UniformSampleCone, x

Percentage Accurate: 56.7% → 99.1%

Time: 6.7s

Alternatives: 16

Speedup: 6.2×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 56.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 99.1

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

8. Applied rewrites 99.1%

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

10. Applied rewrites 99.1%

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

    1. lift-*.f32 N/A

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

    2. count-2-rev N/A

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

    3. lower-+.f32 99.1

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

12. Applied rewrites 99.1%

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

Alternative 2: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\cos \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \leq 0.01810000091791153:\\ \;\;\;\;1 \cdot \sqrt{1 - \mathsf{fma}\left(\left(maxCos + maxCos\right) - 2, ux, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<=
        (* (cos (* (* uy 2.0) PI)) (sqrt (- 1.0 (* t_0 t_0))))
        0.01810000091791153)
     (* 1.0 (sqrt (- 1.0 (fma (- (+ maxCos maxCos) 2.0) ux 1.0))))
     (* 1.0 (sqrt (- 1.0 (* t_0 (- 1.0 ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((cosf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((1.0f - (t_0 * t_0)))) <= 0.01810000091791153f) {
		tmp = 1.0f * sqrtf((1.0f - fmaf(((maxCos + maxCos) - 2.0f), ux, 1.0f)));
	} else {
		tmp = 1.0f * sqrtf((1.0f - (t_0 * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(cos(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0)))) <= Float32(0.01810000091791153))
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - fma(Float32(Float32(maxCos + maxCos) - Float32(2.0)), ux, Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(t_0 * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))) < 0.0181000009

   1. Initial program 36.6%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in ux around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 73.9

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

   8. Applied rewrites 73.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 73.9

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

   10. Applied rewrites 73.9%

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

   if 0.0181000009 < (*.f32 (cos.f32 (*.f32 (*.f32 uy #s(literal 2 binary32)) (PI.f32))) (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))))

   1. Initial program 88.4%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 77.4%

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

   6. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. lift--.f32 74.1

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

   8. Applied rewrites 74.1%

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

Alternative 3: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. lower-+.f32 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f32 N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. mul-1-neg N/A

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

   10. lift-neg.f32 98.3

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

8. Applied rewrites 98.3%

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

9. Final simplification 98.3%

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

Alternative 4: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower--.f32 98.4

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 97.5%

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

    1. lift--.f32 N/A

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

    2. *-lft-identity N/A

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

    3. metadata-eval N/A

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

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

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

    5. +-commutative N/A

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

    6. *-commutative N/A

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

    7. lower-fma.f32 97.5

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

13. Applied rewrites 97.5%

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

14. Final simplification 97.5%

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

Alternative 5: 97.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in maxCos around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f32 N/A

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

   4. lower--.f32 99.1

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

8. Applied rewrites 99.1%

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

10. Applied rewrites 99.1%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 97.5%

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

Alternative 6: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in maxCos around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f32 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

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

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower--.f32 98.4

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in ux around 0

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

11. Applied rewrites 97.5%

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

12. Final simplification 97.5%

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

Alternative 7: 97.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in maxCos around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 97.4

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

8. Applied rewrites 97.4%

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

9. Final simplification 97.4%

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

Alternative 8: 92.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. associate-*r* N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-neg.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in maxCos around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. lower--.f32 92.9

