Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.4% → 99.2%

Time: 4.0s

Alternatives: 13

Speedup: 9.1×

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: 99.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* PI (fma u2 -2.0 0.5)))))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * sinf((((float) M_PI) * fmaf(u2, -2.0f, 0.5f)));
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * sin(Float32(Float32(pi) * fma(u2, Float32(-2.0), Float32(0.5)))))
end

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 99.0

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

5. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. lift-*.f32 N/A

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

   4. lift-PI.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-+.f32 N/A

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

   7. count-2-rev N/A

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

   8. *-commutative N/A

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

   9. count-2-rev N/A

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

   10. lift-+.f32 N/A

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

   11. lift-PI.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lift-*.f32 N/A

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

   14. lift-neg.f32 N/A

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

   15. sin-+PI/2 N/A

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

7. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 0.5 \cdot \pi\right)\right)} \]
8. Step-by-step derivation

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. lift-fma.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(-2 \cdot u2\right) \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \pi\right)} \]

   4. lift-PI.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-2 \cdot u2\right) \cdot \mathsf{PI}\left(\right) + \frac{1}{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   5. lift-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-2 \cdot u2\right) \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{1}{2} \cdot \mathsf{PI}\left(\right)}\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(-2 \cdot u2 + \frac{1}{2}\right)\right)} \]

   7. lower-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(-2 \cdot u2 + \frac{1}{2}\right)\right)} \]

   8. lift-PI.f32 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f32 99.2

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\pi \cdot \color{blue}{\mathsf{fma}\left(u2, -2, 0.5\right)}\right) \]

9. Applied rewrites 99.2%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\pi \cdot \mathsf{fma}\left(u2, -2, 0.5\right)\right)} \]
10. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-PI.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. lower-+.f32 N/A

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

   5. lift-PI.f32 N/A

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

   6. lift-PI.f32 99.0

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

5. Applied rewrites 99.0%

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

Alternative 3: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.054999999701976776)
   (*
    (sqrt (- (log1p (- u1))))
    (fma
     (* u2 u2)
     (fma
      (* (* (* (* PI PI) PI) PI) (* u2 u2))
      0.6666666666666666
      (* (* PI PI) -2.0))
     1.0))
   (* (fma (* (sqrt u1) u1) 0.25 (sqrt u1)) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.054999999701976776f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((u2 * u2), fmaf(((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * ((float) M_PI)) * (u2 * u2)), 0.6666666666666666f, ((((float) M_PI) * ((float) M_PI)) * -2.0f)), 1.0f);
	} else {
		tmp = fmaf((sqrtf(u1) * u1), 0.25f, sqrtf(u1)) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(u2 * u2), fma(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(pi)) * Float32(u2 * u2)), Float32(0.6666666666666666), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(1.0)));
	else
		tmp = Float32(fma(Float32(sqrt(u1) * u1), Float32(0.25), sqrt(u1)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0549999997

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lift-PI.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]

   11. Applied rewrites 91.1%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right), 1\right)} \]

   if 0.0549999997 < u2

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in u1 around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. pow1/2 N/A

         \[\leadsto \left(\frac{{u1}^{2}}{{u1}^{\frac{1}{2}}} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      4. pow-div N/A

         \[\leadsto \left({u1}^{\left(2 - \frac{1}{2}\right)} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. sqrt-pow2 N/A

         \[\leadsto \left({\left(\sqrt{u1}\right)}^{3} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      8. lower-fma.f32 N/A

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

      9. cube-mult N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot \left(\sqrt{u1} \cdot \sqrt{u1}\right), \frac{1}{4}, \sqrt{u1}\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      10. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, \frac{1}{4}, \sqrt{u1}\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      11. lower-*.f32 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, \frac{1}{4}, \sqrt{u1}\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      12. lower-sqrt.f32 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, \frac{1}{4}, \sqrt{u1}\right) \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      13. lower-sqrt.f32 88.3

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

   8. Applied rewrites 88.3%

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

Alternative 4: 97.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.054999999701976776)
   (*
    (sqrt (- (log1p (- u1))))
    (fma
     (* u2 u2)
     (fma
      (* (* (* (* PI PI) PI) PI) (* u2 u2))
      0.6666666666666666
      (* (* PI PI) -2.0))
     1.0))
   (* (sqrt (* (fma 0.5 u1 1.0) u1)) (cos (* (+ PI PI) u2)))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.054999999701976776f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((u2 * u2), fmaf(((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * ((float) M_PI)) * (u2 * u2)), 0.6666666666666666f, ((((float) M_PI) * ((float) M_PI)) * -2.0f)), 1.0f);
	} else {
		tmp = sqrtf((fmaf(0.5f, u1, 1.0f) * u1)) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(u2 * u2), fma(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(pi)) * Float32(u2 * u2)), Float32(0.6666666666666666), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(1.0)));
	else
		tmp = Float32(sqrt(Float32(fma(Float32(0.5), u1, Float32(1.0)) * u1)) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0549999997

