Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.1%

Time: 16.7s

Alternatives: 8

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: 99.1% accurate, 1.0× speedup

Math

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

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(2.0) * Float32(Float32(pi) * u2))) * sqrt(Float32(-log1p(Float32(-u1)))))
end

Derivation

1. Initial program 58.6%

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

   1. sub-neg 58.6%

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

   2. log1p-def 99.2%

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

   3. associate-*l* 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 2: 94.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* u2 (* 2.0 PI)) 0.0005000000237487257)
   (sqrt (- (log1p (- u1))))
   (* (cos (* 2.0 (* PI u2))) (sqrt (+ u1 (* u1 (* u1 0.5)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((u2 * (2.0f * ((float) M_PI))) <= 0.0005000000237487257f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((2.0f * (((float) M_PI) * u2))) * sqrtf((u1 + (u1 * (u1 * 0.5f))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(u2 * Float32(Float32(2.0) * Float32(pi))) <= Float32(0.0005000000237487257))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(2.0) * Float32(Float32(pi) * u2))) * sqrt(Float32(u1 + Float32(u1 * Float32(u1 * Float32(0.5))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 5.00000024e-4

   1. Initial program 59.3%

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

      1. sub-neg 59.3%

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

      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. associate-*l* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in u2 around 0 99.6%

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

   if 5.00000024e-4 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 57.6%

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

   2. Taylor expanded in u1 around 0 86.4%

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

      1. +-commutative 86.4%

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

      2. mul-1-neg 86.4%

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

      3. unsub-neg 86.4%

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

      4. unpow2 86.4%

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

      5. associate-*r* 86.4%

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

   4. Simplified 86.4%

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

   5. Taylor expanded in u2 around inf 86.4%

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

      1. *-commutative 86.4%

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

      2. cancel-sign-sub-inv 86.4%

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

      3. metadata-eval 86.4%

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

      4. unpow2 86.4%

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

      5. associate-*r* 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.0005000000237487257:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(2 \cdot \left(\pi \cdot u2\right)\right) \cdot \sqrt{u1 + u1 \cdot \left(u1 \cdot 0.5\right)}\\ \end{array} \]

Alternative 3: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := u2 \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;t_0 \leq 0.017000000923871994:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* u2 (* 2.0 PI))))
   (if (<= t_0 0.017000000923871994)
     (sqrt (- (log1p (- u1))))
     (* (sqrt u1) (cos t_0)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = u2 * (2.0f * ((float) M_PI));
	float tmp;
	if (t_0 <= 0.017000000923871994f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf(u1) * cosf(t_0);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(u2 * Float32(Float32(2.0) * Float32(pi)))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.017000000923871994))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(u1) * cos(t_0));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 2 (PI.f32)) u2) < 0.0170000009

   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-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 97.1%

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

   if 0.0170000009 < (*.f32 (*.f32 2 (PI.f32)) u2)

   1. Initial program 58.2%

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

      1. add-sqr-sqrt 58.2%

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

      2. pow2 58.2%

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

      3. pow1/2 58.2%

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

      4. sqrt-pow1 58.3%

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

      5. add-sqr-sqrt 58.2%

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

      6. sqrt-unprod 58.3%

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

      7. sqr-neg 58.3%

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

      8. sqrt-unprod 1.5%

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

      9. add-sqr-sqrt 1.5%

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

      10. sub-neg 1.5%

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

      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 73.1%

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

      14. sqr-neg 73.1%

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

      15. sqrt-unprod 73.0%

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

      16. add-sqr-sqrt 73.1%

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

      17. metadata-eval 73.1%

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

   3. Applied egg-rr 73.1%

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

   4. Taylor expanded in u1 around 0 75.3%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;u2 \cdot \left(2 \cdot \pi\right) \leq 0.017000000923871994:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot \left(2 \cdot \pi\right)\right)\\ \end{array} \]

Alternative 4: 79.9% 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 58.6%

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

   1. sub-neg 58.6%

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

   2. log1p-def 99.2%

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

   3. associate-*l* 99.2%

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

3. Simplified 99.2%

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

4. Taylor expanded in u2 around 0 80.4%

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

5. Final simplification 80.4%

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

Alternative 5: 72.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 + u1 \cdot \left(u1 \cdot -0.5\right)}} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt
  (/
   (- (* u1 u1) (* 0.25 (* (* u1 u1) (* u1 u1))))
   (+ u1 (* u1 (* u1 -0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((((u1 * u1) - (0.25f * ((u1 * u1) * (u1 * u1)))) / (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) - (0.25e0 * ((u1 * u1) * (u1 * u1)))) / (u1 + (u1 * (u1 * (-0.5e0))))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(Float32(Float32(u1 * u1) - Float32(Float32(0.25) * Float32(Float32(u1 * u1) * Float32(u1 * u1)))) / Float32(u1 + Float32(u1 * Float32(u1 * Float32(-0.5))))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((((u1 * u1) - (single(0.25) * ((u1 * u1) * (u1 * u1)))) / (u1 + (u1 * (u1 * single(-0.5))))));
end

