UniformSampleCone, y

Percentage Accurate: 57.9% → 98.3%

Time: 8.8s

Alternatives: 11

Speedup: 3.3×

Specification

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

Math

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

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: 57.9% accurate, 1.0× speedup

Math

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

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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.9

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around inf

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-/.f32 N/A

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

   4. lower-+.f32 N/A

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

   5. lower-*.f32 N/A

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

   6. lower-/.f32 97.9

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

9. Applied rewrites 97.9%

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

Alternative 2: 98.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((-((ux - (maxCos * ux)) - 0.0f) * ((1.0f - (fmaf(maxCos, ux, 2.0f) / ux)) * ux))) * sinf((((float) M_PI) * (uy + uy)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(-Float32(Float32(ux - Float32(maxCos * ux)) - Float32(0.0))) * Float32(Float32(Float32(1.0) - Float32(fma(maxCos, ux, Float32(2.0)) / ux)) * ux))) * sin(Float32(Float32(pi) * Float32(uy + uy))))
end

Derivation

1. Initial program 57.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.9

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

3. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate--l- N/A

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

   4. sub-to-mult N/A

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

   5. lower-special-*.f32 N/A

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

   6. lower-special--.f32 N/A

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

   7. lower-special-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lower-fma.f32 98.3

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

5. Applied rewrites 98.3%

   \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \color{blue}{\left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)}} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right) \]
6. Add Preprocessing

Alternative 3: 97.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.9

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3

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

6. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. sub-flip N/A

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

   3. lift--.f32 N/A

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

   4. sub-negate-rev N/A

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

   5. *-lft-identity N/A

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

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

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

   7. lift-*.f32 N/A

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

   8. metadata-eval N/A

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

   9. distribute-rgt-in N/A

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

   10. metadata-eval N/A

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

   11. sub-flip N/A

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

   12. lift--.f32 N/A

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

   13. *-commutative N/A

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

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

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

   15. lift--.f32 N/A

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

   16. sub-negate-rev N/A

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

   17. lower-fma.f32 N/A

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

   18. lower--.f32 N/A

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

   19. metadata-eval 98.3

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

8. Applied rewrites 98.3%

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

Alternative 4: 97.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.9

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

3. Applied rewrites 98.3%

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

4. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 98.3

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

6. Applied rewrites 98.3%

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

7. Taylor expanded in maxCos around 0

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

   1. lower--.f32 97.1

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

9. Applied rewrites 97.1%

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

Alternative 5: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \leq 0.0003100000030826777:\\ \;\;\;\;\sqrt{\left(-\left(maxCos \cdot \left(\frac{ux}{maxCos} - ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= uy 0.0003100000030826777)
   (*
    (sqrt
     (*
      (- (- (* maxCos (- (/ ux maxCos) ux)) 0.0))
      (* (- 1.0 (/ (fma maxCos ux 2.0) ux)) ux)))
    (* 2.0 (* uy PI)))
   (* (sin (* (* uy 2.0) PI)) (sqrt (* ux (- 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (uy <= 0.0003100000030826777f) {
		tmp = sqrtf((-((maxCos * ((ux / maxCos) - ux)) - 0.0f) * ((1.0f - (fmaf(maxCos, ux, 2.0f) / ux)) * ux))) * (2.0f * (uy * ((float) M_PI)));
	} 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(0.0003100000030826777))
		tmp = Float32(sqrt(Float32(Float32(-Float32(Float32(maxCos * Float32(Float32(ux / maxCos) - ux)) - Float32(0.0))) * Float32(Float32(Float32(1.0) - Float32(fma(maxCos, ux, Float32(2.0)) / ux)) * ux))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	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 < 3.10000003e-4

   1. Initial program 57.9%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.9

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

   3. Applied rewrites 98.3%

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

      1. lift--.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate--l- N/A

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

      4. sub-to-mult N/A

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

      5. lower-special-*.f32 N/A

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

      6. lower-special--.f32 N/A

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

      7. lower-special-/.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lower-fma.f32 98.3

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in uy around 0

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

      2. lower-*.f32 N/A

         \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

      3. lower-PI.f32 81.6

         \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   8. Applied rewrites 81.6%

