UniformSampleCone, y

Percentage Accurate: 57.5% → 98.3%

Time: 6.0s

Alternatives: 12

Speedup: 3.3×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.5% 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, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.5

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

3. Applied rewrites 98.3%

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 98.3%

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

Alternative 2: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.5

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

3. Applied rewrites 98.3%

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

5. Applied rewrites 98.3%

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

Alternative 3: 97.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.5

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

3. Applied rewrites 98.3%

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 97.1%

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

Alternative 4: 97.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.5

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

3. Applied rewrites 98.3%

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 97.1%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

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

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

   4. distribute-rgt1-in N/A

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

   5. +-commutative N/A

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

   6. sub-flip N/A

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

   7. lift--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 97.1

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

10. Applied rewrites 97.1%

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

Alternative 5: 95.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;maxCos \leq 1.9999999494757503 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{-1 \cdot \left(ux \cdot \left(ux - 2\right)\right)} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(uy \cdot \left(\pi \cdot \sqrt{\left(2 + ux \cdot \left(maxCos - 1\right)\right) \cdot \left(ux - maxCos \cdot ux\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= maxCos 1.9999999494757503e-5)
   (* (sqrt (* -1.0 (* ux (- ux 2.0)))) (sin (* PI (+ uy uy))))
   (*
    2.0
    (*
     uy
     (* PI (sqrt (* (+ 2.0 (* ux (- maxCos 1.0))) (- ux (* maxCos ux)))))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (maxCos <= 1.9999999494757503e-5f) {
		tmp = sqrtf((-1.0f * (ux * (ux - 2.0f)))) * sinf((((float) M_PI) * (uy + uy)));
	} else {
		tmp = 2.0f * (uy * (((float) M_PI) * sqrtf(((2.0f + (ux * (maxCos - 1.0f))) * (ux - (maxCos * ux))))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (maxCos <= Float32(1.9999999494757503e-5))
		tmp = Float32(sqrt(Float32(Float32(-1.0) * Float32(ux * Float32(ux - Float32(2.0))))) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	else
		tmp = Float32(Float32(2.0) * Float32(uy * Float32(Float32(pi) * sqrt(Float32(Float32(Float32(2.0) + Float32(ux * Float32(maxCos - Float32(1.0)))) * Float32(ux - Float32(maxCos * ux)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (maxCos <= single(1.9999999494757503e-5))
		tmp = sqrt((single(-1.0) * (ux * (ux - single(2.0))))) * sin((single(pi) * (uy + uy)));
	else
		tmp = single(2.0) * (uy * (single(pi) * sqrt(((single(2.0) + (ux * (maxCos - single(1.0)))) * (ux - (maxCos * ux))))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.99999995e-5

   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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.5

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in maxCos around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 92.3

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

   6. Applied rewrites 92.3%

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

   if 1.99999995e-5 < maxCos

   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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.5

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

   3. Applied rewrites 98.3%

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

   5. Applied rewrites 98.3%

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

   6. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-PI.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-+.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower--.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-*.f32 81.5

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

   8. Applied rewrites 81.5%

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

Alternative 6: 90.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := ux \cdot \left(maxCos - 1\right)\\ \mathbf{if}\;uy \leq 0.001449999981559813:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(1 - maxCos, ux, \mathsf{fma}\left(1 - maxCos, ux, t\_0 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)} \cdot \left(2 \cdot \left(uy \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-2 \cdot t\_0} \cdot \sin \left(\pi \cdot \left(uy + uy\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* ux (- maxCos 1.0))))
   (if (<= uy 0.001449999981559813)
     (*
      (sqrt
       (fma
        (- 1.0 maxCos)
        ux
        (fma (- 1.0 maxCos) ux (* t_0 (- ux (* ux maxCos))))))
      (* 2.0 (* uy PI)))
     (* (sqrt (* -2.0 t_0)) (sin (* PI (+ uy uy)))))))

C

float code(float ux, float uy, float maxCos) {
	float t_0 = ux * (maxCos - 1.0f);
	float tmp;
	if (uy <= 0.001449999981559813f) {
		tmp = sqrtf(fmaf((1.0f - maxCos), ux, fmaf((1.0f - maxCos), ux, (t_0 * (ux - (ux * maxCos)))))) * (2.0f * (uy * ((float) M_PI)));
	} else {
		tmp = sqrtf((-2.0f * t_0)) * sinf((((float) M_PI) * (uy + uy)));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	t_0 = Float32(ux * Float32(maxCos - Float32(1.0)))
	tmp = Float32(0.0)
	if (uy <= Float32(0.001449999981559813))
		tmp = Float32(sqrt(fma(Float32(Float32(1.0) - maxCos), ux, fma(Float32(Float32(1.0) - maxCos), ux, Float32(t_0 * Float32(ux - Float32(ux * maxCos)))))) * Float32(Float32(2.0) * Float32(uy * Float32(pi))));
	else
		tmp = Float32(sqrt(Float32(Float32(-2.0) * t_0)) * sin(Float32(Float32(pi) * Float32(uy + uy))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if uy < 0.00144999998

