UniformSampleCone, x

Percentage Accurate: 57.6% → 99.1%

Time: 1.1min

Alternatives: 15

Speedup: 5.9×

• unsoundness detected in the egraph

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.9%

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

6. Applied rewrites 98.9%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. lift-*.f32 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. metadata-eval N/A

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

   11. lift-PI.f32 N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

   14. distribute-lft-out N/A

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

   15. lower-*.f32 N/A

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

   16. lower-fma.f32 99.1%

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

8. Applied rewrites 99.1%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.9%

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

6. Applied rewrites 98.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 98.9%

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

8. Applied rewrites 98.9%

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

Alternative 3: 97.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.9%

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

6. Applied rewrites 98.9%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. distribute-rgt-neg-out N/A

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

   8. lift-*.f32 N/A

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

   9. distribute-rgt-neg-out N/A

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

   10. metadata-eval N/A

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

   11. lift-PI.f32 N/A

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

   12. mult-flip N/A

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

   13. metadata-eval N/A

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

   14. distribute-lft-out N/A

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

   15. lower-*.f32 N/A

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

   16. lower-fma.f32 99.1%

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

8. Applied rewrites 99.1%

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

9. Taylor expanded in maxCos around 0

   \[\leadsto \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{\color{blue}{\left(ux - 2\right)} \cdot \left(ux \cdot \left(maxCos - 1\right)\right)} \]
10. Step-by-step derivation

    1. lower--.f32 97.7%

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

11. Applied rewrites 97.7%

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

Alternative 4: 97.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.9%

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in maxCos around 0

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

   1. lower--.f32 97.6%

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

9. Applied rewrites 97.6%

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

Alternative 5: 96.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.00019500000053085387:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 0.00019500000053085387)
  (sqrt (* (* (- maxCos 1.0) ux) (- (* ux (- 1.0 maxCos)) 2.0)))
  (*
   (sin (* PI (fma uy -2.0 0.5)))
   (sqrt (* -1.0 (* ux (- ux 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00019500000053085387f) {
		tmp = sqrtf((((maxCos - 1.0f) * ux) * ((ux * (1.0f - maxCos)) - 2.0f)));
	} else {
		tmp = sinf((((float) M_PI) * fmaf(uy, -2.0f, 0.5f))) * sqrtf((-1.0f * (ux * (ux - 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00019500000053085387))
		tmp = sqrt(Float32(Float32(Float32(maxCos - Float32(1.0)) * ux) * Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) - Float32(2.0))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * fma(uy, Float32(-2.0), Float32(0.5)))) * sqrt(Float32(Float32(-1.0) * Float32(ux * Float32(ux - Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.95000001e-4

   1. Initial program 57.6%

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

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

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

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

      6. distribute-neg-out N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lift--.f32 N/A

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

      11. sub-flip N/A

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

      12. associate-+l+ N/A

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

      13. associate--r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

      15. sub-negate N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

      16. --rgt-identity N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 79.7%

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

   6. Applied rewrites 79.7%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-rgt-out-- N/A

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

      10. add-flip N/A

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

      11. +-commutative N/A

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

      12. sub-flip N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 79.7%

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

   8. Applied rewrites 79.7%

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

   if 1.95000001e-4 < uy

   1. Initial program 57.6%

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

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

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

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

      6. distribute-neg-out N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lift--.f32 N/A

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

      11. sub-flip N/A

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

      12. associate-+l+ N/A

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

      13. associate--r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

      15. sub-negate N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

      16. --rgt-identity N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 98.9%

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

   6. Applied rewrites 98.9%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f32 N/A

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

      5. lift-*.f32 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift-*.f32 N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. metadata-eval N/A

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

      11. lift-PI.f32 N/A

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

      12. mult-flip N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-out N/A

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

      15. lower-*.f32 N/A

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

      16. lower-fma.f32 99.1%

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

   8. Applied rewrites 99.1%

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

   9. Taylor expanded in maxCos around 0

      \[\leadsto \sin \left(\pi \cdot \mathsf{fma}\left(uy, -2, 0.5\right)\right) \cdot \sqrt{\color{blue}{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}} \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

