Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.7% → 99.1%

Time: 10.8s

Alternatives: 6

Speedup: 4.0×

Specification

\[\left(\left(cosTheta_i > 0.9999 \land cosTheta_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Initial Program: 57.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2)))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(-log((single(1.0) - u1))) * cos(((single(2.0) * single(pi)) * u2));
end

Alternative 1: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\pi \cdot u2\right)\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + \mathsf{fma}\left(-t_0, t_0, {t_0}^{2}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sin (* PI u2))))
   (*
    (sqrt (- (log1p (- u1))))
    (+ (cos (* 2.0 (* PI u2))) (fma (- t_0) t_0 (pow t_0 2.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sinf((((float) M_PI) * u2));
	return sqrtf(-log1pf(-u1)) * (cosf((2.0f * (((float) M_PI) * u2))) + fmaf(-t_0, t_0, powf(t_0, 2.0f)));
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sin(Float32(Float32(pi) * u2))
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) + fma(Float32(-t_0), t_0, (t_0 ^ Float32(2.0)))))
end

Derivation

1. Initial program 60.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 60.8%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]
4. Step-by-step derivation

   1. cos-2 98.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\cos \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right) - \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)} \]

   2. prod-diff 98.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(\cos \left(\pi \cdot u2\right), \cos \left(\pi \cdot u2\right), -\sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)\right)} \]

   3. fma-neg 98.9%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\left(\cos \left(\pi \cdot u2\right) \cdot \cos \left(\pi \cdot u2\right) - \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)} + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)\right) \]

   4. cos-2 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\color{blue}{\cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right) \cdot \sin \left(\pi \cdot u2\right)\right)\right) \]

   5. pow2 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), \color{blue}{{\sin \left(\pi \cdot u2\right)}^{2}}\right)\right) \]

5. Applied egg-rr 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), {\sin \left(\pi \cdot u2\right)}^{2}\right)\right)} \]

6. Final simplification 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) + \mathsf{fma}\left(-\sin \left(\pi \cdot u2\right), \sin \left(\pi \cdot u2\right), {\sin \left(\pi \cdot u2\right)}^{2}\right)\right) \]

Alternative 2: 90.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \mathbf{if}\;t_0 \leq 0.9999300241470337:\\ \;\;\;\;t_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* u2 (* 2.0 PI)))))
   (if (<= t_0 0.9999300241470337)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf((u2 * (2.0f * ((float) M_PI))));
	float tmp;
	if (t_0 <= 0.9999300241470337f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(u2 * Float32(Float32(2.0) * Float32(pi))))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999300241470337))
		tmp = Float32(t_0 * sqrt(u1));
	else
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2)) < 0.999930024

   1. Initial program 58.8%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 58.8%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-udef 98.1%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. add-cbrt-cube 98.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. pow1/3 94.9%

         \[\leadsto \color{blue}{{\left(\left(\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied egg-rr 72.9%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u1 around 0 76.8%

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   if 0.999930024 < (cos.f32 (*.f32 (*.f32 2 (PI.f32)) u2))

   1. Initial program 61.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   2. Step-by-step derivation

      1. sub-neg 61.7%

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. log1p-def 99.5%

         \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. associate-*l* 99.5%

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

   3. Simplified 99.5%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

   4. Taylor expanded in u2 around 0 96.0%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \leq 0.9999300241470337:\\ \;\;\;\;\cos \left(u2 \cdot \left(2 \cdot \pi\right)\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (cos (* 2.0 (* PI u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * cosf((2.0f * (((float) M_PI) * u2)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))))
end

Derivation

1. Initial program 60.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 60.8%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

4. Final simplification 99.1%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \]

Alternative 4: 80.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt (- (log1p (- u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(-log1p(Float32(-u1))))
end

Derivation

1. Initial program 60.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 60.8%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

4. Taylor expanded in u2 around 0 79.5%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

5. Final simplification 79.5%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \]

Alternative 5: 72.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 - \left(u1 \cdot u1\right) \cdot -0.5} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (- u1 (* (* u1 u1) -0.5))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 - ((u1 * u1) * -0.5f)));
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt((u1 - ((u1 * u1) * (-0.5e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 - Float32(Float32(u1 * u1) * Float32(-0.5))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 - ((u1 * u1) * single(-0.5))));
end

Derivation

1. Initial program 60.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 60.8%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

4. Taylor expanded in u2 around 0 79.5%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

5. Taylor expanded in u1 around 0 74.5%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)}} \cdot 1 \]

6. Taylor expanded in u1 around 0 71.2%

   \[\leadsto \sqrt{-\left(-1 \cdot u1 + \color{blue}{-0.5 \cdot {u1}^{2}}\right)} \cdot 1 \]
7. Step-by-step derivation

   1. unpow2 71.2%

      \[\leadsto \sqrt{-\left(-1 \cdot u1 + -0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)}\right)} \cdot 1 \]

8. Simplified 71.2%

   \[\leadsto \sqrt{-\left(-1 \cdot u1 + \color{blue}{-0.5 \cdot \left(u1 \cdot u1\right)}\right)} \cdot 1 \]

9. Taylor expanded in u1 around 0 71.2%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot 1 \]
10. Step-by-step derivation