   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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.5

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

   3. Applied rewrites 98.3%

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

   5. Applied rewrites 98.3%

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

   7. Applied rewrites 98.3%

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

   8. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 81.5

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

   10. Applied rewrites 81.5%

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

   if 0.00144999998 < uy

   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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f32 57.5

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

   3. Applied rewrites 98.3%

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

   4. Taylor expanded in ux around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 76.7

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

   6. Applied rewrites 76.7%

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

Alternative 7: 81.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.5

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

3. Applied rewrites 98.3%

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

5. Applied rewrites 98.3%

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

7. Applied rewrites 98.3%

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

8. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.5

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

10. Applied rewrites 81.5%

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

Alternative 8: 81.5% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.5

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

3. Applied rewrites 98.3%

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 81.5

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

8. Applied rewrites 81.5%

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

Alternative 9: 80.7% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

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. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f32 57.5

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

3. Applied rewrites 98.3%

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

5. Applied rewrites 98.3%

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

6. Taylor expanded in maxCos around 0

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

8. Applied rewrites 97.1%

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

9. Taylor expanded in uy around 0

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

    1. lower-*.f32 N/A

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

    2. lower-*.f32 N/A

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

    3. lower-PI.f32 80.7

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

11. Applied rewrites 80.7%

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

Alternative 10: 75.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0003150000120513141:\\ \;\;\;\;\left(\left(\pi + \pi\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot \left(uy \cdot \pi\right)\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0003150000120513141)
   (* (* (+ PI PI) uy) (sqrt (* ux (- 2.0 (* 2.0 maxCos)))))
   (* (* 2.0 (* uy PI)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

C

float code(float ux, float uy, float maxCos) {
	float tmp;
	if (ux <= 0.0003150000120513141f) {
		tmp = ((((float) M_PI) + ((float) M_PI)) * uy) * sqrtf((ux * (2.0f - (2.0f * maxCos))));
	} else {
		tmp = (2.0f * (uy * ((float) M_PI))) * sqrtf((1.0f - ((1.0f - ux) * (1.0f - ux))));
	}
	return tmp;
}

Julia

function code(ux, uy, maxCos)
	tmp = Float32(0.0)
	if (ux <= Float32(0.0003150000120513141))
		tmp = Float32(Float32(Float32(Float32(pi) + Float32(pi)) * uy) * sqrt(Float32(ux * Float32(Float32(2.0) - Float32(Float32(2.0) * maxCos)))));
	else
		tmp = Float32(Float32(Float32(2.0) * Float32(uy * Float32(pi))) * sqrt(Float32(Float32(1.0) - Float32(Float32(Float32(1.0) - ux) * Float32(Float32(1.0) - ux)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(ux, uy, maxCos)
	tmp = single(0.0);
	if (ux <= single(0.0003150000120513141))
		tmp = ((single(pi) + single(pi)) * uy) * sqrt((ux * (single(2.0) - (single(2.0) * maxCos))));
	else
		tmp = (single(2.0) * (uy * single(pi))) * sqrt((single(1.0) - ((single(1.0) - ux) * (single(1.0) - ux))));
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if ux < 3.15000012e-4

   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. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in ux around 0

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

   7. Applied rewrites 7.1%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. count-2-rev N/A

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

      7. lower-+.f32 7.1

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

   9. Applied rewrites 7.1%

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

   10. Taylor expanded in ux around 0

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

       1. lower-*.f32 N/A

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

       2. lower--.f32 N/A

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

       3. lower-*.f32 66.2

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

   12. Applied rewrites 66.2%

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

   if 3.15000012e-4 < ux

   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. Taylor expanded in uy around 0

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

      1. lower-*.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-PI.f32 50.6

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

   4. Applied rewrites 50.6%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 49.2

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

   7. Applied rewrites 49.2%

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

   8. Taylor expanded in maxCos around 0

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

      1. lower--.f32 49.1

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

   10. Applied rewrites 49.1%

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

Alternative 11: 66.2% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 50.6

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. lower-+.f32 7.1

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

9. Applied rewrites 7.1%

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

10. Taylor expanded in ux around 0

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

    1. lower-*.f32 N/A

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

    2. lower--.f32 N/A

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

    3. lower-*.f32 66.2

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

12. Applied rewrites 66.2%

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

Alternative 12: 7.1% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Derivation

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. Taylor expanded in uy around 0

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

   1. lower-*.f32 N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 50.6

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

4. Applied rewrites 50.6%

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

5. Taylor expanded in ux around 0

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

7. Applied rewrites 7.1%

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

   1. lift-*.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. count-2-rev N/A

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

   7. lower-+.f32 7.1

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

9. Applied rewrites 7.1%

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

Reproduce

herbie shell --seed 2025156 
(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)))))))