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

8. Applied rewrites 92.9%

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

9. Final simplification 92.9%

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

Alternative 9: 78.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ \mathbf{if}\;ux \leq 1.4999999621068127 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot \sqrt{1 - \mathsf{fma}\left(\frac{\left(2 \cdot maxCos\right) \cdot \left(2 \cdot maxCos\right) - 4}{\mathsf{fma}\left(2, maxCos, 2\right)}, ux, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), uy \cdot uy, 1\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma maxCos ux (- 1.0 ux))))
   (if (<= ux 1.4999999621068127e-5)
     (*
      1.0
      (sqrt
       (-
        1.0
        (fma
         (/ (- (* (* 2.0 maxCos) (* 2.0 maxCos)) 4.0) (fma 2.0 maxCos 2.0))
         ux
         1.0))))
     (*
      (fma
       (fma
        (* 0.6666666666666666 (* uy uy))
        (* (* PI PI) (* PI PI))
        (* (* PI PI) -2.0))
       (* uy uy)
       1.0)
      (sqrt (- 1.0 (* t_0 t_0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, (1.0f - ux));
	float tmp;
	if (ux <= 1.4999999621068127e-5f) {
		tmp = 1.0f * sqrtf((1.0f - fmaf(((((2.0f * maxCos) * (2.0f * maxCos)) - 4.0f) / fmaf(2.0f, maxCos, 2.0f)), ux, 1.0f)));
	} else {
		tmp = fmaf(fmaf((0.6666666666666666f * (uy * uy)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (uy * uy), 1.0f) * sqrtf((1.0f - (t_0 * t_0)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = fma(maxCos, ux, Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (ux <= Float32(1.4999999621068127e-5))
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - fma(Float32(Float32(Float32(Float32(Float32(2.0) * maxCos) * Float32(Float32(2.0) * maxCos)) - Float32(4.0)) / fma(Float32(2.0), maxCos, Float32(2.0))), ux, Float32(1.0)))));
	else
		tmp = Float32(fma(fma(Float32(Float32(0.6666666666666666) * Float32(uy * uy)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(uy * uy), Float32(1.0)) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.49999996e-5

   1. Initial program 29.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 28.0%

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

   6. Taylor expanded in ux around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 79.8

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

   8. Applied rewrites 79.8%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. flip-- N/A

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-fma.f32 79.9

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

   10. Applied rewrites 79.9%

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

   if 1.49999996e-5 < ux

   1. Initial program 84.5%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift--.f32 84.5

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lift--.f32 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   7. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(uy \cdot uy\right), {\pi}^{4}, \left(\pi \cdot \pi\right) \cdot -2\right), uy \cdot uy, 1\right)} \cdot \sqrt{1 - \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \]
   8. Step-by-step derivation

      1. lift-PI.f32 N/A

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

      2. lift-pow.f32 N/A

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

      3. sqr-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-*.f32 N/A

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

      7. pow2 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-PI.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. pow2 N/A

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

      12. lift-*.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. lift-PI.f32 78.6

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), uy \cdot uy, 1\right) \cdot \sqrt{1 - \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \]

   9. Applied rewrites 78.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(uy \cdot uy\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), uy \cdot uy, 1\right) \cdot \sqrt{1 - \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 75.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_1 \cdot t\_1 \leq 0.9997000098228455:\\ \;\;\;\;1 \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{1 - \mathsf{fma}\left(\left(maxCos + maxCos\right) - 2, ux, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma maxCos ux (- 1.0 ux))) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_1 t_1) 0.9997000098228455)
     (* 1.0 (sqrt (- 1.0 (* t_0 t_0))))
     (* 1.0 (sqrt (- 1.0 (fma (- (+ maxCos maxCos) 2.0) ux 1.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, (1.0f - ux));
	float t_1 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_1 * t_1) <= 0.9997000098228455f) {
		tmp = 1.0f * sqrtf((1.0f - (t_0 * t_0)));
	} else {
		tmp = 1.0f * sqrtf((1.0f - fmaf(((maxCos + maxCos) - 2.0f), ux, 1.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = fma(maxCos, ux, Float32(Float32(1.0) - ux))
	t_1 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (Float32(t_1 * t_1) <= Float32(0.9997000098228455))
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - fma(Float32(Float32(maxCos + maxCos) - Float32(2.0)), ux, Float32(1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99970001

   1. Initial program 88.0%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 73.7%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-fma.f32 N/A

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

      7. lift--.f32 73.8

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. lift--.f32 73.8

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

   7. Applied rewrites 73.8%

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

   if 0.99970001 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 34.5%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in ux around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 76.4

