UniformSampleCone, y

Percentage Accurate: 57.4% → 98.3%

Time: 7.7s

Alternatives: 18

Speedup: 4.5×

Specification

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

Math

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

FPCore

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

Initial Program: 57.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. --rgt-identity 98.3%

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

5. Applied rewrites 98.3%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-negate N/A

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

   4. sub-flip-reverse N/A

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

   5. lift--.f32 N/A

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

   6. sub-negate-rev N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. add-flip-rev N/A

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

   10. lower--.f32 N/A

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

   11. metadata-eval 98.3%

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

7. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*l* N/A

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

   4. count-2 N/A

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

   5. lift-+.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 98.3%

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

9. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. --rgt-identity 98.3%

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

5. Applied rewrites 98.3%

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

   1. lift-neg.f32 N/A

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

   2. lift--.f32 N/A

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

   3. sub-negate N/A

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

   4. sub-flip-reverse N/A

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

   5. lift--.f32 N/A

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

   6. sub-negate-rev N/A

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

   7. lift--.f32 N/A

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

   8. +-commutative N/A

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

   9. add-flip-rev N/A

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

   10. lower--.f32 N/A

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

   11. metadata-eval 98.3%

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

7. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate--l- N/A

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

   4. sub-flip N/A

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

   5. add-flip N/A

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

   6. metadata-eval N/A

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

   7. sub-negate-rev N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 N/A

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

   11. lower--.f32 98.3%

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

9. Applied rewrites 98.3%

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

Alternative 3: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 98.3%

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

5. Applied rewrites 98.3%

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

Alternative 4: 97.2% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. --rgt-identity 98.3%

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

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

   1. lower--.f32 97.2%

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

8. Applied rewrites 97.2%

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

Alternative 5: 96.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;uy \leq 0.0009500000160187483:\\ \;\;\;\;\left(\sqrt{\left(maxCos \cdot ux - ux\right) \cdot \left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) \cdot \pi, \pi \cdot \pi, \pi + \pi\right)\right) \cdot uy\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= uy 0.0009500000160187483)
  (*
   (*
    (sqrt (* (- (* maxCos ux) ux) (- ux (fma maxCos ux 2.0))))
    (fma
     (* (* (* uy uy) -1.3333333333333333) PI)
     (* PI PI)
     (+ PI PI)))
   uy)
  (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0009500000160187483f) {
		tmp = (sqrtf((((maxCos * ux) - ux) * (ux - fmaf(maxCos, ux, 2.0f)))) * fmaf((((uy * uy) * -1.3333333333333333f) * ((float) M_PI)), (((float) M_PI) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * uy;
	} else {
		tmp = sinf(((uy * 2.0f) * ((float) M_PI))) * sqrtf((ux * (2.0f - ux)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (uy <= Float32(0.0009500000160187483))
		tmp = Float32(Float32(sqrt(Float32(Float32(Float32(maxCos * ux) - ux) * Float32(ux - fma(maxCos, ux, Float32(2.0))))) * fma(Float32(Float32(Float32(uy * uy) * Float32(-1.3333333333333333)) * Float32(pi)), Float32(Float32(pi) * Float32(pi)), Float32(Float32(pi) + Float32(pi)))) * uy);
	else
		tmp = Float32(sin(Float32(Float32(uy * Float32(2.0)) * Float32(pi))) * sqrt(Float32(ux * Float32(Float32(2.0) - ux))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 9.50000016e-4

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

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

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

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

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

      7. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 2 \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

   6. Applied rewrites 89.3%

      \[\leadsto \color{blue}{uy \cdot \mathsf{fma}\left(-1.3333333333333333, {uy}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right), 2 \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)} \]

   8. Applied rewrites 89.2%

      \[\leadsto uy \cdot \mathsf{fma}\left(\left(\pi + \pi\right) \cdot \sqrt{\left|ux - maxCos \cdot ux\right|}, \color{blue}{\sqrt{\left|\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right|}}, \left(\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)}\right) \]

