UniformSampleCone, y

Percentage Accurate: 57.5% → 98.3%

Time: 18.2s

Alternatives: 20

Speedup: 5.0×

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: 57.5% 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, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

4. Applied rewrites 57.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in maxCos around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 98.4

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

10. Applied rewrites 98.4%

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

11. Final simplification 98.4%

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

Alternative 2: 97.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

4. Applied rewrites 57.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in maxCos around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-fma.f32 98.1

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

10. Applied rewrites 98.1%

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

11. Final simplification 98.1%

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

Alternative 3: 97.2% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

4. Applied rewrites 57.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 97.3%

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

11. Final simplification 97.3%

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

Alternative 4: 96.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

4. Applied rewrites 57.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 97.1%

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

11. Final simplification 97.1%

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

Alternative 5: 93.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.06499999761581421:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.06499999761581421)
   (*
    (* uy (fma -1.3333333333333333 (* (* PI (* PI PI)) (* uy uy)) (* 2.0 PI)))
    (sqrt
     (fma
      (fma maxCos -2.0 2.0)
      ux
      (* (* (+ maxCos -1.0) (- 1.0 maxCos)) (* ux ux)))))
   (* (sin (* PI (* uy 2.0))) (sqrt (* 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.06499999761581421f) {
		tmp = (uy * fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * uy)), (2.0f * ((float) M_PI)))) * sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, (((maxCos + -1.0f) * (1.0f - maxCos)) * (ux * ux))));
	} else {
		tmp = sinf((((float) M_PI) * (uy * 2.0f))) * sqrtf((2.0f * ux));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.06499999761581421))
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * uy)), Float32(Float32(2.0) * Float32(pi)))) * sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)) * Float32(ux * ux)))));
	else
		tmp = Float32(sin(Float32(Float32(pi) * Float32(uy * Float32(2.0)))) * sqrt(Float32(Float32(2.0) * ux)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0649999976

   1. Initial program 57.5%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 97.3

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   8. Applied rewrites 97.3%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]
   9. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. lift--.f32 N/A

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

      4. lift-fma.f32 N/A

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

      5. lift-/.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. +-commutative N/A

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

      8. distribute-rgt-in N/A

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

      9. lift-/.f32 N/A

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

      10. div-inv N/A

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

      11. associate-*l* N/A

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

      12. inv-pow N/A

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

      13. lift-*.f32 N/A

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

      14. pow2 N/A

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

      15. pow-prod-up N/A

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

      16. metadata-eval N/A

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

      17. unpow1 N/A

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

      18. *-commutative N/A

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

   10. Applied rewrites 97.4%

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]

   if 0.0649999976 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.3%

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

   3. Taylor expanded in maxCos around 0

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. unpow2 N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f32 N/A

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

      7. lower--.f32 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f32 N/A

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

      10. lower--.f32 54.4

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in ux around 0

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

      1. lower-*.f32 71.7

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

   8. Applied rewrites 71.7%

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

4. Final simplification 92.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.06499999761581421:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi \cdot \left(uy \cdot 2\right)\right) \cdot \sqrt{2 \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;uy \cdot \left(\mathsf{fma}\left(uy, \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.3333333333333333, 2 \cdot \pi\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 2.000000026702864e-10)
   (* (sqrt (* ux (- 2.0 ux))) (sin (* 2.0 (* uy PI))))
   (*
    uy
    (*
     (fma uy (* (* uy (* PI (* PI PI))) -1.3333333333333333) (* 2.0 PI))
     (sqrt
      (*
       ux
       (fma ux (* (+ maxCos -1.0) (- 1.0 maxCos)) (fma maxCos -2.0 2.0))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 2.000000026702864e-10f) {
		tmp = sqrtf((ux * (2.0f - ux))) * sinf((2.0f * (uy * ((float) M_PI))));
	} else {
		tmp = uy * (fmaf(uy, ((uy * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * -1.3333333333333333f), (2.0f * ((float) M_PI))) * sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f)))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(2.000000026702864e-10))
		tmp = Float32(sqrt(Float32(ux * Float32(Float32(2.0) - ux))) * sin(Float32(Float32(2.0) * Float32(uy * Float32(pi)))));
	else
		tmp = Float32(uy * Float32(fma(uy, Float32(Float32(uy * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(-1.3333333333333333)), Float32(Float32(2.0) * Float32(pi))) * sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0)))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 2.00000003e-10

