normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 8.7s

Alternatives: 3

Speedup: 1.5×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 (PI)) u2)))
  0.5))

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 (PI)) u2)))
  0.5))

Alternative 1: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (sqrt (* -0.05555555555555555 (log u1))) (cos (* (* 2.0 (PI)) u2)))
  0.5))

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. unpow1/2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. lower-sqrt.f64 99.4

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \sqrt{\color{blue}{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \sqrt{\color{blue}{\log u1 \cdot -2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-*.f64 99.4

      \[\leadsto \left(\frac{1}{6} \cdot \sqrt{\color{blue}{\log u1 \cdot -2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

4. Applied rewrites 99.4%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\sqrt{\log u1 \cdot -2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + \color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + \color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1 \cdot -2} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot \left(\sqrt{\log u1 \cdot -2} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6} \cdot \left(\sqrt{\log u1 \cdot -2} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)\right)\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{1}{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}\right)\right) \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{0.5 - \left(-0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} - \left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} - \color{blue}{\left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{1}{2}} \]

   5. lower-+.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{1}{2}} \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5} \]
9. Add Preprocessing

Alternative 2: 99.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* -0.05555555555555555 (log u1))) (cos (* (* 2.0 (PI)) u2)) 0.5))

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. unpow1/2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. lower-sqrt.f64 99.4

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

   4. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \sqrt{\color{blue}{-2 \cdot \log u1}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \sqrt{\color{blue}{\log u1 \cdot -2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-*.f64 99.4

      \[\leadsto \left(\frac{1}{6} \cdot \sqrt{\color{blue}{\log u1 \cdot -2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

4. Applied rewrites 99.4%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\sqrt{\log u1 \cdot -2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + \color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} + \color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. associate-*l* N/A

      \[\leadsto \frac{1}{2} + \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1 \cdot -2} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

   6. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left(\frac{1}{6}\right)\right) \cdot \left(\sqrt{\log u1 \cdot -2} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \frac{1}{2} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6} \cdot \left(\sqrt{\log u1 \cdot -2} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)\right)\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{1}{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}\right)\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\frac{1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)}\right)\right) \]

6. Applied rewrites 99.4%

   \[\leadsto \color{blue}{0.5 - \left(-0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} - \left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{1}{2} - \color{blue}{\left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)} \]

   4. +-commutative N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6} \cdot \sqrt{\log u1 \cdot -2}\right)\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + \frac{1}{2}} \]

8. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right), 0.5\right)} \]
9. Add Preprocessing

Alternative 3: 98.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\log u1 \cdot -0.05555555555555555} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (sqrt (* (log u1) -0.05555555555555555)) 0.5))

C

double code(double u1, double u2) {
	return sqrt((log(u1) * -0.05555555555555555)) + 0.5;
}

Fortran

real(8) function code(u1, u2)
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt((log(u1) * (-0.05555555555555555d0))) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt((Math.log(u1) * -0.05555555555555555)) + 0.5;
}

Python

def code(u1, u2):
	return math.sqrt((math.log(u1) * -0.05555555555555555)) + 0.5

Julia

function code(u1, u2)
	return Float64(sqrt(Float64(log(u1) * -0.05555555555555555)) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt((log(u1) * -0.05555555555555555)) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\sqrt{{\left(\log u1 \cdot -2\right)}^{1} \cdot 0.027777777777777776}} + 0.5 \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{{\left(\log u1 \cdot -2\right)}^{1} \cdot \frac{1}{36}}} + \frac{1}{2} \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{\color{blue}{{\left(\log u1 \cdot -2\right)}^{1}} \cdot \frac{1}{36}} + \frac{1}{2} \]

   3. unpow1 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot \frac{1}{36}} + \frac{1}{2} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot \frac{1}{36}} + \frac{1}{2} \]

   5. associate-*l* N/A

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot \frac{1}{36}\right)}} + \frac{1}{2} \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{\log u1 \cdot \color{blue}{\frac{-1}{18}}} + \frac{1}{2} \]

   7. metadata-eval N/A

      \[\leadsto \sqrt{\log u1 \cdot \color{blue}{\left(\frac{1}{36} \cdot -2\right)}} + \frac{1}{2} \]

   8. lower-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left(\frac{1}{36} \cdot -2\right)}} + \frac{1}{2} \]

   9. metadata-eval 98.4

      \[\leadsto \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} + 0.5 \]

6. Applied rewrites 98.4%

   \[\leadsto \sqrt{\color{blue}{\log u1 \cdot -0.05555555555555555}} + 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024332 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 (PI)) u2))) 0.5))