   10. Applied rewrites 89.3%

       \[\leadsto \left(\sqrt{\left(maxCos \cdot ux - ux\right) \cdot \left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) \cdot \pi, \pi \cdot \pi, \pi + \pi\right)\right) \cdot \color{blue}{uy} \]

   if 9.50000016e-4 < uy

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

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

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

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

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

      7. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower--.f32 92.3%

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

   6. Applied rewrites 92.3%

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

Alternative 6: 89.3% accurate, 1.5× speedup

Math

\[uy \cdot \left(\sqrt{\left(maxCos \cdot ux - ux\right) \cdot \left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right)} \cdot \left(\left(\pi + \pi\right) - \left(-uy \cdot uy\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right)\right)\right) \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 uy
 (*
  (sqrt (* (- (* maxCos ux) ux) (- ux (fma maxCos ux 2.0))))
  (-
   (+ PI PI)
   (* (- (* uy uy)) (* (* (* PI PI) PI) -1.3333333333333333))))))

C

float code(float ux, float uy, float maxCos) {
	return uy * (sqrtf((((maxCos * ux) - ux) * (ux - fmaf(maxCos, ux, 2.0f)))) * ((((float) M_PI) + ((float) M_PI)) - (-(uy * uy) * (((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * -1.3333333333333333f))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(uy * Float32(sqrt(Float32(Float32(Float32(maxCos * ux) - ux) * Float32(ux - fma(maxCos, ux, Float32(2.0))))) * Float32(Float32(Float32(pi) + Float32(pi)) - Float32(Float32(-Float32(uy * uy)) * Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(-1.3333333333333333))))))
end

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 2 \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)} \]
5. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

6. Applied rewrites 89.3%

   \[\leadsto \color{blue}{uy \cdot \mathsf{fma}\left(-1.3333333333333333, {uy}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right), 2 \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)} \]

8. Applied rewrites 89.2%

   \[\leadsto uy \cdot \mathsf{fma}\left(\left(\pi + \pi\right) \cdot \sqrt{\left|ux - maxCos \cdot ux\right|}, \color{blue}{\sqrt{\left|\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right|}}, \left(\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)}\right) \]

10. Applied rewrites 89.3%

    \[\leadsto uy \cdot \left(\sqrt{\left(maxCos \cdot ux - ux\right) \cdot \left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right)} \cdot \color{blue}{\left(\left(\pi + \pi\right) - \left(-uy \cdot uy\right) \cdot \left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot -1.3333333333333333\right)\right)}\right) \]
11. Add Preprocessing

Alternative 7: 89.3% accurate, 1.5× speedup

Math

\[\left(\sqrt{\left(maxCos \cdot ux - ux\right) \cdot \left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) \cdot \pi, \pi \cdot \pi, \pi + \pi\right)\right) \cdot uy \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (*
  (sqrt (* (- (* maxCos ux) ux) (- ux (fma maxCos ux 2.0))))
  (fma (* (* (* uy uy) -1.3333333333333333) PI) (* PI PI) (+ PI PI)))
 uy))

C

float code(float ux, float uy, float maxCos) {
	return (sqrtf((((maxCos * ux) - ux) * (ux - fmaf(maxCos, ux, 2.0f)))) * fmaf((((uy * uy) * -1.3333333333333333f) * ((float) M_PI)), (((float) M_PI) * ((float) M_PI)), (((float) M_PI) + ((float) M_PI)))) * uy;
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(sqrt(Float32(Float32(Float32(maxCos * ux) - ux) * Float32(ux - fma(maxCos, ux, Float32(2.0))))) * fma(Float32(Float32(Float32(uy * uy) * Float32(-1.3333333333333333)) * Float32(pi)), Float32(Float32(pi) * Float32(pi)), Float32(Float32(pi) + Float32(pi)))) * uy)
end

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{uy \cdot \left(\frac{-4}{3} \cdot \left({uy}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right) + 2 \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)} \]
5. Step-by-step derivation

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

6. Applied rewrites 89.3%

   \[\leadsto \color{blue}{uy \cdot \mathsf{fma}\left(-1.3333333333333333, {uy}^{2} \cdot \left({\pi}^{3} \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right), 2 \cdot \left(\pi \cdot \sqrt{\left(ux - maxCos \cdot ux\right) \cdot \left(\left(2 + maxCos \cdot ux\right) - ux\right)}\right)\right)} \]

8. Applied rewrites 89.2%

   \[\leadsto uy \cdot \mathsf{fma}\left(\left(\pi + \pi\right) \cdot \sqrt{\left|ux - maxCos \cdot ux\right|}, \color{blue}{\sqrt{\left|\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right|}}, \left(\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot \pi\right)\right) \cdot \sqrt{\left(\mathsf{fma}\left(maxCos, ux, 2\right) - ux\right) \cdot \left(ux - maxCos \cdot ux\right)}\right) \]

