Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.4% → 99.0%

Time: 14.7s

Alternatives: 18

Speedup: 21.0×

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: 57.4% 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, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.5%

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

4. Applied egg-rr 99.1%

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

Alternative 2: 90.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\left(2 \cdot \pi\right) \cdot u2\right)\\ \mathbf{if}\;t\_0 \leq 0.9999905228614807:\\ \;\;\;\;t\_0 \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cos (* (* 2.0 PI) u2))))
   (if (<= t_0 0.9999905228614807)
     (* t_0 (sqrt u1))
     (sqrt (- (log1p (- u1)))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = cosf(((2.0f * ((float) M_PI)) * u2));
	float tmp;
	if (t_0 <= 0.9999905228614807f) {
		tmp = t_0 * sqrtf(u1);
	} else {
		tmp = sqrtf(-log1pf(-u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))
	tmp = Float32(0.0)
	if (t_0 <= Float32(0.9999905228614807))
		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 #s(literal 2 binary32) (PI.f32)) u2)) < 0.999990523

   1. Initial program 57.9%

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

   4. Applied egg-rr 98.7%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-sqrt.f32 76.7

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

   7. Simplified 76.7%

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

   if 0.999990523 < (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))

   1. Initial program 60.3%

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

   4. Applied egg-rr 99.3%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 58.2

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

   6. Applied egg-rr 58.2%

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

   7. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \]

      5. lower-log1p.f32 N/A

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

      6. lower-neg.f32 97.6

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

   9. Simplified 97.6%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.9999905228614807:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{u1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 59.5%

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

   1. sub-neg N/A

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

   2. lower-log1p.f32 N/A

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

   3. lower-neg.f32 99.0

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

4. Applied egg-rr 99.0%

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

   1. lift-PI.f32 N/A

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

   2. associate-*r* N/A

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

   3. lift-*.f32 N/A

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

   4. count-2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right) \cdot u2} + \mathsf{PI}\left(\right) \cdot u2\right) \]

   6. lift-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot u2 + \color{blue}{\mathsf{PI}\left(\right) \cdot u2}\right) \]

   7. distribute-lft-out N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

   9. lower-+.f32 99.0

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

6. Applied egg-rr 99.0%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \cos \color{blue}{\left(\pi \cdot \left(u2 + u2\right)\right)} \]
7. Add Preprocessing

Alternative 4: 97.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.009999999776482582)
   (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))
   (*
    (cos (* PI (+ u2 u2)))
    (sqrt
     (*
      (- u1)
      (fma u1 (fma u1 (fma u1 -0.25 -0.3333333333333333) -0.5) -1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.009999999776482582f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	} else {
		tmp = cosf((((float) M_PI) * (u2 + u2))) * sqrtf((-u1 * fmaf(u1, fmaf(u1, fmaf(u1, -0.25f, -0.3333333333333333f), -0.5f), -1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.009999999776482582))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(Float32(-u1) * fma(u1, fma(u1, fma(u1, Float32(-0.25), Float32(-0.3333333333333333)), Float32(-0.5)), Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.00999999978

   1. Initial program 60.0%

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

   4. Applied egg-rr 99.3%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 58.0

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

   6. Applied egg-rr 58.0%

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

   7. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(\frac{1}{1 - u1}\right)} + -2 \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{1 - u1}\right)} + \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(1 + {u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

   9. Simplified 99.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}} \]

   if 0.00999999978 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.3%

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

      1. sub-neg N/A

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

      2. lower-log1p.f32 N/A

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

      3. lower-neg.f32 98.4

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

   4. Applied egg-rr 98.4%

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

      1. lift-PI.f32 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f32 N/A

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

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right) \cdot u2} + \mathsf{PI}\left(\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot u2 + \color{blue}{\mathsf{PI}\left(\right) \cdot u2}\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      9. lower-+.f32 98.4