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

   9. Taylor expanded in maxCos around inf

      \[\leadsto \sqrt{\left(-\left(\color{blue}{maxCos \cdot \left(\frac{ux}{maxCos} - ux\right)} - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
   10. Step-by-step derivation

       1. lower-*.f32 N/A

          \[\leadsto \sqrt{\left(-\left(maxCos \cdot \color{blue}{\left(\frac{ux}{maxCos} - ux\right)} - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

       2. lower--.f32 N/A

          \[\leadsto \sqrt{\left(-\left(maxCos \cdot \left(\frac{ux}{maxCos} - \color{blue}{ux}\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

       3. lower-/.f32 81.5

          \[\leadsto \sqrt{\left(-\left(maxCos \cdot \left(\frac{ux}{maxCos} - ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   11. Applied rewrites 81.5%

       \[\leadsto \sqrt{\left(-\left(\color{blue}{maxCos \cdot \left(\frac{ux}{maxCos} - ux\right)} - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   if 3.10000003e-4 < uy

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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. metadata-eval N/A

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

      3. lift-*.f32 N/A

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

      4. sqr-neg-rev N/A

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

      5. difference-of-squares N/A

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

      6. +-commutative N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. sub-negate N/A

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

      11. associate-+l- N/A

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

      12. associate-+l- N/A

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

      13. lift--.f32 N/A

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

      14. lift-+.f32 N/A

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

      15. lower-*.f32 N/A

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

   3. Applied rewrites 98.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \pi\right) \cdot \sqrt{\color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 0\right) \cdot \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) - -1\right)}} \]

   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: 81.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(-\left(maxCos \cdot \left(\frac{ux}{maxCos} - ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \end{array} \]

FPCore

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

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((-((maxCos * ((ux / maxCos) - ux)) - 0.0f) * ((1.0f - (fmaf(maxCos, ux, 2.0f) / ux)) * ux))) * (2.0f * (uy * ((float) M_PI)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(Float32(-Float32(Float32(maxCos * Float32(Float32(ux / maxCos) - ux)) - Float32(0.0))) * Float32(Float32(Float32(1.0) - Float32(fma(maxCos, ux, Float32(2.0)) / ux)) * ux))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))))
end

Derivation

1. Initial program 57.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.9

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

3. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate--l- N/A

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

   4. sub-to-mult N/A

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

   5. lower-special-*.f32 N/A

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

   6. lower-special--.f32 N/A

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

   7. lower-special-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lower-fma.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

   \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   3. lower-PI.f32 81.6

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

8. Applied rewrites 81.6%

   \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \]

9. Taylor expanded in maxCos around inf

   \[\leadsto \sqrt{\left(-\left(\color{blue}{maxCos \cdot \left(\frac{ux}{maxCos} - ux\right)} - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
10. Step-by-step derivation

    1. lower-*.f32 N/A

       \[\leadsto \sqrt{\left(-\left(maxCos \cdot \color{blue}{\left(\frac{ux}{maxCos} - ux\right)} - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

    2. lower--.f32 N/A

       \[\leadsto \sqrt{\left(-\left(maxCos \cdot \left(\frac{ux}{maxCos} - \color{blue}{ux}\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

    3. lower-/.f32 81.5

       \[\leadsto \sqrt{\left(-\left(maxCos \cdot \left(\frac{ux}{maxCos} - ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

11. Applied rewrites 81.5%

    \[\leadsto \sqrt{\left(-\left(\color{blue}{maxCos \cdot \left(\frac{ux}{maxCos} - ux\right)} - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]
12. Add Preprocessing

Alternative 7: 81.6% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.9

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

3. Applied rewrites 98.3%

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

   1. lift--.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate--l- N/A

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

   4. sub-to-mult N/A

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

   5. lower-special-*.f32 N/A

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

   6. lower-special--.f32 N/A

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

   7. lower-special-/.f32 N/A

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

   8. lift-*.f32 N/A

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

   9. lower-fma.f32 98.3

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

   \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right)\right)}\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) \]

   3. lower-PI.f32 81.6

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

8. Applied rewrites 81.6%

   \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)} \cdot \color{blue}{\left(2 \cdot \left(uy \cdot \pi\right)\right)} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)}} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \color{blue}{\left(\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right) \cdot ux\right)}} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   3. lift--.f32 N/A