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lift-PI.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]

   11. Applied rewrites 91.1%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right), 1\right)} \]

   if 0.0549999997 < u2

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 99.0

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

   5. Applied rewrites 99.0%

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

   6. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. +-commutative N/A

         \[\leadsto \sqrt{\left(\frac{1}{2} \cdot u1 + 1\right) \cdot u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right) \]

      4. lower-fma.f32 88.1

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

   8. Applied rewrites 88.1%

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

Alternative 5: 95.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.05999999865889549:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 0.5 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.05999999865889549)
   (*
    (sqrt (- (log1p (- u1))))
    (fma
     (* u2 u2)
     (fma
      (* (* (* (* PI PI) PI) PI) (* u2 u2))
      0.6666666666666666
      (* (* PI PI) -2.0))
     1.0))
   (* (sqrt u1) (sin (fma (* -2.0 u2) PI (* 0.5 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.05999999865889549f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((u2 * u2), fmaf(((((((float) M_PI) * ((float) M_PI)) * ((float) M_PI)) * ((float) M_PI)) * (u2 * u2)), 0.6666666666666666f, ((((float) M_PI) * ((float) M_PI)) * -2.0f)), 1.0f);
	} else {
		tmp = sqrtf(u1) * sinf(fmaf((-2.0f * u2), ((float) M_PI), (0.5f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.05999999865889549))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(u2 * u2), fma(Float32(Float32(Float32(Float32(Float32(pi) * Float32(pi)) * Float32(pi)) * Float32(pi)) * Float32(u2 * u2)), Float32(0.6666666666666666), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * sin(fma(Float32(Float32(-2.0) * u2), Float32(pi), Float32(Float32(0.5) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0599999987

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lift-PI.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \left(-2 \cdot {\mathsf{PI}\left(\right)}^{2} + \frac{2}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{4}\right)\right)\right)} \]

   11. Applied rewrites 91.1%

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(u2 \cdot u2, \mathsf{fma}\left(\left(\left(\left(\pi \cdot \pi\right) \cdot \pi\right) \cdot \pi\right) \cdot \left(u2 \cdot u2\right), 0.6666666666666666, \left(\pi \cdot \pi\right) \cdot -2\right), 1\right)} \]

   if 0.0599999987 < u2

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 99.0

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

   5. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. lift-neg.f32 N/A

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

      15. sin-+PI/2 N/A

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

   7. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 0.5 \cdot \pi\right)\right)} \]

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 76.5%

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

Alternative 6: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.02240999974310398:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 0.5 \cdot \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.02240999974310398)
   (* (sqrt (- (log1p (- u1)))) (fma (* -2.0 (* u2 u2)) (* PI PI) 1.0))
   (* (sqrt u1) (sin (fma (* -2.0 u2) PI (* 0.5 PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.02240999974310398f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = sqrtf(u1) * sinf(fmaf((-2.0f * u2), ((float) M_PI), (0.5f * ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.02240999974310398))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * sin(fma(Float32(Float32(-2.0) * u2), Float32(pi), Float32(Float32(0.5) * Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0224099997

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. pow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      4. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \pi\right)\right) + 1\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {u2}^{2}\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \]

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 88.0

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

   6. Applied rewrites 88.0%

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

   if 0.0224099997 < u2

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-PI.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-PI.f32 99.0

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

   5. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. lift-*.f32 N/A

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

      4. lift-PI.f32 N/A

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

      5. lift-PI.f32 N/A

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

      6. lift-+.f32 N/A

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

      7. count-2-rev N/A

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

      8. *-commutative N/A

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

      9. count-2-rev N/A

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

      10. lift-+.f32 N/A

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

      11. lift-PI.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-*.f32 N/A

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

      14. lift-neg.f32 N/A

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

      15. sin-+PI/2 N/A

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

   7. Applied rewrites 99.2%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(-2 \cdot u2, \pi, 0.5 \cdot \pi\right)\right)} \]

   8. Taylor expanded in u1 around 0

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

   10. Applied rewrites 76.5%

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

Alternative 7: 94.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u2 \leq 0.02240999974310398:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(u2 \cdot \left(\pi + \pi\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u2 0.02240999974310398)
   (* (sqrt (- (log1p (- u1)))) (fma (* -2.0 (* u2 u2)) (* PI PI) 1.0))
   (* (sqrt u1) (cos (* u2 (+ PI PI))))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if (u2 <= 0.02240999974310398f) {
		tmp = sqrtf(-log1pf(-u1)) * fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 1.0f);
	} else {
		tmp = sqrtf(u1) * cosf((u2 * (((float) M_PI) + ((float) M_PI))));
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (u2 <= Float32(0.02240999974310398))
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)));
	else
		tmp = Float32(sqrt(u1) * cos(Float32(u2 * Float32(Float32(pi) + Float32(pi)))));
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if u2 < 0.0224099997