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) - \frac{1}{2}, -1\right)}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot \left(\frac{-1}{4} \cdot u1 - \frac{1}{3}\right) + \color{blue}{\frac{-1}{2}}, -1\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{4} \cdot u1 - \frac{1}{3}, \frac{-1}{2}\right)}, -1\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{4} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, \frac{-1}{2}\right), -1\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{4}} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), \frac{-1}{2}\right), -1\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{4} + \color{blue}{\frac{-1}{3}}, \frac{-1}{2}\right), -1\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      11. lower-fma.f32 93.1

          \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right)}, -0.5\right), -1\right)} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) \]

   9. Simplified 93.1%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.009999999776482582:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.25, -0.3333333333333333\right), -0.5\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.04660499840974808:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.04660499840974808)
   (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))
   (*
    (cos (* PI (+ u2 u2)))
    (sqrt (fma (* u1 u1) (fma u1 (fma u1 0.25 0.3333333333333333) 0.5) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.04660499840974808f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	} else {
		tmp = cosf((((float) M_PI) * (u2 + u2))) * sqrtf(fmaf((u1 * u1), fmaf(u1, fmaf(u1, 0.25f, 0.3333333333333333f), 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.04660499840974808))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(fma(Float32(u1 * u1), fma(u1, fma(u1, Float32(0.25), Float32(0.3333333333333333)), Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0466049984

   1. Initial program 59.5%

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

   4. Applied egg-rr 99.3%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 57.4

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

   6. Applied egg-rr 57.4%

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

   7. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(\frac{1}{1 - u1}\right)} + -2 \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{1 - u1}\right)} + \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(1 + {u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

   9. Simplified 99.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}} \]

   if 0.0466049984 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 59.3%

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

      1. sub-neg N/A

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

      2. lower-log1p.f32 N/A

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

      3. lower-neg.f32 98.4

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

   4. Applied egg-rr 98.4%

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

      1. lift-PI.f32 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f32 N/A

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

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right) \cdot u2} + \mathsf{PI}\left(\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot u2 + \color{blue}{\mathsf{PI}\left(\right) \cdot u2}\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      9. lower-+.f32 98.4

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

   6. Applied egg-rr 98.4%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) + u1 \cdot 1}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot u1\right) \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)} + u1 \cdot 1} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \sqrt{\color{blue}{{u1}^{2}} \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + u1 \cdot 1} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. *-rgt-identity N/A

         \[\leadsto \sqrt{{u1}^{2} \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + \color{blue}{u1}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      6. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left({u1}^{2}, \frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      9. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      10. lower-fma.f32 N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{\mathsf{fma}\left(u1, \frac{1}{3} + \frac{1}{4} \cdot u1, \frac{1}{2}\right)}, u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, \frac{1}{2}\right), u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      12. *-commutative N/A

          \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{1}{4}} + \frac{1}{3}, \frac{1}{2}\right), u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      13. lower-fma.f32 91.7

          \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right)}, 0.5\right), u1\right)} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) \]

   9. Simplified 91.7%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.04660499840974808:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, 0.25, 0.3333333333333333\right), 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 97.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot u2\\ \mathbf{if}\;t\_0 \leq 0.07999999821186066:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos t\_0 \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 PI) u2)))
   (if (<= t_0 0.07999999821186066)
     (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))
     (*
      (cos t_0)
      (sqrt (- (* u1 (fma u1 (fma u1 -0.3333333333333333 -0.5) -1.0))))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = (2.0f * ((float) M_PI)) * u2;
	float tmp;
	if (t_0 <= 0.07999999821186066f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	} else {
		tmp = cosf(t_0) * sqrtf(-(u1 * fmaf(u1, fmaf(u1, -0.3333333333333333f, -0.5f), -1.0f)));
	}
	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(0.07999999821186066))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	else
		tmp = Float32(cos(t_0) * sqrt(Float32(-Float32(u1 * fma(u1, fma(u1, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0799999982

   1. Initial program 59.8%

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

   4. Applied egg-rr 99.3%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 57.6

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(\frac{1}{1 - u1}\right)} + -2 \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{1 - u1}\right)} + \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(1 + {u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

   9. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}} \]

   if 0.0799999982 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.4%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3} \cdot u1 - \frac{1}{2}, -1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{3} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 89.3