10. Applied rewrites 89.3%

    \[\leadsto \left(\sqrt{\left(maxCos \cdot ux - ux\right) \cdot \left(ux - \mathsf{fma}\left(maxCos, ux, 2\right)\right)} \cdot \mathsf{fma}\left(\left(\left(uy \cdot uy\right) \cdot -1.3333333333333333\right) \cdot \pi, \pi \cdot \pi, \pi + \pi\right)\right) \cdot \color{blue}{uy} \]
11. Add Preprocessing

Alternative 8: 81.7% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (*
  (+ uy uy)
  (sqrt (- 0.0 (* (- ux (fma maxCos ux 2.0)) (- ux (* maxCos ux))))))
 PI))

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.5%

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

4. Applied rewrites 50.5%

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

   1. lift-pow.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. sum-square-pow N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower--.f32 53.0%

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

6. Applied rewrites 53.0%

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

8. Applied rewrites 81.6%

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

Alternative 9: 81.6% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (*
 (* PI (+ uy uy))
 (sqrt (- 0.0 (* (- ux (fma maxCos ux 2.0)) (- ux (* maxCos ux)))))))

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.5%

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

4. Applied rewrites 50.5%

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

   1. lift-pow.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. sum-square-pow N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower--.f32 53.0%

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

6. Applied rewrites 53.0%

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

8. Applied rewrites 81.7%

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

Alternative 10: 81.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. sub-negate-rev N/A

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

   3. lift-*.f32 N/A

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

   4. sqr-neg-rev N/A

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

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

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

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

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

   7. lower-*.f32 N/A

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lower--.f32 N/A

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

   10. lower-+.f32 N/A

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

   11. lower-*.f32 81.6%

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

6. Applied rewrites 81.6%

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

Alternative 11: 76.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00017499999376013875:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\left(\pi \cdot \sqrt{\left|ux\right|}\right) \cdot \sqrt{\left|\mathsf{fma}\left(maxCos, 2, -2\right)\right|}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1\right) - ux, ux - 1, 1\right)}\right)\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= ux 0.00017499999376013875)
  (*
   2.0
   (*
    uy
    (* (* PI (sqrt (fabs ux))) (sqrt (fabs (fma maxCos 2.0 -2.0))))))
  (*
   2.0
   (*
    uy
    (* PI (sqrt (fma (- (fma maxCos ux 1.0) ux) (- ux 1.0) 1.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00017499999376013875f) {
		tmp = 2.0f * (uy * ((((float) M_PI) * sqrtf(fabsf(ux))) * sqrtf(fabsf(fmaf(maxCos, 2.0f, -2.0f)))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(fmaf((fmaf(maxCos, ux, 1.0f) - ux), (ux - 1.0f), 1.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00017499999376013875))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(Float32(pi) * sqrt(abs(ux))) * sqrt(abs(fma(maxCos, Float32(2.0), Float32(-2.0)))))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(fma(Float32(fma(maxCos, ux, Float32(1.0)) - ux), Float32(ux - Float32(1.0)), Float32(1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 1.74999994e-4

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 66.2%

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

   7. Applied rewrites 66.2%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. sqrt-prod N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-fabs.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. lift--.f32 N/A

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

      12. fabs-sub N/A

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

      13. lower-fabs.f32 N/A

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

      14. sub-flip N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

      18. metadata-eval 66.2%

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

   9. Applied rewrites 66.2%

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

   if 1.74999994e-4 < ux

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

      1. lift-pow.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. associate--l+ N/A

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

      5. +-commutative N/A

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

      6. sum-square-pow N/A

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

      7. metadata-eval N/A

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

      8. lower-+.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f32 N/A

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

      11. lower--.f32 N/A

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

      12. lower--.f32 N/A

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

      13. lower-*.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower--.f32 53.0%