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

   4. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

      2. pow2 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      3. lift-*.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      4. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

      5. lift-PI.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \pi\right)\right) + 1\right) \]

      6. associate-*r* N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {u2}^{2}\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \]

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. pow2 N/A

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

      10. lift-*.f32 88.0

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

   6. Applied rewrites 88.0%

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

   if 0.0224099997 < u2

   1. Initial program 57.4%

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

   2. Taylor expanded in u1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f32 N/A

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

      7. lift-PI.f32 N/A

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

      8. lift-*.f32 N/A

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

      9. lift-cos.f32 76.4

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 76.4

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

      13. lift-PI.f32 N/A

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

      14. lift-*.f32 N/A

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

      15. count-2-rev N/A

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

      16. lower-+.f32 N/A

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

      17. lift-PI.f32 N/A

          \[\leadsto \sqrt{u1} \cdot \cos \left(u2 \cdot \left(\pi + \mathsf{PI}\left(\right)\right)\right) \]

      18. lift-PI.f32 76.4

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

   4. Applied rewrites 76.4%

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

Alternative 8: 88.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \pi \cdot \pi, 1\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (fma (* -2.0 (* u2 u2)) (* PI PI) 1.0)))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * fmaf((-2.0f * (u2 * u2)), (((float) M_PI) * ((float) M_PI)), 1.0f);
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(pi) * Float32(pi)), Float32(1.0)))
end

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.0

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

3. Applied rewrites 99.0%

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

4. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} \]
5. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \color{blue}{1}\right) \]

   2. pow2 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

   3. lift-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

   4. lift-PI.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \mathsf{PI}\left(\right)\right)\right) + 1\right) \]

   5. lift-PI.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(-2 \cdot \left({u2}^{2} \cdot \left(\pi \cdot \pi\right)\right) + 1\right) \]

   6. associate-*r* N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(\left(-2 \cdot {u2}^{2}\right) \cdot \left(\pi \cdot \pi\right) + 1\right) \]

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. pow2 N/A

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

   10. lift-*.f32 88.0

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

6. Applied rewrites 88.0%

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

Alternative 9: 80.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot 1 \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) 1.0))

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-log1pf(-u1)) * 1.0f;
}

Julia

function code(cosTheta_i, u1, u2)
	return Float32(sqrt(Float32(-log1p(Float32(-u1)))) * Float32(1.0))
end

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lower-+.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   6. lower-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   12. count-2-rev N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   13. lower-+.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   14. lift-PI.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lift-PI.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 80.0%

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

Alternative 10: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.054999999701976776)
     (* (fma (* (sqrt u1) u1) 0.25 (sqrt u1)) 1.0)
     t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.054999999701976776f) {
		tmp = fmaf((sqrtf(u1) * u1), 0.25f, sqrtf(u1)) * 1.0f;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.054999999701976776))
		tmp = Float32(fma(Float32(sqrt(u1) * u1), Float32(0.25), sqrt(u1)) * Float32(1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0549999997

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lift-PI.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in u1 around 0

      \[\leadsto \color{blue}{\left(\sqrt{u1} + \frac{1}{4} \cdot \frac{{u1}^{2}}{\sqrt{u1}}\right)} \cdot 1 \]
   10. Step-by-step derivation

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow1/2 N/A

          \[\leadsto \left(\frac{{u1}^{2}}{{u1}^{\frac{1}{2}}} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot 1 \]

       4. pow-div N/A

          \[\leadsto \left({u1}^{\left(2 - \frac{1}{2}\right)} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot 1 \]

       5. metadata-eval N/A

          \[\leadsto \left({u1}^{\frac{3}{2}} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot 1 \]

       6. metadata-eval N/A

          \[\leadsto \left({u1}^{\left(\frac{3}{2}\right)} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot 1 \]

       7. sqrt-pow2 N/A

          \[\leadsto \left({\left(\sqrt{u1}\right)}^{3} \cdot \frac{1}{4} + \sqrt{u1}\right) \cdot 1 \]

       8. lower-fma.f32 N/A

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

       9. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot \left(\sqrt{u1} \cdot \sqrt{u1}\right), \frac{1}{4}, \sqrt{u1}\right) \cdot 1 \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot {\left(\sqrt{u1}\right)}^{2}, \frac{1}{4}, \sqrt{u1}\right) \cdot 1 \]