   1. Initial program 58.7%

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

   3. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-+l+ N/A

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

      6. mul-1-neg N/A

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

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

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

      8. lower-fma.f32 N/A

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f32 N/A

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

      7. lower-sin.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 98.4

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

   8. Applied rewrites 98.4%

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

   if 2.00000003e-10 < maxCos

   1. Initial program 50.5%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 91.5

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   8. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   10. Applied rewrites 91.9%

       \[\leadsto \color{blue}{\left(\mathsf{fma}\left(uy, \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.3333333333333333, 2 \cdot \pi\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot uy} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 2.000000026702864 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{ux \cdot \left(2 - ux\right)} \cdot \sin \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;uy \cdot \left(\mathsf{fma}\left(uy, \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.3333333333333333, 2 \cdot \pi\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* uy (fma -1.3333333333333333 (* (* PI (* PI PI)) (* uy uy)) (* 2.0 PI)))
  (sqrt
   (fma
    (fma maxCos -2.0 2.0)
    ux
    (* (* (+ maxCos -1.0) (- 1.0 maxCos)) (* ux ux))))))

C

float code(float ux, float uy, float maxCos) {
	return (uy * fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * uy)), (2.0f * ((float) M_PI)))) * sqrtf(fmaf(fmaf(maxCos, -2.0f, 2.0f), ux, (((maxCos + -1.0f) * (1.0f - maxCos)) * (ux * ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * uy)), Float32(Float32(2.0) * Float32(pi)))) * sqrt(fma(fma(maxCos, Float32(-2.0), Float32(2.0)), ux, Float32(Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)) * Float32(ux * ux)))))
end

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. cube-mult N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 87.1

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

8. Applied rewrites 87.1%

   \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]
9. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. lift--.f32 N/A

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

   4. lift-fma.f32 N/A

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

   5. lift-/.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. +-commutative N/A

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

   8. distribute-rgt-in N/A

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

   9. lift-/.f32 N/A

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

   10. div-inv N/A

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

   11. associate-*l* N/A

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

   12. inv-pow N/A

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

   13. lift-*.f32 N/A

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

   14. pow2 N/A

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

   15. pow-prod-up N/A

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

   16. metadata-eval N/A

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

   17. unpow1 N/A

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

   18. *-commutative N/A

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

10. Applied rewrites 87.1%

    \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(ux \cdot ux\right) \cdot \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right)\right)}} \]

11. Final simplification 87.1%

    \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, -2, 2\right), ux, \left(\left(maxCos + -1\right) \cdot \left(1 - maxCos\right)\right) \cdot \left(ux \cdot ux\right)\right)} \]
12. Add Preprocessing

Alternative 8: 88.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ uy \cdot \left(\mathsf{fma}\left(uy, \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.3333333333333333, 2 \cdot \pi\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  uy
  (*
   (fma uy (* (* uy (* PI (* PI PI))) -1.3333333333333333) (* 2.0 PI))
   (sqrt
    (*
     ux
     (fma ux (* (+ maxCos -1.0) (- 1.0 maxCos)) (fma maxCos -2.0 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	return uy * (fmaf(uy, ((uy * (((float) M_PI) * (((float) M_PI) * ((float) M_PI)))) * -1.3333333333333333f), (2.0f * ((float) M_PI))) * sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f)))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(uy * Float32(fma(uy, Float32(Float32(uy * Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi)))) * Float32(-1.3333333333333333)), Float32(Float32(2.0) * Float32(pi))) * sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0)))))))
end

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. cube-mult N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 87.1

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

8. Applied rewrites 87.1%

   \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

10. Applied rewrites 87.1%

    \[\leadsto \color{blue}{\left(\mathsf{fma}\left(uy, \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.3333333333333333, 2 \cdot \pi\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot uy} \]

11. Final simplification 87.1%

    \[\leadsto uy \cdot \left(\mathsf{fma}\left(uy, \left(uy \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right) \cdot -1.3333333333333333, 2 \cdot \pi\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \]
12. Add Preprocessing

Alternative 9: 88.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (sqrt
   (* ux (fma ux (* (+ maxCos -1.0) (- 1.0 maxCos)) (fma maxCos -2.0 2.0))))
  (* uy (fma -1.3333333333333333 (* (* PI (* PI PI)) (* uy uy)) (* 2.0 PI)))))