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

   8. Applied rewrites 76.4%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 76.4

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

   10. Applied rewrites 76.4%

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

Alternative 11: 77.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(maxCos, ux, 1 - ux\right)\\ \mathbf{if}\;ux \leq 1.4999999621068127 \cdot 10^{-5}:\\ \;\;\;\;1 \cdot \sqrt{1 - \mathsf{fma}\left(\frac{\left(2 \cdot maxCos\right) \cdot \left(2 \cdot maxCos\right) - 4}{\mathsf{fma}\left(2, maxCos, 2\right)}, ux, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot -2, uy \cdot uy, 1\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (fma maxCos ux (- 1.0 ux))))
   (if (<= ux 1.4999999621068127e-5)
     (*
      1.0
      (sqrt
       (-
        1.0
        (fma
         (/ (- (* (* 2.0 maxCos) (* 2.0 maxCos)) 4.0) (fma 2.0 maxCos 2.0))
         ux
         1.0))))
     (* (fma (* (* PI PI) -2.0) (* uy uy) 1.0) (sqrt (- 1.0 (* t_0 t_0)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = fmaf(maxCos, ux, (1.0f - ux));
	float tmp;
	if (ux <= 1.4999999621068127e-5f) {
		tmp = 1.0f * sqrtf((1.0f - fmaf(((((2.0f * maxCos) * (2.0f * maxCos)) - 4.0f) / fmaf(2.0f, maxCos, 2.0f)), ux, 1.0f)));
	} else {
		tmp = fmaf(((((float) M_PI) * ((float) M_PI)) * -2.0f), (uy * uy), 1.0f) * sqrtf((1.0f - (t_0 * t_0)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = fma(maxCos, ux, Float32(Float32(1.0) - ux))
	tmp = Float32(0.0)
	if (ux <= Float32(1.4999999621068127e-5))
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - fma(Float32(Float32(Float32(Float32(Float32(2.0) * maxCos) * Float32(Float32(2.0) * maxCos)) - Float32(4.0)) / fma(Float32(2.0), maxCos, Float32(2.0))), ux, Float32(1.0)))));
	else
		tmp = Float32(fma(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0)), Float32(uy * uy), Float32(1.0)) * sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.49999996e-5

   1. Initial program 29.3%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 28.0%

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

   6. Taylor expanded in ux around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 79.8

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

   8. Applied rewrites 79.8%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. flip-- N/A

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-fma.f32 79.9

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

   10. Applied rewrites 79.9%

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

   if 1.49999996e-5 < ux

   1. Initial program 84.5%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. lift--.f32 84.5

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

      8. lift-+.f32 N/A

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

      9. lift--.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. lift--.f32 84.5

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in uy around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f32 N/A

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

   7. Applied rewrites 78.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(uy \cdot uy\right), {\pi}^{4}, \left(\pi \cdot \pi\right) \cdot -2\right), uy \cdot uy, 1\right)} \cdot \sqrt{1 - \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \]

   8. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. pow2 N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-*.f32 75.9

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

   10. Applied rewrites 75.9%

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

Alternative 12: 76.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;1 \cdot \sqrt{1 - \mathsf{fma}\left(\frac{\left(2 \cdot maxCos\right) \cdot \left(2 \cdot maxCos\right) - 4}{\mathsf{fma}\left(2, maxCos, 2\right)}, ux, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \mathsf{fma}\left(maxCos - 1, ux, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (*
    1.0
    (sqrt
     (-
      1.0
      (fma
       (/ (- (* (* 2.0 maxCos) (* 2.0 maxCos)) 4.0) (fma 2.0 maxCos 2.0))
       ux
       1.0))))
   (*
    1.0
    (sqrt
     (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (fma (- maxCos 1.0) ux 1.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = 1.0f * sqrtf((1.0f - fmaf(((((2.0f * maxCos) * (2.0f * maxCos)) - 4.0f) / fmaf(2.0f, maxCos, 2.0f)), ux, 1.0f)));
	} else {
		tmp = 1.0f * sqrtf((1.0f - (((1.0f - ux) + (ux * maxCos)) * fmaf((maxCos - 1.0f), ux, 1.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - fma(Float32(Float32(Float32(Float32(Float32(2.0) * maxCos) * Float32(Float32(2.0) * maxCos)) - Float32(4.0)) / fma(Float32(2.0), maxCos, Float32(2.0))), ux, Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) * fma(Float32(maxCos - Float32(1.0)), ux, Float32(1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 34.0%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in ux around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 76.7