       11. sqrt-pow2 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot {u1}^{\left(\frac{2}{2}\right)}, \frac{1}{4}, \sqrt{u1}\right) \cdot 1 \]

       12. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot {u1}^{1}, \frac{1}{4}, \sqrt{u1}\right) \cdot 1 \]

       13. unpow1 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, \frac{1}{4}, \sqrt{u1}\right) \cdot 1 \]

       14. lower-*.f32 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, \frac{1}{4}, \sqrt{u1}\right) \cdot 1 \]

       15. lower-sqrt.f32 N/A

           \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, \frac{1}{4}, \sqrt{u1}\right) \cdot 1 \]

       16. lower-sqrt.f32 72.9

           \[\leadsto \mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right) \cdot 1 \]

   11. Applied rewrites 72.9%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{u1} \cdot u1, 0.25, \sqrt{u1}\right)} \cdot 1 \]

   if 0.0549999997 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 49.2

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

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 78.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.054999999701976776:\\ \;\;\;\;\sqrt{-\mathsf{fma}\left(-0.5, u1, -1\right) \cdot u1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.054999999701976776)
     (* (sqrt (- (* (fma -0.5 u1 -1.0) u1))) 1.0)
     t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.054999999701976776f) {
		tmp = sqrtf(-(fmaf(-0.5f, u1, -1.0f) * u1)) * 1.0f;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.054999999701976776))
		tmp = Float32(sqrt(Float32(-Float32(fma(Float32(-0.5), u1, Float32(-1.0)) * u1))) * Float32(1.0));
	else
		tmp = t_0;
	end
	return tmp
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0549999997

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lift-PI.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. sub-flip N/A

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

       4. metadata-eval N/A

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

       5. lower-fma.f32 72.7

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

   11. Applied rewrites 72.7%

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

   if 0.0549999997 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 49.2

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

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.014000000432133675:\\ \;\;\;\;\sqrt{u1} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.014000000432133675)
     (* (sqrt u1) 1.0)
     t_0)))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_0 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.014000000432133675f) {
		tmp = sqrtf(u1) * 1.0f;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_0 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.014000000432133675))
		tmp = Float32(sqrt(u1) * Float32(1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(cosTheta_i, u1, u2)
	t_0 = sqrt(-log((single(1.0) - u1)));
	tmp = single(0.0);
	if ((t_0 * cos(((single(2.0) * single(pi)) * u2))) <= single(0.014000000432133675))
		tmp = sqrt(u1) * single(1.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Derivation

1. Split input into 2 regimes

2. if (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2))) < 0.0140000004

   1. Initial program 57.4%

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

      1. lift--.f32 N/A

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

      2. lift-log.f32 N/A

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

      3. sub-flip N/A

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

      4. lower-log1p.f32 N/A

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

      5. lower-neg.f32 99.0

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

   3. Applied rewrites 99.0%

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

      1. lift-cos.f32 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lower-+.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      6. lower-neg.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      11. lift-*.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. count-2-rev N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-+.f32 N/A

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lift-PI.f32 N/A

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

      15. lift-PI.f32 N/A

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

      16. lower-/.f32 N/A

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

      17. lift-PI.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u2 around 0

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in u1 around 0

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

   11. Applied rewrites 64.7%

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

   if 0.0140000004 < (*.f32 (sqrt.f32 (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))) (cos.f32 (*.f32 (*.f32 #s(literal 2 binary32) (PI.f32)) u2)))

   1. Initial program 57.4%

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

   2. Taylor expanded in u2 around 0

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      3. lift-neg.f32 N/A

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

      4. lift-sqrt.f32 49.2

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

   4. Applied rewrites 49.2%

      \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 64.7% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot 1 \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(u1) * 1.0f;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = sqrt(u1) * 1.0e0
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.4%

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

   1. lift--.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. sub-flip N/A

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

   4. lower-log1p.f32 N/A

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

   5. lower-neg.f32 99.0

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

3. Applied rewrites 99.0%

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

   1. lift-cos.f32 N/A

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

   2. cos-neg-rev N/A

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

   3. sin-+PI/2-rev N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   4. lower-sin.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   5. lower-+.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(\mathsf{neg}\left(\left(2 \cdot \pi\right) \cdot u2\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   6. lower-neg.f32 N/A

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

   7. lift-*.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \left(2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   11. lift-*.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   12. count-2-rev N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   13. lower-+.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(-u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   14. lift-PI.f32 N/A

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

   15. lift-PI.f32 N/A

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

   16. lower-/.f32 N/A

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

   17. lift-PI.f32 99.1

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in u2 around 0

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 80.0%

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

9. Taylor expanded in u1 around 0

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

11. Applied rewrites 64.7%

    \[\leadsto \sqrt{\color{blue}{u1}} \cdot 1 \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025134 
(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))))