C

float code(float ux, float uy, float maxCos) {
	return sqrtf((ux * fmaf(ux, ((maxCos + -1.0f) * (1.0f - maxCos)), fmaf(maxCos, -2.0f, 2.0f)))) * (uy * fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * uy)), (2.0f * ((float) M_PI))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(sqrt(Float32(ux * fma(ux, Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), fma(maxCos, Float32(-2.0), Float32(2.0))))) * Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * uy)), Float32(Float32(2.0) * Float32(pi)))))
end

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

   8. lower-fma.f32 N/A

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. cube-mult N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 87.1

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \]

8. Applied rewrites 87.1%

   \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \]

9. Final simplification 87.1%

   \[\leadsto \sqrt{ux \cdot \mathsf{fma}\left(ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), \mathsf{fma}\left(maxCos, -2, 2\right)\right)} \cdot \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \]
10. Add Preprocessing

Alternative 10: 83.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* uy (fma -1.3333333333333333 (* (* PI (* PI PI)) (* uy uy)) (* 2.0 PI)))
  (sqrt (* (* ux ux) (+ -1.0 (/ 2.0 ux))))))

C

float code(float ux, float uy, float maxCos) {
	return (uy * fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * uy)), (2.0f * ((float) M_PI)))) * sqrtf(((ux * ux) * (-1.0f + (2.0f / ux))));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * uy)), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(Float32(ux * ux) * Float32(Float32(-1.0) + Float32(Float32(2.0) / ux)))))
end

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. cube-mult N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 87.1

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

8. Applied rewrites 87.1%

   \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

9. Taylor expanded in maxCos around 0

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

    1. lower-*.f32 N/A

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

    2. unpow2 N/A

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

    3. lower-*.f32 N/A

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

    6. lower-+.f32 N/A

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

    7. associate-*r/ N/A

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

    8. metadata-eval N/A

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

    9. lower-/.f32 83.4

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(\color{blue}{\frac{2}{ux}} + -1\right)} \]

11. Applied rewrites 83.4%

    \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} + -1\right)}} \]

12. Final simplification 83.4%

    \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \left(-1 + \frac{2}{ux}\right)} \]
13. Add Preprocessing

Alternative 11: 84.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0025599999353289604:\\ \;\;\;\;\left(\pi \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot \left(uy \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right)\right)\right) \cdot \left(uy \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0025599999353289604)
   (*
    (*
     PI
     (sqrt
      (fma
       (* ux ux)
       (* (+ maxCos -1.0) (- 1.0 maxCos))
       (* ux (fma maxCos -2.0 2.0)))))
    (* uy 2.0))
   (*
    (fma 2.0 PI (* -1.3333333333333333 (* (* PI (* PI PI)) (* uy uy))))
    (* uy (sqrt (* ux (fma -2.0 maxCos 2.0)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0025599999353289604f) {
		tmp = (((float) M_PI) * sqrtf(fmaf((ux * ux), ((maxCos + -1.0f) * (1.0f - maxCos)), (ux * fmaf(maxCos, -2.0f, 2.0f))))) * (uy * 2.0f);
	} else {
		tmp = fmaf(2.0f, ((float) M_PI), (-1.3333333333333333f * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * uy)))) * (uy * sqrtf((ux * fmaf(-2.0f, maxCos, 2.0f))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0025599999353289604))
		tmp = Float32(Float32(Float32(pi) * sqrt(fma(Float32(ux * ux), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0)))))) * Float32(uy * Float32(2.0)));
	else
		tmp = Float32(fma(Float32(2.0), Float32(pi), Float32(Float32(-1.3333333333333333) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * uy)))) * Float32(uy * sqrt(Float32(ux * fma(Float32(-2.0), maxCos, Float32(2.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00255999994

   1. Initial program 58.2%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 97.1

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

   8. Applied rewrites 97.1%

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

   10. Applied rewrites 97.2%

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

   if 0.00255999994 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.9%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 64.0

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   8. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   9. Taylor expanded in ux around 0

