Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.0%

Time: 13.2s

Alternatives: 5

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: 57.6% 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: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.9%

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

   1. sub-neg 57.9%

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

   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. Simplified 99.1%

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

4. Final simplification 99.1%

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

Alternative 2: 95.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.0020000000949949026)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt (* u1 (- (- -1.0) (* u1 -0.5))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.0020000000949949026f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf((u1 * (-(-1.0f) - (u1 * -0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.0020000000949949026))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(u1 * Float32(Float32(-Float32(-1.0)) - Float32(u1 * Float32(-0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.00200000009

   1. Initial program 58.0%

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

      1. sub-neg 58.0%

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

      2. log1p-def 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in u2 around 0 98.8%

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

   if 0.00200000009 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 57.7%

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

   2. Taylor expanded in u1 around 0 85.6%

      \[\leadsto \sqrt{-\color{blue}{\left(-1 \cdot u1 + -0.5 \cdot {u1}^{2}\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
   3. Step-by-step derivation

      1. *-commutative 85.6%

         \[\leadsto \sqrt{-\left(\color{blue}{u1 \cdot -1} + -0.5 \cdot {u1}^{2}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      2. *-commutative 85.6%

         \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{{u1}^{2} \cdot -0.5}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      3. unpow2 85.6%

         \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. associate-*l* 85.6%

         \[\leadsto \sqrt{-\left(u1 \cdot -1 + \color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. distribute-lft-out 85.7%

         \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + u1 \cdot -0.5\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Simplified 85.7%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \left(-1 + u1 \cdot -0.5\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0020000000949949026:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1 \cdot \left(\left(--1\right) - u1 \cdot -0.5\right)}\\ \end{array} \]

Alternative 3: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t_0 \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos t_0 \cdot \sqrt{u1}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.017999999225139618)
     (sqrt (- (log1p (- u1))))
     (* (cos t_0) (sqrt u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.017999999225139618f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf(t_0) * sqrtf(u1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(Float32(2.0) * Float32(pi)) * u2)
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.017999999225139618))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(t_0) * sqrt(u1));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0179999992

   1. Initial program 58.9%

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

      1. sub-neg 58.9%

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

      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. Simplified 99.5%

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

   4. Taylor expanded in u2 around 0 94.5%

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

   if 0.0179999992 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 55.4%

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

      1. add-sqr-sqrt 55.3%

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

      2. pow2 55.3%

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

      3. pow1/2 55.3%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(-\log \left(1 - u1\right)\right)}^{0.5}}}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      4. sqrt-pow1 55.3%

         \[\leadsto {\color{blue}{\left({\left(-\log \left(1 - u1\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      5. add-sqr-sqrt 55.3%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{-\log \left(1 - u1\right)} \cdot \sqrt{-\log \left(1 - u1\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      6. sqrt-unprod 55.3%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{\left(-\log \left(1 - u1\right)\right) \cdot \left(-\log \left(1 - u1\right)\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      7. sqr-neg 55.3%

         \[\leadsto {\left({\left(\sqrt{\color{blue}{\log \left(1 - u1\right) \cdot \log \left(1 - u1\right)}}\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      8. sqrt-unprod 1.8%

         \[\leadsto {\left({\color{blue}{\left(\sqrt{\log \left(1 - u1\right)} \cdot \sqrt{\log \left(1 - u1\right)}\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      9. add-sqr-sqrt 1.8%

         \[\leadsto {\left({\color{blue}{\log \left(1 - u1\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      10. sub-neg 1.8%

          \[\leadsto {\left({\log \color{blue}{\left(1 + \left(-u1\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      11. log1p-udef -0.0%

          \[\leadsto {\left({\color{blue}{\left(\mathsf{log1p}\left(-u1\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      12. add-sqr-sqrt -0.0%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{-u1} \cdot \sqrt{-u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      13. sqrt-unprod 74.1%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-u1\right) \cdot \left(-u1\right)}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      14. sqr-neg 74.1%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\sqrt{\color{blue}{u1 \cdot u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      15. sqrt-unprod 74.2%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{\sqrt{u1} \cdot \sqrt{u1}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      16. add-sqr-sqrt 74.1%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(\color{blue}{u1}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

      17. metadata-eval 74.1%

          \[\leadsto {\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{\color{blue}{0.25}}\right)}^{2} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   3. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   4. Taylor expanded in u1 around 0 76.1%

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.017999999225139618:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \end{array} \]

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 57.9%

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

   1. sub-neg 57.9%

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

   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. Simplified 99.1%

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

4. Taylor expanded in u2 around 0 79.2%

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

5. Final simplification 79.2%

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

Alternative 5: 64.8% 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 57.9%

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

   1. sub-neg 57.9%

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

   2. log1p-udef 99.1%

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

   3. add-cbrt-cube 99.0%

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

   4. pow1/3 95.5%

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

3. Applied egg-rr 72.2%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. unpow1/3 74.1%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Simplified 74.1%

   \[\leadsto \color{blue}{\sqrt[3]{{\left(\mathsf{log1p}\left(u1\right)\right)}^{1.5}}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Taylor expanded in u2 around 0 37.0%

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

   1. log1p-def 63.2%

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

8. Simplified 63.2%

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

9. Taylor expanded in u1 around 0 64.6%

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

10. Final simplification 64.6%

    \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023326 
(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))))