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\color{blue}{\left(1 - \frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}\right)} \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   4. lift-/.f32 N/A

      \[\leadsto \sqrt{\left(-\left(\left(ux - maxCos \cdot ux\right) - 0\right)\right) \cdot \left(\left(1 - \color{blue}{\frac{\mathsf{fma}\left(maxCos, ux, 2\right)}{ux}}\right) \cdot ux\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right) \]

   5. sub-to-mult-rev N/A

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

   6. lift-fma.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. associate--l- N/A

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

   9. lift--.f32 N/A

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

   10. lift--.f32 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f32 81.6

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

   13. lift--.f32 N/A

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

   14. lift--.f32 N/A

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

   15. associate--l- N/A

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

   16. lift-*.f32 N/A

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

   17. lift-fma.f32 N/A

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

   18. lower--.f32 81.6

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

   19. lift-neg.f32 N/A

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

   20. lift--.f32 N/A

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

   21. --rgt-identity N/A

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

   22. lift--.f32 N/A

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

10. Applied rewrites 81.6%

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

Alternative 8: 81.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 51.0

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

4. Applied rewrites 51.0%

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

   1. lift-*.f32 N/A

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

   2. pow2 N/A

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

   3. lift-+.f32 N/A

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

   4. lift--.f32 N/A

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

   5. lift-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f32 N/A

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

   8. associate--r- N/A

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

   9. lift--.f32 N/A

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

   10. sub-square-pow N/A

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

   11. pow2 N/A

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

   12. lower-+.f32 N/A

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

   13. metadata-eval N/A

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

   14. lower--.f32 N/A

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

   15. lower-*.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lift--.f32 N/A

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

   18. sub-negate-rev N/A

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

   19. lift--.f32 N/A

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

   20. sub-negate-rev N/A

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

6. Applied rewrites 53.2%

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

8. Applied rewrites 81.6%

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

Alternative 9: 75.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.019999999552965164:\\ \;\;\;\;\left(\left(uy + uy\right) \cdot \pi\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.019999999552965164)
     (* (* (+ uy uy) PI) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))
     (* (* 2.0 (* uy PI)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux))))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = (1.0f - ux) + (ux * maxCos);
	float tmp;
	if (sqrtf((1.0f - (t_0 * t_0))) <= 0.019999999552965164f) {
		tmp = ((uy + uy) * ((float) M_PI)) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(Float32(Float32(1.0) - ux) + Float32(ux * maxCos))
	tmp = Float32(0.0)
	if (sqrt(Float32(Float32(1.0) - Float32(t_0 * t_0))) <= Float32(0.019999999552965164))
		tmp = Float32(Float32(Float32(uy + uy) * Float32(pi)) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	t_0 = (single(1.0) - ux) + (ux * maxCos);
	tmp = single(0.0);
	if (sqrt((single(1.0) - (t_0 * t_0))) <= single(0.019999999552965164))
		tmp = ((uy + uy) * single(pi)) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = (single(2.0) * (uy * single(pi))) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0199999996

   1. Initial program 57.9%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 7.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. lift-*.f32 7.1

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. count-2 N/A

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

      10. lift-+.f32 7.1

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

   9. Applied rewrites 7.1%

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

   10. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 66.0

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

   12. Applied rewrites 66.0%

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

   if 0.0199999996 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 57.9%

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

   2. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 51.0

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

   4. Applied rewrites 51.0%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 49.7

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

   7. Applied rewrites 49.7%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 49.5

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

   10. Applied rewrites 49.5%

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

Alternative 10: 66.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 51.0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 7.1

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2 N/A

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

   10. lift-+.f32 7.1

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

9. Applied rewrites 7.1%

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

10. Taylor expanded in ux around 0

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

    1. lower-*.f32 N/A

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

    2. lower--.f32 N/A

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

    3. lower-*.f32 66.0

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

12. Applied rewrites 66.0%

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

Alternative 11: 7.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.9%

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

2. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 51.0

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

4. Applied rewrites 51.0%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. lift-*.f32 7.1

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. count-2 N/A

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

   10. lift-+.f32 7.1

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

9. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025151 
(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)))))))