Beckmann Sample, near normal, slope_x

Percentage Accurate: 57.6% → 99.1%

Time: 3.6s

Alternatives: 9

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Initial Program: 57.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

MATLAB

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

Alternative 1: 99.1% accurate, 1.0× speedup

Math

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

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 57.6%

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

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   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 99.1

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

3. Applied rewrites 99.1%

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

   1. lift-*.f32 N/A

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

   2. count-2-rev N/A

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

   3. lift-+.f32 99.1

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

5. Applied rewrites 99.1%

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

Alternative 2: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.006500000134110451:\\ \;\;\;\;\sqrt{u1 \cdot \left(1 + 0.5 \cdot u1\right)} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
      0.006500000134110451)
   (* (sqrt (* u1 (+ 1.0 (* 0.5 u1)))) (cos (* (+ PI PI) u2)))
   (*
    (sqrt (- (log1p (- u1))))
    (fma
     (fma
      (* 0.6666666666666666 (* u2 u2))
      (* (* PI PI) (* PI PI))
      (* (* PI PI) -2.0))
     (* u2 u2)
     1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.006500000134110451f) {
		tmp = sqrtf((u1 * (1.0f + (0.5f * u1)))) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (u2 * u2), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.006500000134110451))
		tmp = Float32(sqrt(Float32(u1 * Float32(Float32(1.0) + Float32(Float32(0.5) * u1)))) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0)));
	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))) < 0.00650000013

   1. Initial program 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      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 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 99.1

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

   5. Applied rewrites 99.1%

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

   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. lower-*.f32 N/A

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

      2. lower-+.f32 N/A

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

      3. lower-*.f32 88.2

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

   8. Applied rewrites 88.2%

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

   if 0.00650000013 < (*.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 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      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 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 91.7

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + {u2}^{2} \cdot \mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right)\right) \]

   6. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 91.7

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right), \color{blue}{{u2}^{2}}, 1\right) \]

   8. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sqrt{-\log \left(1 - u1\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.00023999999393709004:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<=
      (* (sqrt (- (log (- 1.0 u1)))) (cos (* (* 2.0 PI) u2)))
      0.00023999999393709004)
   (* (sqrt u1) (cos (* (+ PI PI) u2)))
   (*
    (sqrt (- (log1p (- u1))))
    (fma
     (fma
      (* 0.6666666666666666 (* u2 u2))
      (* (* PI PI) (* PI PI))
      (* (* PI PI) -2.0))
     (* u2 u2)
     1.0))))

C

float code(float cosTheta_i, float u1, float u2) {
	float tmp;
	if ((sqrtf(-logf((1.0f - u1))) * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.00023999999393709004f) {
		tmp = sqrtf(u1) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = sqrtf(-log1pf(-u1)) * fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (u2 * u2), 1.0f);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	tmp = Float32(0.0)
	if (Float32(sqrt(Float32(-log(Float32(Float32(1.0) - u1)))) * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.00023999999393709004))
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(sqrt(Float32(-log1p(Float32(-u1)))) * fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0)));
	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))) < 2.39999994e-4

   1. Initial program 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      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 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.6%

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

   if 2.39999994e-4 < (*.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 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      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 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 91.7

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + {u2}^{2} \cdot \mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right)\right) \]

   6. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f32 91.7

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right), \color{blue}{{u2}^{2}}, 1\right) \]

   8. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.011599999852478504:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.011599999852478504)
     (* (sqrt u1) (cos (* (+ PI PI) u2)))
     (*
      t_0
      (fma
       (fma
        (* 0.6666666666666666 (* u2 u2))
        (* (* PI PI) (* PI PI))
        (* (* PI PI) -2.0))
       (* u2 u2)
       1.0)))))

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))) <= 0.011599999852478504f) {
		tmp = sqrtf(u1) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = t_0 * fmaf(fmaf((0.6666666666666666f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * (((float) M_PI) * ((float) M_PI))), ((((float) M_PI) * ((float) M_PI)) * -2.0f)), (u2 * u2), 1.0f);
	}
	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(0.011599999852478504))
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = Float32(t_0 * fma(fma(Float32(Float32(0.6666666666666666) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(Float32(pi) * Float32(pi))), Float32(Float32(Float32(pi) * Float32(pi)) * Float32(-2.0))), Float32(u2 * u2), Float32(1.0)));
	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))) < 0.0115999999