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

   6. Applied rewrites 53.0%

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

      1. lift--.f32 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

   8. Applied rewrites 50.5%

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

   9. Taylor expanded in maxCos around 0

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

       1. lower--.f32 49.4%

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

   11. Applied rewrites 49.4%

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

Alternative 12: 75.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9994000196456909:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux - 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\left(\pi \cdot \sqrt{\left|ux\right|}\right) \cdot \sqrt{\left|\mathsf{fma}\left(maxCos, 2, -2\right)\right|}\right)\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
  (if (<= (* t_0 t_0) 0.9994000196456909)
    (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (- 1.0 ux) (- ux 1.0)))))))
    (*
     2.0
     (*
      uy
      (*
       (* PI (sqrt (fabs ux)))
       (sqrt (fabs (fma maxCos 2.0 -2.0)))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9994000196456909f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((1.0f - ux) * (ux - 1.0f))))));
	} else {
		tmp = 2.0f * (uy * ((((float) M_PI) * sqrtf(fabsf(ux))) * sqrtf(fabsf(fmaf(maxCos, 2.0f, -2.0f)))));
	}
	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(t_0 * t_0) <= Float32(0.9994000196456909))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux - Float32(1.0))))))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(Float32(pi) * sqrt(abs(ux))) * sqrt(abs(fma(maxCos, Float32(2.0), Float32(-2.0)))))));
	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))) < 0.99940002

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower--.f32 49.2%

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

   9. Applied rewrites 49.2%

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

   if 0.99940002 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 66.2%

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

   7. Applied rewrites 66.2%

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

      1. lift-*.f32 N/A

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

      2. lift-sqrt.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. sqrt-prod N/A

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

      5. associate-*r* N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-fabs.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

      11. lift--.f32 N/A

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

      12. fabs-sub N/A

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

      13. lower-fabs.f32 N/A

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

      14. sub-flip N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f32 N/A

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

      18. metadata-eval 66.2%

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

   9. Applied rewrites 66.2%

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

Alternative 13: 75.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;ux \leq 0.00031999999191612005:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\sqrt{\left|ux\right|} \cdot \left(\sqrt{\left|\mathsf{fma}\left(maxCos, 2, -2\right)\right|} \cdot \pi\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux - 1\right)}\right)\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (if (<= ux 0.00031999999191612005)
  (*
   2.0
   (*
    uy
    (* (sqrt (fabs ux)) (* (sqrt (fabs (fma maxCos 2.0 -2.0))) PI))))
  (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (- 1.0 ux) (- ux 1.0)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.00031999999191612005f) {
		tmp = 2.0f * (uy * (sqrtf(fabsf(ux)) * (sqrtf(fabsf(fmaf(maxCos, 2.0f, -2.0f))) * ((float) M_PI))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((1.0f - ux) * (ux - 1.0f))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.00031999999191612005))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(sqrt(abs(ux)) * Float32(sqrt(abs(fma(maxCos, Float32(2.0), Float32(-2.0)))) * Float32(pi)))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux - Float32(1.0))))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.19999992e-4

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 66.2%

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

   7. Applied rewrites 66.2%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lift-sqrt.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. sqrt-prod N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f32 N/A

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

      8. lower-sqrt.f32 N/A

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

      9. lower-fabs.f32 N/A

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

      10. lower-*.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lift--.f32 N/A

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

      13. fabs-sub N/A

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

      14. lower-fabs.f32 N/A

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

      15. sub-flip N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

      18. lower-fma.f32 N/A

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

      19. metadata-eval 66.2%

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

   9. Applied rewrites 66.2%

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

   if 3.19999992e-4 < ux

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower--.f32 49.2%

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

   9. Applied rewrites 49.2%

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

Alternative 14: 75.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9994000196456909:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux - 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \left(\sqrt{\left|\mathsf{fma}\left(maxCos, 2, -2\right)\right|} \cdot \sqrt{\left|ux\right|}\right)\right)\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
  (if (<= (* t_0 t_0) 0.9994000196456909)
    (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (- 1.0 ux) (- ux 1.0)))))))
    (*
     2.0
     (*
      uy
      (*
       PI
       (* (sqrt (fabs (fma maxCos 2.0 -2.0))) (sqrt (fabs ux)))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9994000196456909f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((1.0f - ux) * (ux - 1.0f))))));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * (sqrtf(fabsf(fmaf(maxCos, 2.0f, -2.0f))) * sqrtf(fabsf(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(t_0 * t_0) <= Float32(0.9994000196456909))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux - Float32(1.0))))))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * Float32(sqrt(abs(fma(maxCos, Float32(2.0), Float32(-2.0)))) * sqrt(abs(ux))))));
	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))) < 0.99940002

