) ux)))))
  (* (cos (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00022000000171829015f) {
		tmp = sqrtf((ux - fmaf(maxCos, ux, (((maxCos * ux) - ux) * ((1.0f + (maxCos * ux)) - ux)))));
	} else {
		tmp = cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00022000000171829015))
		tmp = sqrt(Float32(ux - fma(maxCos, ux, Float32(Float32(Float32(maxCos * ux) - ux) * Float32(Float32(Float32(1.0) + Float32(maxCos * ux)) - ux)))));
	else
		tmp = Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 2.20000002e-4

   1. Initial program 58.0%

      \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. sqr-neg-rev N/A

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

      3. sqr-neg-rev N/A

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

      4. remove-double-neg N/A

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

      5. remove-double-neg N/A

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

      6. lift-+.f32 N/A

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

      7. lift--.f32 N/A

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

      8. sub-flip N/A

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

      9. associate-+l+ N/A

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

      10. add-flip-rev N/A

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

      11. distribute-lft-in N/A

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

      12. *-rgt-identity N/A

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

      13. lower-+.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. +-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f32 N/A

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

      19. lower-*.f32 N/A

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

   3. Applied rewrites 58.8%

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

   4. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-fma.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower-+.f32 N/A

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

      9. lower-*.f32 79.7%

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

   6. Applied rewrites 79.7%

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

   if 2.20000002e-4 < uy

   1. Initial program 58.0%

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

      1. lift--.f32 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-sqr-1 N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. lower-*.f32 N/A

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

   3. Applied rewrites 98.9%

      \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(-\left(\left(ux - maxCos \cdot ux\right) - 2\right)\right)}} \]

   4. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 92.6%

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

   6. Applied rewrites 92.6%

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

Alternative 8: 79.7% accurate, 2.5× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. sqr-neg-rev N/A

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

   3. sqr-neg-rev N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-+.f32 N/A

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

   7. lift--.f32 N/A

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

   8. sub-flip N/A

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

   9. associate-+l+ N/A

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

   10. add-flip-rev N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. lower-+.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-*.f32 N/A

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

3. Applied rewrites 58.8%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 79.7%

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

6. Applied rewrites 79.7%

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

Alternative 9: 79.7% accurate, 3.1× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

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 49.6%

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

4. Applied rewrites 49.6%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. lower-pow.f32 N/A

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

   14. pow2 N/A

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

   15. difference-of-squares-rev N/A

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

6. Applied rewrites 79.7%

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

   1. lift--.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. associate--r- N/A

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

   7. metadata-eval N/A

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

   8. lift--.f32 N/A

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

   9. associate--l+ N/A

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

   10. lift--.f32 N/A

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

   11. associate--r- N/A

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

   12. lift--.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. *-commutative N/A

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

   15. lift-*.f32 N/A

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

   16. lift-+.f32 N/A

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

   17. lower-+.f32 79.7%

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

   18. lift-+.f32 N/A

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

   19. lift-*.f32 N/A

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

   20. *-commutative N/A

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

   21. lift-*.f32 N/A

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

   22. +-commutative N/A

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

   23. lift-*.f32 N/A

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

   24. lower-fma.f32 79.7%

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

8. Applied rewrites 79.7%

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

Alternative 10: 79.7% accurate, 3.5× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

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 49.6%

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

4. Applied rewrites 49.6%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. lower-pow.f32 N/A

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

   14. pow2 N/A

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

   15. difference-of-squares-rev N/A

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

6. Applied rewrites 79.7%

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

   1. lift--.f32 N/A

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

   2. lift-fma.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. associate--l+ N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 79.7%

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

8. Applied rewrites 79.7%

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

Alternative 11: 79.7% accurate, 3.5× speedup

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

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 58.0%

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

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 49.6%

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

4. Applied rewrites 49.6%

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

   1. lift--.f32 N/A

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

   2. metadata-eval N/A

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

   3. lift--.f32 N/A

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

   4. lift-+.f32 N/A

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

   5. +-commutative N/A

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

   6. associate-+r- N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. lift-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. lift-+.f32 N/A

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

   13. lower-pow.f32 N/A

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

   14. pow2 N/A

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

   15. difference-of-squares-rev N/A

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

6. Applied rewrites 79.7%

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

7. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 79.7%

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

9. Applied rewrites 79.7%

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

Alternative 12: 75.4% accurate, 4.8× speedup

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

Math

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

FPCore

(FPCore (ux uy maxCos)
  :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)))
  (sqrt (- ux (* -1.0 (* ux (- 1.0 ux))))))

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. sqr-neg-rev N/A

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

   3. sqr-neg-rev N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-+.f32 N/A

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

   7. lift--.f32 N/A

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

   8. sub-flip N/A

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

   9. associate-+l+ N/A

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

   10. add-flip-rev N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. lower-+.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-*.f32 N/A

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

3. Applied rewrites 58.8%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 79.7%

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

6. Applied rewrites 79.7%

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

7. Taylor expanded in maxCos around 0

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

   1. lower--.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower--.f32 75.4%

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

9. Applied rewrites 75.4%

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

Alternative 13: 64.1% accurate, 5.9× speedup

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

Math

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

FPCore

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

C

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 58.0%

   \[\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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. sqr-neg-rev N/A

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

   3. sqr-neg-rev N/A

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

   4. remove-double-neg N/A

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

   5. remove-double-neg N/A

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

   6. lift-+.f32 N/A

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

   7. lift--.f32 N/A

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

   8. sub-flip N/A

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

   9. associate-+l+ N/A

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

   10. add-flip-rev N/A

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

   11. distribute-lft-in N/A

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

   12. *-rgt-identity N/A

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

   13. lower-+.f32 N/A

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

   14. lift-+.f32 N/A

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

   15. +-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

   18. lower-fma.f32 N/A

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

   19. lower-*.f32 N/A

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

3. Applied rewrites 58.8%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-fma.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-+.f32 N/A

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

   9. lower-*.f32 79.7%

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

6. Applied rewrites 79.7%

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

7. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 64.1%

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

9. Applied rewrites 64.1%

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

Reproduce

herbie shell --seed 2025356 
(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)))))))