   1. Initial program 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      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 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.6%

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

   if 0.0115999999 < (*.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 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      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 99.1

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

   3. Applied rewrites 99.1%

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

   4. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-*.f32 N/A

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

      3. lower-pow.f32 N/A

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

      4. lower-fma.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-PI.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lower-pow.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 91.7

          \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \left(1 + {u2}^{2} \cdot \mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right)\right) \]

   6. Applied rewrites 91.7%

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \color{blue}{\left(1 + {u2}^{2} \cdot \mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. lift-log1p.f32 N/A

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

      2. lift-neg.f32 N/A

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

      3. sub-flip-reverse N/A

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

      4. lift-log.f32 N/A

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

      5. lift--.f32 54.2

         \[\leadsto \sqrt{-\log \color{blue}{\left(1 - u1\right)}} \cdot \left(1 + {u2}^{2} \cdot \mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right)\right) \]

      6. lift-+.f32 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f32 N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left({u2}^{2} \cdot \mathsf{fma}\left(-2, {\pi}^{2}, \frac{2}{3} \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right) + 1\right) \]

      9. *-commutative N/A

         \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\mathsf{fma}\left(-2, {\pi}^{2}, \frac{2}{3} \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right) \cdot {u2}^{2} + 1\right) \]

      10. lower-fma.f32 54.2

          \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(-2, {\pi}^{2}, 0.6666666666666666 \cdot \left({u2}^{2} \cdot {\pi}^{4}\right)\right), \color{blue}{{u2}^{2}}, 1\right) \]

   8. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\pi \cdot \pi\right) \cdot -2\right), u2 \cdot u2, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 88.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.011599999852478504:\\ \;\;\;\;\sqrt{u1} \cdot \cos \left(\left(\pi + \pi\right) \cdot u2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot t\_0, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.011599999852478504)
     (* (sqrt u1) (cos (* (+ PI PI) u2)))
     (fma (* -2.0 (* u2 u2)) (* (* PI PI) t_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))) <= 0.011599999852478504f) {
		tmp = sqrtf(u1) * cosf(((((float) M_PI) + ((float) M_PI)) * u2));
	} else {
		tmp = fmaf((-2.0f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * t_0), 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(0.011599999852478504))
		tmp = Float32(sqrt(u1) * cos(Float32(Float32(Float32(pi) + Float32(pi)) * u2)));
	else
		tmp = fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * t_0), 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))) < 0.0115999999

   1. Initial program 57.6%

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

      1. lift-log.f32 N/A

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

      2. lift--.f32 N/A

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

      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 99.1

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

   3. Applied rewrites 99.1%

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

      1. lift-*.f32 N/A

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

      2. count-2-rev N/A

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

      3. lift-+.f32 99.1

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

   5. Applied rewrites 99.1%

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

   6. Taylor expanded in u1 around 0

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

   8. Applied rewrites 76.6%

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

   if 0.0115999999 < (*.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 57.6%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-neg.f32 N/A

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

   4. Applied rewrites 53.0%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 53.0

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

      8. lift-pow.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 53.0

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

      11. lift-pow.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 53.0

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

   6. Applied rewrites 53.0%

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

Alternative 6: 82.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -2 \cdot \left(u2 \cdot u2\right)\\ t_1 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_1 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.011599999852478504:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left(\pi \cdot \pi\right) \cdot \sqrt{u1}, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, \left(\pi \cdot \pi\right) \cdot t\_1, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* -2.0 (* u2 u2))) (t_1 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_1 (cos (* (* 2.0 PI) u2))) 0.011599999852478504)
     (fma t_0 (* (* PI PI) (sqrt u1)) (sqrt u1))
     (fma t_0 (* (* PI PI) t_1) t_1))))

C

float code(float cosTheta_i, float u1, float u2) {
	float t_0 = -2.0f * (u2 * u2);
	float t_1 = sqrtf(-logf((1.0f - u1)));
	float tmp;
	if ((t_1 * cosf(((2.0f * ((float) M_PI)) * u2))) <= 0.011599999852478504f) {
		tmp = fmaf(t_0, ((((float) M_PI) * ((float) M_PI)) * sqrtf(u1)), sqrtf(u1));
	} else {
		tmp = fmaf(t_0, ((((float) M_PI) * ((float) M_PI)) * t_1), t_1);
	}
	return tmp;
}