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

       2. lower-*.f32 N/A

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

       3. lower--.f32 92.8%

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

   11. Applied rewrites 92.8%

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

Alternative 6: 96.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.00019500000053085387:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 0.00019500000053085387)
  (sqrt (* (* (- maxCos 1.0) ux) (- (* ux (- 1.0 maxCos)) 2.0)))
  (* (cos (* 2.0 (* uy PI))) (sqrt (* -1.0 (* ux (- ux 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.00019500000053085387f) {
		tmp = sqrtf((((maxCos - 1.0f) * ux) * ((ux * (1.0f - maxCos)) - 2.0f)));
	} else {
		tmp = cosf((2.0f * (uy * ((float) M_PI)))) * sqrtf((-1.0f * (ux * (ux - 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.00019500000053085387))
		tmp = sqrt(Float32(Float32(Float32(maxCos - Float32(1.0)) * ux) * Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) - Float32(2.0))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(uy * Float32(pi)))) * sqrt(Float32(Float32(-1.0) * Float32(ux * Float32(ux - Float32(2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (uy <= single(0.00019500000053085387))
		tmp = sqrt((((maxCos - single(1.0)) * ux) * ((ux * (single(1.0) - maxCos)) - single(2.0))));
	else
		tmp = cos((single(2.0) * (uy * single(pi)))) * sqrt((single(-1.0) * (ux * (ux - single(2.0)))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if uy < 1.95000001e-4

   1. Initial program 57.6%

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

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

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

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

      6. distribute-neg-out N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lift--.f32 N/A

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

      11. sub-flip N/A

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

      12. associate-+l+ N/A

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

      13. associate--r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

      15. sub-negate N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

      16. --rgt-identity N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 79.7%

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

   6. Applied rewrites 79.7%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-rgt-out-- N/A

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

      10. add-flip N/A

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

      11. +-commutative N/A

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

      12. sub-flip N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 79.7%

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

   8. Applied rewrites 79.7%

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

   if 1.95000001e-4 < uy

   1. Initial program 57.6%

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

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

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

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

      6. distribute-neg-out N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lift--.f32 N/A

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

      11. sub-flip N/A

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

      12. associate-+l+ N/A

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

      13. associate--r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

      15. sub-negate N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

      16. --rgt-identity N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 79.7%

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

   6. Applied rewrites 79.7%

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

   8. Applied rewrites 21.5%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower-*.f32 N/A

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

       2. lower-cos.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-*.f32 N/A

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

       5. lower-PI.f32 N/A

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

       6. lower-sqrt.f32 N/A

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

       7. lower-*.f32 N/A

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

       8. lower-*.f32 N/A

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

       9. lower--.f32 92.7%

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

   11. Applied rewrites 92.7%

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

Alternative 7: 90.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \leq 0.9999964833259583:\\ \;\;\;\;\cos \left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{-2 \cdot \left(maxCos \cdot ux - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 2\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= (cos (* (* uy 2.0) PI)) 0.9999964833259583)
  (* (cos (* (+ uy uy) PI)) (sqrt (* -2.0 (- (* maxCos ux) ux))))
  (sqrt (* (* (- maxCos 1.0) ux) (- (* ux (- 1.0 maxCos)) 2.0)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (cosf(((uy * 2.0f) * ((float) M_PI))) <= 0.9999964833259583f) {
		tmp = cosf(((uy + uy) * ((float) M_PI))) * sqrtf((-2.0f * ((maxCos * ux) - ux)));
	} else {
		tmp = sqrtf((((maxCos - 1.0f) * ux) * ((ux * (1.0f - maxCos)) - 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) <= Float32(0.9999964833259583))
		tmp = Float32(cos(Float32(Float32(uy + uy) * Float32(pi))) * sqrt(Float32(Float32(-2.0) * Float32(Float32(maxCos * ux) - ux))));
	else
		tmp = sqrt(Float32(Float32(Float32(maxCos - Float32(1.0)) * ux) * Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) - Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (cos(((uy * single(2.0)) * single(pi))) <= single(0.9999964833259583))
		tmp = cos(((uy + uy) * single(pi))) * sqrt((single(-2.0) * ((maxCos * ux) - ux)));
	else
		tmp = sqrt((((maxCos - single(1.0)) * ux) * ((ux * (single(1.0) - maxCos)) - single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999996483