         \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Simplified 89.3%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.07999999821186066:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 97.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.07999999821186066:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.07999999821186066)
   (* (sqrt (- (log1p (- u1)))) (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0))
   (*
    (cos (* PI (+ u2 u2)))
    (sqrt (fma u1 (* u1 (fma u1 0.3333333333333333 0.5)) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.07999999821186066f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f);
	} else {
		tmp = cosf((((float) M_PI) * (u2 + u2))) * sqrtf(fmaf(u1, (u1 * fmaf(u1, 0.3333333333333333f, 0.5f)), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.07999999821186066))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(fma(u1, Float32(u1 * fma(u1, Float32(0.3333333333333333), Float32(0.5))), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 0.0799999982

   1. Initial program 59.8%

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

   4. Applied egg-rr 99.3%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 57.6

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

   6. Applied egg-rr 57.6%

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

   7. Taylor expanded in u2 around 0

      \[\leadsto \color{blue}{\sqrt{\log \left(\frac{1}{1 - u1}\right)} + -2 \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{1 - u1}\right)} + \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      2. distribute-rgt1-in N/A

         \[\leadsto \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(1 + \color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \left(1 + \color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      6. *-commutative N/A

         \[\leadsto \left(1 + {u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)} \]

      7. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{\log \left(\frac{1}{1 - u1}\right)}} \]

   9. Simplified 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \sqrt{-\mathsf{log1p}\left(-u1\right)}} \]

   if 0.0799999982 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 58.4%

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

      1. sub-neg N/A

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

      2. lower-log1p.f32 N/A

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

      3. lower-neg.f32 98.3

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

   4. Applied egg-rr 98.3%

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

      1. lift-PI.f32 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f32 N/A

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

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right) \cdot u2} + \mathsf{PI}\left(\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot u2 + \color{blue}{\mathsf{PI}\left(\right) \cdot u2}\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      9. lower-+.f32 98.3

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

   6. Applied egg-rr 98.3%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) + 1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) + u1 \cdot 1}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) + \color{blue}{u1}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right), u1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)}, u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot \color{blue}{\left(\frac{1}{3} \cdot u1 + \frac{1}{2}\right)}, u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot \left(\color{blue}{u1 \cdot \frac{1}{3}} + \frac{1}{2}\right), u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      8. lower-fma.f32 89.2

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot \color{blue}{\mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right)}, u1\right)} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) \]

   9. Simplified 89.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.07999999821186066:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 96.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.0003499999875202775)
   (sqrt (- (log1p (- u1))))
   (*
    (cos (* PI (+ u2 u2)))
    (sqrt (fma u1 (* u1 (fma u1 0.3333333333333333 0.5)) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.0003499999875202775f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((((float) M_PI) * (u2 + u2))) * sqrtf(fmaf(u1, (u1 * fmaf(u1, 0.3333333333333333f, 0.5f)), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.0003499999875202775))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(fma(u1, Float32(u1 * fma(u1, Float32(0.3333333333333333), Float32(0.5))), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.49999988e-4

   1. Initial program 62.8%

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

   4. Applied egg-rr 99.5%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 61.0

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \]

      5. lower-log1p.f32 N/A

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

      6. lower-neg.f32 99.5

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

   9. Simplified 99.5%

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

   if 3.49999988e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

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

      1. sub-neg N/A

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

      2. lower-log1p.f32 N/A

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

      3. lower-neg.f32 98.5

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

   4. Applied egg-rr 98.5%

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

      1. lift-PI.f32 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f32 N/A

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

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right) \cdot u2} + \mathsf{PI}\left(\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot u2 + \color{blue}{\mathsf{PI}\left(\right) \cdot u2}\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      9. lower-+.f32 98.5

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) + 1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) + u1 \cdot 1}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot \left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) + \color{blue}{u1}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right), u1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)}, u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot \color{blue}{\left(\frac{1}{3} \cdot u1 + \frac{1}{2}\right)}, u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot \left(\color{blue}{u1 \cdot \frac{1}{3}} + \frac{1}{2}\right), u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      8. lower-fma.f32 92.2