Julia

function code(cosTheta_i, u1, u2)
	t_0 = Float32(Float32(-2.0) * Float32(u2 * u2))
	t_1 = sqrt(Float32(-log(Float32(Float32(1.0) - u1))))
	tmp = Float32(0.0)
	if (Float32(t_1 * cos(Float32(Float32(Float32(2.0) * Float32(pi)) * u2))) <= Float32(0.011599999852478504))
		tmp = fma(t_0, Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(u1)), sqrt(u1));
	else
		tmp = fma(t_0, Float32(Float32(Float32(pi) * Float32(pi)) * t_1), t_1);
	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))) < 0.0115999999

   1. Initial program 57.6%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-neg.f32 N/A

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 69.7

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

   7. Applied rewrites 69.7%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f32 N/A

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

   9. Applied rewrites 69.7%

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

   if 0.0115999999 < (*.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 57.6%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-neg.f32 N/A

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

   4. Applied rewrites 53.0%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f32 N/A

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

      7. lower-*.f32 53.0

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

      8. lift-pow.f32 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f32 53.0

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

      11. lift-pow.f32 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f32 53.0

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

   6. Applied rewrites 53.0%

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

Alternative 7: 80.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{if}\;t\_0 \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \leq 0.013500000350177288:\\ \;\;\;\;\mathsf{fma}\left(-2 \cdot \left(u2 \cdot u2\right), \left(\pi \cdot \pi\right) \cdot \sqrt{u1}, \sqrt{u1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (sqrt (- (log (- 1.0 u1))))))
   (if (<= (* t_0 (cos (* (* 2.0 PI) u2))) 0.013500000350177288)
     (fma (* -2.0 (* u2 u2)) (* (* PI PI) (sqrt u1)) (sqrt u1))
     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))) <= 0.013500000350177288f) {
		tmp = fmaf((-2.0f * (u2 * u2)), ((((float) M_PI) * ((float) M_PI)) * sqrtf(u1)), sqrtf(u1));
	} 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(0.013500000350177288))
		tmp = fma(Float32(Float32(-2.0) * Float32(u2 * u2)), Float32(Float32(Float32(pi) * Float32(pi)) * sqrt(u1)), sqrt(u1));
	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))) < 0.0135000004

   1. Initial program 57.6%

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

   2. Taylor expanded in u2 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-neg.f32 N/A

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

      4. lower-log.f32 N/A

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

      5. lower--.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-*.f32 N/A

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

      8. lower-pow.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lower-pow.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. lower-neg.f32 N/A

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

   4. Applied rewrites 53.0%

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

   5. Taylor expanded in u1 around 0

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

      1. lower-+.f32 N/A

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

      2. lower-sqrt.f32 N/A

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

      3. lower-*.f32 N/A

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

      4. lower-*.f32 N/A

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

      5. lower-pow.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. lower-pow.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 69.7

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

   7. Applied rewrites 69.7%

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

      1. lift-+.f32 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. lift-*.f32 N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f32 N/A

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

   9. Applied rewrites 69.7%

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

   if 0.0135000004 < (*.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 57.6%

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

   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. lower-sqrt.f32 N/A

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

      2. lower-neg.f32 N/A

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

      3. lower-log.f32 N/A

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

      4. lower--.f32 49.6

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

   4. Applied rewrites 49.6%

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

Alternative 8: 49.6% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

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(-log((1.0e0 - u1)))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. lower--.f32 49.6

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

4. Applied rewrites 49.6%

   \[\leadsto \color{blue}{\sqrt{-\log \left(1 - u1\right)}} \]
5. Add Preprocessing

Alternative 9: 6.6% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\log 1} \end{array} \]

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return sqrtf(-logf(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(-log(1.0e0))
end function

Julia

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

MATLAB

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

Derivation

1. Initial program 57.6%

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

2. Taylor expanded in u2 around 0

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

   1. lower-sqrt.f32 N/A

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

   2. lower-neg.f32 N/A

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

   3. lower-log.f32 N/A

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

   4. lower--.f32 49.6

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

4. Applied rewrites 49.6%

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

5. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{-\log 1} \]
6. Step-by-step derivation

7. Applied rewrites 6.6%

   \[\leadsto \sqrt{-\log 1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025140 
(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))))