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

   8. Applied rewrites 76.7%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. flip-- N/A

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

      5. lower-/.f32 N/A

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

      6. metadata-eval N/A

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

      7. lower--.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-fma.f32 76.7

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

   10. Applied rewrites 76.7%

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

   if 1.10000001e-4 < ux

   1. Initial program 87.6%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift--.f32 74.3

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

   8. Applied rewrites 74.3%

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

Alternative 13: 76.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011000000085914508:\\ \;\;\;\;1 \cdot \sqrt{1 - \mathsf{fma}\left(\left(maxCos + maxCos\right) - 2, ux, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \mathsf{fma}\left(maxCos - 1, ux, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011000000085914508)
   (* 1.0 (sqrt (- 1.0 (fma (- (+ maxCos maxCos) 2.0) ux 1.0))))
   (*
    1.0
    (sqrt
     (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (fma (- maxCos 1.0) ux 1.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00011000000085914508f) {
		tmp = 1.0f * sqrtf((1.0f - fmaf(((maxCos + maxCos) - 2.0f), ux, 1.0f)));
	} else {
		tmp = 1.0f * sqrtf((1.0f - (((1.0f - ux) + (ux * maxCos)) * fmaf((maxCos - 1.0f), ux, 1.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00011000000085914508))
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - fma(Float32(Float32(maxCos + maxCos) - Float32(2.0)), ux, Float32(1.0)))));
	else
		tmp = Float32(Float32(1.0) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos)) * fma(Float32(maxCos - Float32(1.0)), ux, Float32(1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.10000001e-4

   1. Initial program 34.0%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 31.0%

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

   6. Taylor expanded in ux around 0

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

      1. pow2 N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 N/A

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

      8. lower--.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 76.7

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

   8. Applied rewrites 76.7%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 76.7

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

   10. Applied rewrites 76.7%

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

   if 1.10000001e-4 < ux

   1. Initial program 87.6%

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

   3. Taylor expanded in uy around 0

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 N/A

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

      6. lift--.f32 74.3

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

   8. Applied rewrites 74.3%

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

Alternative 14: 65.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 49.0%

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

6. Taylor expanded in ux around 0

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

   1. pow2 N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. pow2 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 64.8

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

8. Applied rewrites 64.8%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 64.8

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

10. Applied rewrites 64.8%

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

Alternative 15: 62.5% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 49.0%

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

6. Taylor expanded in ux around 0

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

   1. pow2 N/A

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

   2. *-commutative N/A

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

   3. +-commutative N/A

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

   4. pow2 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f32 N/A

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

   8. lower--.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 64.8

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

8. Applied rewrites 64.8%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 62.5%

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

Alternative 16: 6.6% accurate, 8.2× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Julia

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

MATLAB

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

Derivation

1. Initial program 56.9%

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

3. Taylor expanded in uy around 0

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

5. Applied rewrites 49.0%

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

6. Taylor expanded in ux around 0

   \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{1}} \]
7. Step-by-step derivation

   1. pow2 6.6

      \[\leadsto 1 \cdot \sqrt{1 - 1} \]

   2. *-commutative 6.6

      \[\leadsto 1 \cdot \sqrt{1 - 1} \]

   3. +-commutative 6.6

      \[\leadsto 1 \cdot \sqrt{1 - 1} \]

   4. pow2 6.6

      \[\leadsto 1 \cdot \sqrt{1 - 1} \]

8. Applied rewrites 6.6%

   \[\leadsto 1 \cdot \sqrt{1 - \color{blue}{1}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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)))))))