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, u1 \cdot \color{blue}{\mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right)}, u1\right)} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) \]

   9. Simplified 92.2%

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}} \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, 0.3333333333333333, 0.5\right), u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 94.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.0003499999875202775)
   (sqrt (- (log1p (- u1))))
   (* (cos (* PI (+ u2 u2))) (sqrt (* (- u1) (fma u1 -0.5 -1.0))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.0003499999875202775f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((((float) M_PI) * (u2 + u2))) * sqrtf((-u1 * fmaf(u1, -0.5f, -1.0f)));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.0003499999875202775))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(Float32(Float32(-u1) * fma(u1, Float32(-0.5), Float32(-1.0)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.49999988e-4

   1. Initial program 62.8%

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

   4. Applied egg-rr 99.5%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 61.0

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \]

      5. lower-log1p.f32 N/A

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

      6. lower-neg.f32 99.5

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

   9. Simplified 99.5%

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

   if 3.49999988e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

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

      1. sub-neg N/A

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

      2. lower-log1p.f32 N/A

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

      3. lower-neg.f32 98.5

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

   4. Applied egg-rr 98.5%

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

      1. lift-PI.f32 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f32 N/A

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

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right) \cdot u2} + \mathsf{PI}\left(\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot u2 + \color{blue}{\mathsf{PI}\left(\right) \cdot u2}\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      9. lower-+.f32 98.5

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(\frac{-1}{2} \cdot u1 - 1\right)}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(\frac{-1}{2} \cdot u1 + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(\color{blue}{u1 \cdot \frac{-1}{2}} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \frac{-1}{2} + \color{blue}{-1}\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. lower-fma.f32 88.6

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

   9. Simplified 88.6%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\left(-u1\right) \cdot \mathsf{fma}\left(u1, -0.5, -1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 94.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.0003499999875202775)
   (sqrt (- (log1p (- u1))))
   (* (cos (* PI (+ u2 u2))) (sqrt (fma u1 (* u1 0.5) u1)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.0003499999875202775f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = cosf((((float) M_PI) * (u2 + u2))) * sqrtf(fmaf(u1, (u1 * 0.5f), u1));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.0003499999875202775))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(cos(Float32(Float32(pi) * Float32(u2 + u2))) * sqrt(fma(u1, Float32(u1 * Float32(0.5)), u1)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.49999988e-4

   1. Initial program 62.8%

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

   4. Applied egg-rr 99.5%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 61.0

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \]

      5. lower-log1p.f32 N/A

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

      6. lower-neg.f32 99.5

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

   9. Simplified 99.5%

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

   if 3.49999988e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

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

      1. sub-neg N/A

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

      2. lower-log1p.f32 N/A

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

      3. lower-neg.f32 98.5

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

   4. Applied egg-rr 98.5%

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

      1. lift-PI.f32 N/A

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

      2. associate-*r* N/A

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

      3. lift-*.f32 N/A

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

      4. count-2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\color{blue}{\mathsf{PI}\left(\right) \cdot u2} + \mathsf{PI}\left(\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot u2 + \color{blue}{\mathsf{PI}\left(\right) \cdot u2}\right) \]

      7. distribute-lft-out N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)\right)} \cdot \cos \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

      9. lower-+.f32 98.5

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + \frac{1}{2} \cdot u1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]
   8. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{u1 \cdot \color{blue}{\left(\frac{1}{2} \cdot u1 + 1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      2. distribute-lft-in N/A

         \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(\frac{1}{2} \cdot u1\right) + u1 \cdot 1}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      3. *-rgt-identity N/A

         \[\leadsto \sqrt{u1 \cdot \left(\frac{1}{2} \cdot u1\right) + \color{blue}{u1}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(u1, \frac{1}{2} \cdot u1, u1\right)}} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{1}{2}}, u1\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right) \]