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower--.f32 49.2%

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

   9. Applied rewrites 49.2%

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

   if 0.99940002 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 66.2%

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

   7. Applied rewrites 66.2%

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

      1. lift-sqrt.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. sqrt-prod N/A

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

      5. lower-*.f32 N/A

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

      6. lower-sqrt.f32 N/A

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

      7. lift--.f32 N/A

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

      8. fabs-sub N/A

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

      9. lower-fabs.f32 N/A

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

      10. sub-flip N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f32 N/A

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

      14. metadata-eval N/A

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

      15. lower-sqrt.f32 N/A

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

      16. lower-fabs.f32 66.2%

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

   9. Applied rewrites 66.2%

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

Alternative 15: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9994000196456909:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{1 + \left(1 - ux\right) \cdot \left(ux - 1\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy + uy\right) \cdot \left(\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (ux uy maxCos)
  :precision binary32
  (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
  (if (<= (* t_0 t_0) 0.9994000196456909)
    (* 2.0 (* uy (* PI (sqrt (+ 1.0 (* (- 1.0 ux) (- ux 1.0)))))))
    (* (+ uy uy) (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) PI)))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if ((t_0 * t_0) <= 0.9994000196456909f) {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf((1.0f + ((1.0f - ux) * (ux - 1.0f))))));
	} else {
		tmp = (uy + uy) * (sqrtf((fmaf(-2.0f, maxCos, 2.0f) * ux)) * ((float) M_PI));
	}
	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(t_0 * t_0) <= Float32(0.9994000196456909))
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(1.0) + Float32(Float32(Float32(1.0) - ux) * Float32(ux - Float32(1.0))))))));
	else
		tmp = Float32(Float32(uy + uy) * Float32(sqrt(Float32(fma(Float32(-2.0), maxCos, Float32(2.0)) * ux)) * Float32(pi)));
	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))) < 0.99940002

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 N/A

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

      4. lower-sqrt.f32 N/A

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

      5. lower-+.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 N/A

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

      8. lower--.f32 49.2%

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

   9. Applied rewrites 49.2%

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

   if 0.99940002 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 57.4%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower--.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower--.f32 N/A

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

      9. lower-+.f32 N/A

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

      10. lower-*.f32 50.5%

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

   4. Applied rewrites 50.5%

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

   5. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-PI.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 66.2%

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

   7. Applied rewrites 66.2%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. lower-*.f32 66.2%

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lower-+.f32 66.2%

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 66.2%

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

   9. Applied rewrites 66.2%

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

Alternative 16: 66.2% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.5%

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 66.2%

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

7. Applied rewrites 66.2%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lower-*.f32 66.2%

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lower-+.f32 66.2%

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 66.2%

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

9. Applied rewrites 66.2%

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

Alternative 17: 63.5% accurate, 4.5× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.5%

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

4. Applied rewrites 50.5%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower-PI.f32 N/A

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

   3. lower-sqrt.f32 N/A

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

   4. lower-*.f32 N/A

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

   5. lower--.f32 N/A

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

   6. lower-*.f32 66.2%

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

7. Applied rewrites 66.2%

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

8. Taylor expanded in maxCos around 0

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

   1. lower-*.f32 63.5%

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

10. Applied rewrites 63.5%

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

Alternative 18: 7.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. lower-PI.f32 N/A

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

   5. lower-sqrt.f32 N/A

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

   6. lower--.f32 N/A

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

   7. lower-pow.f32 N/A

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

   8. lower--.f32 N/A

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

   9. lower-+.f32 N/A

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

   10. lower-*.f32 50.5%

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

4. Applied rewrites 50.5%

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

   1. lift-pow.f32 N/A

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

   2. lift--.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. associate--l+ N/A

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

   5. +-commutative N/A

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

   6. sum-square-pow N/A

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

   7. metadata-eval N/A

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

   8. lower-+.f32 N/A

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

   9. unpow2 N/A

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

   10. lower-fma.f32 N/A

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

   11. lower--.f32 N/A

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

   12. lower--.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. lower-*.f32 N/A

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

   15. lower--.f32 53.0%

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

6. Applied rewrites 53.0%

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

7. Taylor expanded in ux around 0

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

9. Applied rewrites 7.1%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

11. Applied rewrites 7.1%

    \[\leadsto \left(uy + uy\right) \cdot \color{blue}{\left(\sqrt{1 - 1} \cdot \pi\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025323 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :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)))
  (* (sin (* (* uy 2.0) PI)) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))