   1. Initial program 57.6%

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

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

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

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

      6. distribute-neg-out N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lift--.f32 N/A

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

      11. sub-flip N/A

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

      12. associate-+l+ N/A

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

      13. associate--r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

      15. sub-negate N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

      16. --rgt-identity N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in ux around 0

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

   6. Applied rewrites 76.4%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 76.4%

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

   8. Applied rewrites 76.4%

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

   if 0.999996483 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.6%

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

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

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

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

      6. distribute-neg-out N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lift--.f32 N/A

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

      11. sub-flip N/A

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

      12. associate-+l+ N/A

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

      13. associate--r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

      15. sub-negate N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

      16. --rgt-identity N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 79.7%

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

   6. Applied rewrites 79.7%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-rgt-out-- N/A

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

      10. add-flip N/A

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

      11. +-commutative N/A

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

      12. sub-flip N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 79.7%

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

   8. Applied rewrites 79.7%

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

Alternative 8: 89.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left(uy \cdot 2\right) \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq 0.9999399781227112:\\ \;\;\;\;t\_0 \cdot \sqrt{ux \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot \left(ux \cdot \left(1 - maxCos\right) - 2\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (cos (* (* uy 2.0) PI))))
  (if (<= t_0 0.9999399781227112)
    (* t_0 (sqrt (* ux 2.0)))
    (sqrt (* (* (- maxCos 1.0) ux) (- (* ux (- 1.0 maxCos)) 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = cosf(((uy * 2.0f) * ((float) M_PI)));
	float tmp;
	if (t_0 <= 0.9999399781227112f) {
		tmp = t_0 * sqrtf((ux * 2.0f));
	} else {
		tmp = sqrtf((((maxCos - 1.0f) * ux) * ((ux * (1.0f - maxCos)) - 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999399781227112))
		tmp = Float32(t_0 * sqrt(Float32(ux * Float32(2.0))));
	else
		tmp = sqrt(Float32(Float32(Float32(maxCos - Float32(1.0)) * ux) * Float32(Float32(ux * Float32(Float32(1.0) - maxCos)) - Float32(2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = cos(((uy * single(2.0)) * single(pi)));
	tmp = single(0.0);
	if (t_0 <= single(0.9999399781227112))
		tmp = t_0 * sqrt((ux * single(2.0)));
	else
		tmp = sqrt((((maxCos - single(1.0)) * ux) * ((ux * (single(1.0) - maxCos)) - single(2.0))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) < 0.999939978

   1. Initial program 57.6%

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

   2. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 N/A

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

      3. lower-*.f32 76.4%

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in maxCos around 0

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

   7. Applied rewrites 72.8%

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

   if 0.999939978 < (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32)))

   1. Initial program 57.6%

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

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

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

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

      6. distribute-neg-out N/A

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

      7. sub-flip N/A

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

      8. sub-negate-rev N/A

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

      9. lift-+.f32 N/A

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

      10. lift--.f32 N/A

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

      11. sub-flip N/A

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

      12. associate-+l+ N/A

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

      13. associate--r+ N/A

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

      14. metadata-eval N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

      15. sub-negate N/A

          \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

      16. --rgt-identity N/A

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

   3. Applied rewrites 98.9%

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

   4. Taylor expanded in uy around 0

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

      1. lower-sqrt.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 79.7%

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

   6. Applied rewrites 79.7%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. add-flip N/A

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

      5. mul-1-neg N/A

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

      6. metadata-eval N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. distribute-rgt-out-- N/A

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

      10. add-flip N/A

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

      11. +-commutative N/A

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

      12. sub-flip N/A

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

      13. lift--.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 79.7%

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

   8. Applied rewrites 79.7%

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

Alternative 9: 79.7% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt((((maxcos - 1.0e0) * ux) * ((ux * (1.0e0 - maxcos)) - 2.0e0)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 79.7%

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

6. Applied rewrites 79.7%

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

   1. lift--.f32 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. add-flip N/A