Derivation

1. Initial program 58.6%

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

2. Taylor expanded in u1 around 0 87.9%

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

   1. +-commutative 87.9%

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

   2. mul-1-neg 87.9%

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

   3. unsub-neg 87.9%

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

   4. unpow2 87.9%

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

   5. associate-*r* 87.9%

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

4. Simplified 87.9%

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

5. Taylor expanded in u2 around 0 72.8%

   \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 72.8%

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

   2. metadata-eval 72.8%

      \[\leadsto \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

   3. unpow2 72.8%

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

   4. associate-*r* 72.8%

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

7. Simplified 72.8%

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

   1. flip-+ 72.8%

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

   2. *-commutative 72.8%

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

   3. *-commutative 72.8%

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

   4. *-commutative 72.8%

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

   5. *-commutative 72.8%

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

   6. *-commutative 72.8%

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

   7. *-commutative 72.8%

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

9. Applied egg-rr 72.8%

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

    1. associate-*r* 72.8%

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

    2. unpow2 72.8%

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

    3. *-commutative 72.8%

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

    4. associate-*r* 72.8%

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

    5. unpow2 72.8%

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

    6. *-commutative 72.8%

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

    7. swap-sqr 72.8%

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

    8. metadata-eval 72.8%

       \[\leadsto \sqrt{\frac{u1 \cdot u1 - \color{blue}{0.25} \cdot \left({u1}^{2} \cdot {u1}^{2}\right)}{u1 - u1 \cdot \left(u1 \cdot 0.5\right)}} \]

    9. unpow2 72.8%

       \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\color{blue}{\left(u1 \cdot u1\right)} \cdot {u1}^{2}\right)}{u1 - u1 \cdot \left(u1 \cdot 0.5\right)}} \]

    10. unpow2 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}\right)}{u1 - u1 \cdot \left(u1 \cdot 0.5\right)}} \]

    11. associate-*r* 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 - \color{blue}{\left(u1 \cdot u1\right) \cdot 0.5}}} \]

    12. unpow2 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 - \color{blue}{{u1}^{2}} \cdot 0.5}} \]

    13. *-commutative 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 - \color{blue}{0.5 \cdot {u1}^{2}}}} \]

    14. cancel-sign-sub-inv 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{\color{blue}{u1 + \left(-0.5\right) \cdot {u1}^{2}}}} \]

    15. metadata-eval 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 + \color{blue}{-0.5} \cdot {u1}^{2}}} \]

    16. *-commutative 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 + \color{blue}{{u1}^{2} \cdot -0.5}}} \]

    17. unpow2 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 + \color{blue}{\left(u1 \cdot u1\right)} \cdot -0.5}} \]

    18. associate-*r* 72.8%

        \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 + \color{blue}{u1 \cdot \left(u1 \cdot -0.5\right)}}} \]

11. Simplified 72.8%

    \[\leadsto \sqrt{\color{blue}{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 + u1 \cdot \left(u1 \cdot -0.5\right)}}} \]

12. Final simplification 72.8%

    \[\leadsto \sqrt{\frac{u1 \cdot u1 - 0.25 \cdot \left(\left(u1 \cdot u1\right) \cdot \left(u1 \cdot u1\right)\right)}{u1 + u1 \cdot \left(u1 \cdot -0.5\right)}} \]

Alternative 6: 72.6% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 \cdot \left(1 + u1 \cdot 0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (* u1 (+ 1.0 (* u1 0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 * (1.0f + (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 * (1.0e0 + (u1 * 0.5e0))))
end function

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(u1 * Float32(0.5)))))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = sqrt((u1 * (single(1.0) + (u1 * single(0.5)))));
end

Derivation

1. Initial program 58.6%

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

2. Taylor expanded in u1 around 0 87.9%

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

   1. +-commutative 87.9%

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

   2. mul-1-neg 87.9%

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

   3. unsub-neg 87.9%

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

   4. unpow2 87.9%

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

   5. associate-*r* 87.9%

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

4. Simplified 87.9%

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

5. Taylor expanded in u2 around 0 72.8%

   \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 72.8%

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

   2. metadata-eval 72.8%

      \[\leadsto \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

   3. unpow2 72.8%

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

   4. associate-*r* 72.8%

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

7. Simplified 72.8%

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

   1. distribute-rgt1-in 72.7%

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

   2. *-commutative 72.7%

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

9. Applied egg-rr 72.7%

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

10. Final simplification 72.7%

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

Alternative 7: 72.7% accurate, 3.8× speedup

Math

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

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(u1 * Float32(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 58.6%

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

2. Taylor expanded in u1 around 0 87.9%

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

   1. +-commutative 87.9%

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

   2. mul-1-neg 87.9%

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

   3. unsub-neg 87.9%

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

   4. unpow2 87.9%

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

   5. associate-*r* 87.9%

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

4. Simplified 87.9%

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

5. Taylor expanded in u2 around 0 72.8%

   \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 72.8%

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

   2. metadata-eval 72.8%

      \[\leadsto \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

   3. unpow2 72.8%

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

   4. associate-*r* 72.8%

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

7. Simplified 72.8%

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

8. Final simplification 72.8%

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

Alternative 8: 64.6% 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 58.6%

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

2. Taylor expanded in u1 around 0 87.9%

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

   1. +-commutative 87.9%

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

   2. mul-1-neg 87.9%

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

   3. unsub-neg 87.9%

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

   4. unpow2 87.9%

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

   5. associate-*r* 87.9%

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

4. Simplified 87.9%

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

5. Taylor expanded in u2 around 0 72.8%

   \[\leadsto \color{blue}{\sqrt{u1 - -0.5 \cdot {u1}^{2}}} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 72.8%

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

   2. metadata-eval 72.8%

      \[\leadsto \sqrt{u1 + \color{blue}{0.5} \cdot {u1}^{2}} \]

   3. unpow2 72.8%

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

   4. associate-*r* 72.8%

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

7. Simplified 72.8%

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

8. Taylor expanded in u1 around 0 64.3%

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

9. Final simplification 64.3%

   \[\leadsto \sqrt{u1} \]

Reproduce

herbie shell --seed 2023285 
(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))))