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

       1. associate-*r* N/A

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

       2. lower-*.f32 N/A

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

       3. lower-*.f32 N/A

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

       4. lower-sqrt.f32 N/A

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

       5. lower-*.f32 N/A

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

       6. +-commutative N/A

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

       7. lower-fma.f32 N/A

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

       8. +-commutative N/A

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

       9. lower-fma.f32 N/A

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

       10. lower-PI.f32 N/A

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

       11. lower-*.f32 N/A

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

       12. lower-*.f32 N/A

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

       13. unpow2 N/A

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

       14. lower-*.f32 N/A

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

       15. cube-mult N/A

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

       16. lower-*.f32 N/A

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

       17. lower-PI.f32 N/A

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

       18. lower-*.f32 N/A

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

       19. lower-PI.f32 N/A

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

       20. lower-PI.f32 57.2

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

   11. Applied rewrites 57.2%

       \[\leadsto \color{blue}{\left(\sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)} \cdot uy\right) \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0025599999353289604:\\ \;\;\;\;\left(\pi \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot \left(uy \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right)\right)\right) \cdot \left(uy \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0025599999353289604:\\ \;\;\;\;\left(\pi \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot \left(uy \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.0025599999353289604)
   (*
    (*
     PI
     (sqrt
      (fma
       (* ux ux)
       (* (+ maxCos -1.0) (- 1.0 maxCos))
       (* ux (fma maxCos -2.0 2.0)))))
    (* uy 2.0))
   (*
    (* uy (fma -1.3333333333333333 (* (* PI (* PI PI)) (* uy uy)) (* 2.0 PI)))
    (sqrt (* ux (fma -2.0 maxCos 2.0))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if ((uy * 2.0f) <= 0.0025599999353289604f) {
		tmp = (((float) M_PI) * sqrtf(fmaf((ux * ux), ((maxCos + -1.0f) * (1.0f - maxCos)), (ux * fmaf(maxCos, -2.0f, 2.0f))))) * (uy * 2.0f);
	} else {
		tmp = (uy * fmaf(-1.3333333333333333f, ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * uy)), (2.0f * ((float) M_PI)))) * sqrtf((ux * fmaf(-2.0f, maxCos, 2.0f)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (Float32(uy * Float32(2.0)) <= Float32(0.0025599999353289604))
		tmp = Float32(Float32(Float32(pi) * sqrt(fma(Float32(ux * ux), Float32(Float32(maxCos + Float32(-1.0)) * Float32(Float32(1.0) - maxCos)), Float32(ux * fma(maxCos, Float32(-2.0), Float32(2.0)))))) * Float32(uy * Float32(2.0)));
	else
		tmp = Float32(Float32(uy * fma(Float32(-1.3333333333333333), Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * uy)), Float32(Float32(2.0) * Float32(pi)))) * sqrt(Float32(ux * fma(Float32(-2.0), maxCos, Float32(2.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00255999994

   1. Initial program 58.2%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 97.1

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

   8. Applied rewrites 97.1%

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

   10. Applied rewrites 97.2%

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

   if 0.00255999994 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.9%

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

   3. Taylor expanded in ux around inf

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

      1. lower-*.f32 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f32 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

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

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

      9. metadata-eval N/A

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

      10. associate-+l+ N/A

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-fma.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f32 N/A

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

      6. cube-mult N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-PI.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-*.f32 N/A

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

      13. lower-PI.f32 64.0

          \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   8. Applied rewrites 64.0%

      \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

   9. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f32 57.1

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

   11. Applied rewrites 57.1%

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.0025599999353289604:\\ \;\;\;\;\left(\pi \cdot \sqrt{\mathsf{fma}\left(ux \cdot ux, \left(maxCos + -1\right) \cdot \left(1 - maxCos\right), ux \cdot \mathsf{fma}\left(maxCos, -2, 2\right)\right)}\right) \cdot \left(uy \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right), 2 \cdot \pi\right)\right) \cdot \sqrt{ux \cdot \mathsf{fma}\left(-2, maxCos, 2\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 83.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(uy \cdot ux\right) \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (*
   (* uy ux)
   (fma 2.0 PI (* -1.3333333333333333 (* (* PI (* PI PI)) (* uy uy)))))
  (sqrt (+ -1.0 (/ 2.0 ux)))))

C

float code(float ux, float uy, float maxCos) {
	return ((uy * ux) * fmaf(2.0f, ((float) M_PI), (-1.3333333333333333f * ((((float) M_PI) * (((float) M_PI) * ((float) M_PI))) * (uy * uy))))) * sqrtf((-1.0f + (2.0f / ux)));
}

Julia

function code(ux, uy, maxCos)
	return Float32(Float32(Float32(uy * ux) * fma(Float32(2.0), Float32(pi), Float32(Float32(-1.3333333333333333) * Float32(Float32(Float32(pi) * Float32(Float32(pi) * Float32(pi))) * Float32(uy * uy))))) * sqrt(Float32(Float32(-1.0) + Float32(Float32(2.0) / ux))))
end