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

   5. mul-1-neg N/A

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

   6. metadata-eval N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-out N/A

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

   9. distribute-rgt-out-- N/A

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

   10. add-flip N/A

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

   11. +-commutative N/A

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

   12. sub-flip N/A

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

   13. lift--.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 79.7%

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

8. Applied rewrites 79.7%

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

Alternative 10: 79.7% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 79.7%

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

6. Applied rewrites 79.7%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift--.f32 N/A

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

   5. sub-flip N/A

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

   6. +-commutative N/A

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

   7. distribute-rgt1-in N/A

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

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

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

   9. lift-*.f32 N/A

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

   10. associate--l- N/A

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

   11. lower--.f32 N/A

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

   12. lift-*.f32 N/A

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

   13. lower-fma.f32 79.7%

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

8. Applied rewrites 79.7%

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

   1. remove-double-neg N/A

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

   2. lift--.f32 N/A

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

   3. sub-negate-rev N/A

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

   4. sub-flip N/A

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

   5. distribute-neg-in N/A

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

   6. lift-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. distribute-lft-neg-out N/A

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

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

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

   10. lift-neg.f32 N/A

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

   11. distribute-rgt1-in N/A

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

   12. +-commutative N/A

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

   13. sub-flip N/A

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

   14. lift--.f32 N/A

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

   15. lift-neg.f32 N/A

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

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

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

   17. distribute-lft-neg-out N/A

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

   18. lift--.f32 N/A

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

   19. sub-negate-rev N/A

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

   20. lift--.f32 N/A

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

   21. lower-*.f32 79.7%

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

10. Applied rewrites 79.7%

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

Alternative 11: 75.3% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(((-1.0e0) * (ux * (ux - 2.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 79.7%

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

6. Applied rewrites 79.7%

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

7. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 75.3%

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

9. Applied rewrites 75.3%

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

Alternative 12: 64.3% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(4) function code(ux, uy, maxcos)
use fmin_fmax_functions
    real(4), intent (in) :: ux
    real(4), intent (in) :: uy
    real(4), intent (in) :: maxcos
    code = sqrt(((-2.0e0) * (ux * (maxcos - 1.0e0))))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 79.7%

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

6. Applied rewrites 79.7%

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

7. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 64.3%

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

9. Applied rewrites 64.3%

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

Alternative 13: 21.5% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

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(((uy + uy) - (uy * uy)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 79.7%

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

6. Applied rewrites 79.7%

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

8. Applied rewrites 21.5%

   \[\leadsto \sqrt{\left(uy + uy\right) - uy \cdot uy} \]
9. Add Preprocessing

Alternative 14: 21.5% accurate, 7.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 79.7%

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

6. Applied rewrites 79.7%

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

8. Applied rewrites 21.5%

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

Alternative 15: 21.5% accurate, 11.3× speedup

Math

\[\sqrt{2 \cdot uy} \]

FPCore

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

C

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

Fortran

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

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

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

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

   6. distribute-neg-out N/A

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

   7. sub-flip N/A

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

   8. sub-negate-rev N/A

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

   9. lift-+.f32 N/A

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

   10. lift--.f32 N/A

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

   11. sub-flip N/A

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

   12. associate-+l+ N/A

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

   13. associate--r+ N/A

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

   14. metadata-eval N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \left(\mathsf{neg}\left(\left(\color{blue}{0} - \left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right)\right)\right)\right)} \]

   15. sub-negate N/A

       \[\leadsto \cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\left(\left(\mathsf{neg}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right) - 1\right) \cdot \color{blue}{\left(\left(\left(\mathsf{neg}\left(ux\right)\right) + ux \cdot maxCos\right) - 0\right)}} \]

   16. --rgt-identity N/A

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

3. Applied rewrites 98.9%

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

4. Taylor expanded in uy around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 79.7%

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

6. Applied rewrites 79.7%

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

8. Applied rewrites 21.5%

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

9. Taylor expanded in uy around 0

   \[\leadsto \sqrt{2 \cdot uy} \]
10. Step-by-step derivation

    1. lower-sqrt.f32 N/A

       \[\leadsto \sqrt{2 \cdot uy} \]

    2. lower-*.f32 21.5%

       \[\leadsto \sqrt{2 \cdot uy} \]

11. Applied rewrites 21.5%

    \[\leadsto \sqrt{2 \cdot uy} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025322 
(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)))))))