      6. lower-*.f32 88.5

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

   9. Simplified 88.5%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\pi \cdot \left(u2 + u2\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot 0.5, u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot u2 \leq 0.0003499999875202775:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (* (* 2.0 PI) u2) 0.0003499999875202775)
   (sqrt (- (log1p (- u1))))
   (*
    (sqrt (- (* u1 (fma u1 (fma u1 -0.3333333333333333 -0.5) -1.0))))
    (fma (* u2 u2) (* -2.0 (* PI PI)) 1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (((2.0f * ((float) M_PI)) * u2) <= 0.0003499999875202775f) {
		tmp = sqrtf(-log1pf(-u1));
	} else {
		tmp = sqrtf(-(u1 * fmaf(u1, fmaf(u1, -0.3333333333333333f, -0.5f), -1.0f))) * fmaf((u2 * u2), (-2.0f * (((float) M_PI) * ((float) M_PI))), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(Float32(Float32(2.0) * Float32(pi)) * u2) <= Float32(0.0003499999875202775))
		tmp = sqrt(Float32(-log1p(Float32(-u1))));
	else
		tmp = Float32(sqrt(Float32(-Float32(u1 * fma(u1, fma(u1, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))))) * fma(Float32(u2 * u2), Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi))), Float32(1.0)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2) < 3.49999988e-4

   1. Initial program 62.8%

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

   4. Applied egg-rr 99.5%

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

      1. lift-+.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. diff-log N/A

         \[\leadsto \sqrt{\color{blue}{\log \left(\frac{1 + u1}{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. clear-num N/A

         \[\leadsto \sqrt{\log \color{blue}{\left(\frac{1}{\frac{1 + u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}{1 + u1}}\right)}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. lift-*.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{u1 \cdot \left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lift-neg.f32 N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + u1 \cdot \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. distribute-rgt-neg-out N/A

         \[\leadsto \sqrt{\log \left(\frac{1}{\frac{1 + \color{blue}{\left(\mathsf{neg}\left(u1 \cdot u1\right)\right)}}{1 + u1}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. sub-neg N/A

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

      10. metadata-eval N/A

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

      11. lift-+.f32 N/A

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

      12. flip-- N/A

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

      13. lower-log.f32 N/A

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

      14. lower-/.f32 N/A

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

      15. lower--.f32 61.0

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

   6. Applied egg-rr 61.0%

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

   7. Taylor expanded in u2 around 0

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

      1. lower-sqrt.f32 N/A

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

      2. log-rec N/A

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

      3. lower-neg.f32 N/A

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

      4. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\log \color{blue}{\left(1 + \left(\mathsf{neg}\left(u1\right)\right)\right)}\right)} \]

      5. lower-log1p.f32 N/A

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

      6. lower-neg.f32 99.5

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

   9. Simplified 99.5%

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

   if 3.49999988e-4 < (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)

   1. Initial program 55.7%

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

   3. Taylor expanded in u1 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   4. Step-by-step derivation

      1. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3} \cdot u1 - \frac{1}{2}, -1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. sub-neg N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{3} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. metadata-eval N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. lower-fma.f32 92.2

         \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   5. Simplified 92.2%

      \[\leadsto \sqrt{-\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) \]

      3. associate-*r* N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) \]

      4. *-commutative N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      7. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

      8. lower-*.f32 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{-2 \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

      9. unpow2 N/A

         \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      10. lower-*.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

      11. lower-PI.f32 N/A

          \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

      12. lower-PI.f32 72.3

          \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

   8. Simplified 72.3%

      \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 82.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \sqrt{u1 - \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (fma u2 (* u2 (* -2.0 (* PI PI))) 1.0)
  (sqrt (- u1 (* (* u1 u1) (fma u1 -0.3333333333333333 -0.5))))))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf(u2, (u2 * (-2.0f * (((float) M_PI) * ((float) M_PI)))), 1.0f) * sqrtf((u1 - ((u1 * u1) * fmaf(u1, -0.3333333333333333f, -0.5f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(u2, Float32(u2 * Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi)))), Float32(1.0)) * sqrt(Float32(u1 - Float32(Float32(u1 * u1) * fma(u1, Float32(-0.3333333333333333), Float32(-0.5))))))
end

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3} \cdot u1 - \frac{1}{2}, -1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{3} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-fma.f32 89.2