    1. neg-mul-1 71.2%

       \[\leadsto \sqrt{-\left(\color{blue}{\left(-u1\right)} + -0.5 \cdot {u1}^{2}\right)} \cdot 1 \]

    2. +-commutative 71.2%

       \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} + \left(-u1\right)\right)}} \cdot 1 \]

    3. fma-def 71.2%

       \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(-0.5, {u1}^{2}, -u1\right)}} \cdot 1 \]

    4. fma-neg 71.2%

       \[\leadsto \sqrt{-\color{blue}{\left(-0.5 \cdot {u1}^{2} - u1\right)}} \cdot 1 \]

    5. *-commutative 71.2%

       \[\leadsto \sqrt{-\left(\color{blue}{{u1}^{2} \cdot -0.5} - u1\right)} \cdot 1 \]

    6. unpow2 71.2%

       \[\leadsto \sqrt{-\left(\color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5 - u1\right)} \cdot 1 \]

11. Simplified 71.2%

    \[\leadsto \sqrt{-\color{blue}{\left(\left(u1 \cdot u1\right) \cdot -0.5 - u1\right)}} \cdot 1 \]

12. Final simplification 71.2%

    \[\leadsto \sqrt{u1 - \left(u1 \cdot u1\right) \cdot -0.5} \]

Alternative 6: 64.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2) :precision binary32 (sqrt u1))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1);
}

Fortran

real(4) function code(costheta_i, u1, u2)
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(u1)
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt(u1);
end

Derivation

1. Initial program 60.8%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
2. Step-by-step derivation

   1. sub-neg 60.8%

      \[\leadsto \sqrt{-\log \color{blue}{\left(1 + \left(-u1\right)\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   2. log1p-def 99.1%

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(-u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. associate-*l* 99.1%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

3. Simplified 99.1%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \left(2 \cdot \left(\pi \cdot u2\right)\right)} \]

4. Taylor expanded in u2 around 0 79.5%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{1} \]

5. Taylor expanded in u1 around 0 74.5%

   \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)}} \cdot 1 \]
6. Step-by-step derivation

   1. expm1-log1p-u 74.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-\left(-1 \cdot u1 + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)}\right)\right)} \cdot 1 \]

   2. expm1-udef 63.6%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{-\left(-1 \cdot u1 + \left(-0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)\right)}\right)} - 1\right)} \cdot 1 \]

   3. fma-def 63.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{-\color{blue}{\mathsf{fma}\left(-1, u1, -0.3333333333333333 \cdot {u1}^{3} + -0.5 \cdot {u1}^{2}\right)}}\right)} - 1\right) \cdot 1 \]

   4. fma-def 63.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{-\mathsf{fma}\left(-1, u1, \color{blue}{\mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot {u1}^{2}\right)}\right)}\right)} - 1\right) \cdot 1 \]

   5. unpow2 63.6%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\sqrt{-\mathsf{fma}\left(-1, u1, \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \color{blue}{\left(u1 \cdot u1\right)}\right)\right)}\right)} - 1\right) \cdot 1 \]

7. Applied egg-rr 63.6%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{-\mathsf{fma}\left(-1, u1, \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)\right)}\right)} - 1\right)} \cdot 1 \]
8. Step-by-step derivation

   1. expm1-def 74.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{-\mathsf{fma}\left(-1, u1, \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)\right)}\right)\right)} \cdot 1 \]

   2. expm1-log1p 74.5%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{fma}\left(-1, u1, \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)\right)}} \cdot 1 \]

   3. neg-sub0 74.5%

      \[\leadsto \sqrt{\color{blue}{0 - \mathsf{fma}\left(-1, u1, \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)\right)}} \cdot 1 \]

   4. metadata-eval 74.5%

      \[\leadsto \sqrt{\color{blue}{\log 1} - \mathsf{fma}\left(-1, u1, \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)\right)} \cdot 1 \]

   5. fma-udef 74.5%

      \[\leadsto \sqrt{\log 1 - \color{blue}{\left(-1 \cdot u1 + \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)\right)}} \cdot 1 \]

   6. neg-mul-1 74.5%

      \[\leadsto \sqrt{\log 1 - \left(\color{blue}{\left(-u1\right)} + \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)\right)} \cdot 1 \]

   7. associate--r+ 74.5%

      \[\leadsto \sqrt{\color{blue}{\left(\log 1 - \left(-u1\right)\right) - \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)}} \cdot 1 \]

   8. metadata-eval 74.5%

      \[\leadsto \sqrt{\left(\color{blue}{0} - \left(-u1\right)\right) - \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)} \cdot 1 \]

   9. neg-sub0 74.5%

      \[\leadsto \sqrt{\color{blue}{\left(-\left(-u1\right)\right)} - \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)} \cdot 1 \]

   10. remove-double-neg 74.5%

       \[\leadsto \sqrt{\color{blue}{u1} - \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)} \cdot 1 \]

9. Simplified 74.5%

   \[\leadsto \color{blue}{\sqrt{u1 - \mathsf{fma}\left(-0.3333333333333333, {u1}^{3}, -0.5 \cdot \left(u1 \cdot u1\right)\right)}} \cdot 1 \]

10. Taylor expanded in u1 around 0 63.2%

    \[\leadsto \sqrt{\color{blue}{u1}} \cdot 1 \]

11. Final simplification 63.2%

    \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023228 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_x"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2))))