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-fma.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f32 N/A

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

   6. cube-mult N/A

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

   7. lower-*.f32 N/A

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

   8. lower-PI.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-PI.f32 N/A

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

   11. lower-PI.f32 N/A

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

   12. lower-*.f32 N/A

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

   13. lower-PI.f32 87.1

       \[\leadsto \left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \color{blue}{\pi}\right)\right) \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

8. Applied rewrites 87.1%

   \[\leadsto \color{blue}{\left(uy \cdot \mathsf{fma}\left(-1.3333333333333333, \left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right), 2 \cdot \pi\right)\right)} \cdot \sqrt{\left(ux \cdot ux\right) \cdot \mathsf{fma}\left(maxCos + -1, 1 - maxCos, \frac{\mathsf{fma}\left(maxCos, -2, 2\right)}{ux}\right)} \]

9. Taylor expanded in maxCos around 0

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

    1. lower-*.f32 N/A

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

11. Applied rewrites 83.3%

    \[\leadsto \color{blue}{\left(\left(ux \cdot uy\right) \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(uy \cdot uy\right) \cdot \left(\pi \cdot \left(\pi \cdot \pi\right)\right)\right)\right)\right) \cdot \sqrt{\frac{2}{ux} + -1}} \]

12. Final simplification 83.3%

    \[\leadsto \left(\left(uy \cdot ux\right) \cdot \mathsf{fma}\left(2, \pi, -1.3333333333333333 \cdot \left(\left(\pi \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(uy \cdot uy\right)\right)\right)\right) \cdot \sqrt{-1 + \frac{2}{ux}} \]
13. Add Preprocessing

Alternative 14: 81.3% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 79.2

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

8. Applied rewrites 79.2%

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

10. Applied rewrites 79.2%

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

11. Final simplification 79.2%

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

Alternative 15: 81.2% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

4. Applied rewrites 57.0%

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

5. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. lower--.f32 N/A

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

7. Applied rewrites 98.4%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 79.2

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

10. Applied rewrites 79.2%

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

Alternative 16: 81.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 79.2

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

8. Applied rewrites 79.2%

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

10. Applied rewrites 79.2%

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

12. Applied rewrites 79.2%

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

13. Final simplification 79.2%

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

Alternative 17: 81.2% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around 0

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

   1. lower-*.f32 N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

   5. associate-+l+ N/A

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

   6. mul-1-neg N/A

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

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

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

   8. lower-fma.f32 N/A

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 79.2

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

8. Applied rewrites 79.2%

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

Alternative 18: 80.3% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 79.2

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

8. Applied rewrites 79.2%

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

10. Applied rewrites 79.2%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 78.2%

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

14. Final simplification 78.2%

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

Alternative 19: 77.0% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

1. Initial program 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 79.2

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

8. Applied rewrites 79.2%

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

10. Applied rewrites 79.2%

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

11. Taylor expanded in maxCos around 0

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

    1. associate-*r* N/A

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

    2. lower-*.f32 N/A

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

    3. lower-*.f32 N/A

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

    4. lower-*.f32 N/A

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

    5. lower-PI.f32 N/A

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

    6. lower-sqrt.f32 N/A

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

    7. +-commutative N/A

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

    8. mul-1-neg N/A

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

    9. unpow2 N/A

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

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

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

    11. mul-1-neg N/A

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

    12. distribute-rgt-in N/A

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

    13. lower-*.f32 N/A

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

    14. mul-1-neg N/A

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

    15. unsub-neg N/A

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

    16. lower--.f32 76.1

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

13. Applied rewrites 76.1%

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

14. Final simplification 76.1%

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

Alternative 20: 63.0% accurate, 5.0× speedup

Math

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

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(Float32(2.0) * Float32(uy * 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 57.1%

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

3. Taylor expanded in ux around inf

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

   1. lower-*.f32 N/A

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

   2. unpow2 N/A

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

   3. lower-*.f32 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

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

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

   9. metadata-eval N/A

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

   10. associate-+l+ N/A

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 79.2

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

8. Applied rewrites 79.2%

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

9. Taylor expanded in ux around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f32 64.2

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

11. Applied rewrites 64.2%

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

12. Taylor expanded in maxCos around 0

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

    1. lower-*.f32 62.6

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

14. Applied rewrites 62.6%

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

Reproduce

herbie shell --seed 2024216 
(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)))))))