      \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Simplified 89.2%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)} + -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + -1 \cdot u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lift-neg.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), u1, \mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lower-*.f32 89.3

      \[\leadsto \sqrt{-\mathsf{fma}\left(\color{blue}{u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, u1, -u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Applied egg-rr 89.3%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), u1, -u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

8. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)} + -2 \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)}\right)} \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)} + \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)}} \]

   2. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \cdot \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)}} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \cdot \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(1 + \color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2}\right) \cdot \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)} \]

   5. associate-*r* N/A

      \[\leadsto \left(1 + \color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)}\right) \cdot \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)} \]

   6. *-commutative N/A

      \[\leadsto \left(1 + {u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{u1 - {u1}^{2} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)}} \]

10. Simplified 80.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(u2, u2 \cdot \left(-2 \cdot \left(\pi \cdot \pi\right)\right), 1\right) \cdot \sqrt{u1 - \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}} \]
11. Add Preprocessing

Alternative 13: 82.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (- (* u1 (fma u1 (fma u1 -0.3333333333333333 -0.5) -1.0))))
  (fma (* u2 u2) (* -2.0 (* PI PI)) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-(u1 * fmaf(u1, fmaf(u1, -0.3333333333333333f, -0.5f), -1.0f))) * fmaf((u2 * u2), (-2.0f * (((float) M_PI) * ((float) M_PI))), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-Float32(u1 * fma(u1, fma(u1, Float32(-0.3333333333333333), Float32(-0.5)), Float32(-1.0))))) * fma(Float32(u2 * u2), Float32(Float32(-2.0) * Float32(Float32(pi) * Float32(pi))), Float32(1.0)))
end

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3} \cdot u1 - \frac{1}{2}, -1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{3} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-fma.f32 89.2

      \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Simplified 89.2%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \left(\color{blue}{\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot -2} + 1\right) \]

   3. associate-*r* N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \left(\color{blue}{{u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot -2\right)} + 1\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \left({u2}^{2} \cdot \color{blue}{\left(-2 \cdot {\mathsf{PI}\left(\right)}^{2}\right)} + 1\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, -2 \cdot {\mathsf{PI}\left(\right)}^{2}, 1\right) \]

   8. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{-2 \cdot {\mathsf{PI}\left(\right)}^{2}}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

   10. lower-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}, 1\right) \]

   11. lower-PI.f32 N/A

       \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), -1\right)\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right), 1\right) \]

   12. lower-PI.f32 79.8

       \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \color{blue}{\pi}\right), 1\right) \]

8. Simplified 79.8%

   \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, -2 \cdot \left(\pi \cdot \pi\right), 1\right)} \]
9. Add Preprocessing

Alternative 14: 78.6% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, -0.3333333333333333, 0.5\right), u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (fma (* -2.0 (* u2 u2)) (* PI PI) 1.0)
  (sqrt (fma (* u1 u1) (fma u1 -0.3333333333333333 0.5) u1))))

C

float code(float cosTheta_i, float u1, float u2) {
	return fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 1.0f) * sqrtf(fmaf((u1 * u1), fmaf(u1, -0.3333333333333333f, 0.5f), u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)) * sqrt(fma(Float32(u1 * u1), fma(u1, Float32(-0.3333333333333333), Float32(0.5)), u1)))
end

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3} \cdot u1 - \frac{1}{2}, -1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{3} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-fma.f32 89.2

      \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Simplified 89.2%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)} + -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + -1 \cdot u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lift-neg.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), u1, \mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lower-*.f32 89.3

      \[\leadsto \sqrt{-\mathsf{fma}\left(\color{blue}{u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, u1, -u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Applied egg-rr 89.3%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), u1, -u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

9. Applied egg-rr 84.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right), \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, 0.5\right), u1\right)}, \left(-\left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, 0.5\right), u1\right)}\right)} \]

10. Taylor expanded in u2 around 0

    \[\leadsto \color{blue}{\sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} + -2 \cdot \left(\left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)}\right)} \]
11. Step-by-step derivation

    1. associate-*r* N/A

       \[\leadsto \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} + \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)}} \]

    2. distribute-rgt1-in N/A

       \[\leadsto \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)}} \]

    3. lower-*.f32 N/A

       \[\leadsto \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)}} \]

    4. associate-*r* N/A

       \[\leadsto \left(\color{blue}{\left(-2 \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    5. lower-fma.f32 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    6. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{-2 \cdot {u2}^{2}}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    7. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    8. lower-*.f32 N/A

       \[\leadsto \mathsf{fma}\left(-2 \cdot \color{blue}{\left(u2 \cdot u2\right)}, {\mathsf{PI}\left(\right)}^{2}, 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    9. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    10. lower-*.f32 N/A

        \[\leadsto \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    11. lower-PI.f32 N/A

        \[\leadsto \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    12. lower-PI.f32 N/A

        \[\leadsto \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) \cdot \sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)} \]

    13. lower-sqrt.f32 N/A

        \[\leadsto \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \color{blue}{\sqrt{u1 + {u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right)}} \]

    14. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\color{blue}{{u1}^{2} \cdot \left(\frac{1}{2} + \frac{-1}{3} \cdot u1\right) + u1}} \]

    15. lower-fma.f32 N/A

        \[\leadsto \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left({u1}^{2}, \frac{1}{2} + \frac{-1}{3} \cdot u1, u1\right)}} \]

12. Simplified 76.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \cdot \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, -0.3333333333333333, 0.5\right), u1\right)}} \]
13. Add Preprocessing

Alternative 15: 75.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1 - \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (- u1 (* (* u1 u1) (fma u1 -0.3333333333333333 -0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf((u1 - ((u1 * u1) * fmaf(u1, -0.3333333333333333f, -0.5f))));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(Float32(u1 - Float32(Float32(u1 * u1) * fma(u1, Float32(-0.3333333333333333), Float32(-0.5)))))
end

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3} \cdot u1 - \frac{1}{2}, -1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{3} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-fma.f32 89.2

      \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Simplified 89.2%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)} + -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + -1 \cdot u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lift-neg.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), u1, \mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lower-*.f32 89.3

      \[\leadsto \sqrt{-\mathsf{fma}\left(\color{blue}{u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, u1, -u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Applied egg-rr 89.3%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), u1, -u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

8. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower--.f32 N/A

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

   3. lower-*.f32 N/A

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

   4. unpow2 N/A

      \[\leadsto \sqrt{u1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \sqrt{u1 - \color{blue}{\left(u1 \cdot u1\right)} \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right)} \]

   6. sub-neg N/A

      \[\leadsto \sqrt{u1 - \left(u1 \cdot u1\right) \cdot \color{blue}{\left(\frac{-1}{3} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{u1 - \left(u1 \cdot u1\right) \cdot \left(\color{blue}{u1 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)} \]

   8. metadata-eval N/A

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

   9. lower-fma.f32 71.8

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

10. Simplified 71.8%

    \[\leadsto \color{blue}{\sqrt{u1 - \left(u1 \cdot u1\right) \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}} \]
11. Add Preprocessing

Alternative 16: 72.2% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, -0.3333333333333333, 0.5\right), u1\right)} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (sqrt (fma (* u1 u1) (fma u1 -0.3333333333333333 0.5) u1)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(fmaf((u1 * u1), fmaf(u1, -0.3333333333333333f, 0.5f), u1));
}

Julia

function code(cosTheta_i, u1, u2)
	return sqrt(fma(Float32(u1 * u1), fma(u1, Float32(-0.3333333333333333), Float32(0.5)), u1))
end

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. lower-*.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) - 1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \left(\frac{-1}{3} \cdot u1 - \frac{1}{2}\right) + \color{blue}{-1}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3} \cdot u1 - \frac{1}{2}, -1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. sub-neg N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\frac{-1}{3} \cdot u1 + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, \color{blue}{u1 \cdot \frac{-1}{3}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \mathsf{fma}\left(u1, u1 \cdot \frac{-1}{3} + \color{blue}{\frac{-1}{2}}, -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-fma.f32 89.2

      \[\leadsto \sqrt{-u1 \cdot \mathsf{fma}\left(u1, \color{blue}{\mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, -1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

5. Simplified 89.2%

   \[\leadsto \sqrt{-\color{blue}{u1 \cdot \mathsf{fma}\left(u1, \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), -1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]
6. Step-by-step derivation

   1. lift-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(u1 \cdot \left(u1 \cdot \color{blue}{\mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)} + -1\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + -1 \cdot u1\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. neg-mul-1 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. lift-neg.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\left(\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right)\right) \cdot u1 + \color{blue}{\left(\mathsf{neg}\left(u1\right)\right)}\right)\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{neg}\left(\color{blue}{\mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, \frac{-1}{3}, \frac{-1}{2}\right), u1, \mathsf{neg}\left(u1\right)\right)}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. lower-*.f32 89.3

      \[\leadsto \sqrt{-\mathsf{fma}\left(\color{blue}{u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right)}, u1, -u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

7. Applied egg-rr 89.3%

   \[\leadsto \sqrt{-\color{blue}{\mathsf{fma}\left(u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, -0.5\right), u1, -u1\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \]

9. Applied egg-rr 84.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right), \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, 0.5\right), u1\right)}, \left(-\left(0.5 + -0.5 \cdot \cos \left(\pi \cdot \left(u2 + u2\right)\right)\right)\right) \cdot \sqrt{\mathsf{fma}\left(u1, u1 \cdot \mathsf{fma}\left(u1, -0.3333333333333333, 0.5\right), u1\right)}\right)} \]

10. Taylor expanded in u2 around 0

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

    1. lower-sqrt.f32 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f32 N/A

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

    4. unpow2 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{2} + \frac{-1}{3} \cdot u1, u1\right)} \]

    5. lower-*.f32 N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot u1}, \frac{1}{2} + \frac{-1}{3} \cdot u1, u1\right)} \]

    6. +-commutative N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{\frac{-1}{3} \cdot u1 + \frac{1}{2}}, u1\right)} \]

    7. *-commutative N/A

       \[\leadsto \sqrt{\mathsf{fma}\left(u1 \cdot u1, \color{blue}{u1 \cdot \frac{-1}{3}} + \frac{1}{2}, u1\right)} \]

    8. lower-fma.f32 68.7

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

12. Simplified 68.7%

    \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(u1 \cdot u1, \mathsf{fma}\left(u1, -0.3333333333333333, 0.5\right), u1\right)}} \]
13. Add Preprocessing

Alternative 17: 64.9% accurate, 10.5× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return u1 / 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 = u1 / sqrt(u1)
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 59.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 3.4

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

5. Simplified 3.4%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{-1 \cdot \sqrt{u1}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-sqrt.f32 5.6

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

8. Simplified 5.6%

   \[\leadsto \color{blue}{-\sqrt{u1}} \]
9. Step-by-step derivation

   1. lift-sqrt.f32 N/A

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

   2. neg-sub0 N/A

      \[\leadsto \color{blue}{0 - \sqrt{u1}} \]

   3. metadata-eval N/A

      \[\leadsto \color{blue}{\log 1} - \sqrt{u1} \]

   4. flip-- N/A

      \[\leadsto \color{blue}{\frac{\log 1 \cdot \log 1 - \sqrt{u1} \cdot \sqrt{u1}}{\log 1 + \sqrt{u1}}} \]

10. Applied egg-rr 62.6%

    \[\leadsto \color{blue}{\frac{u1}{\sqrt{u1}}} \]
11. Add Preprocessing

Alternative 18: 65.0% accurate, 21.0× 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 59.5%

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

3. Taylor expanded in u1 around 0

   \[\leadsto \color{blue}{\left(\sqrt{u1} \cdot {\left(\sqrt{-1}\right)}^{2}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 3.4

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

5. Simplified 3.4%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{-1 \cdot \sqrt{u1}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-sqrt.f32 5.6

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

8. Simplified 5.6%

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

10. Applied egg-rr 62.5%

    \[\leadsto \color{blue}{\sqrt{u1}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024207 
(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))))