normal distribution

Percentage Accurate: 99.4% → 99.5%

Time: 6.8s

Alternatives: 4

Speedup: 1.4×

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: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(0.16666666666666666 \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (*
   (* 0.16666666666666666 (* (sqrt 2.0) (sqrt (- (log u1)))))
   (cos (* (* 2.0 (PI)) u2)))
  0.5))

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in u1 around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. log-rec N/A

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

   6. lower-neg.f64 N/A

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

   7. lower-log.f64 99.6

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

5. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. metadata-eval 99.6

      \[\leadsto \left(\color{blue}{0.16666666666666666} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

7. Applied rewrites 99.6%

   \[\leadsto \left(\color{blue}{0.16666666666666666} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (sqrt (* (log u1) -2.0)) 0.16666666666666666)
  (cos (* u2 (* (PI) 2.0)))
  0.5))

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
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 99.5

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

   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 99.5

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

4. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\sqrt{\log u1 \cdot -2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
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. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 99.5

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

   7. lift-/.f64 N/A

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

   8. metadata-eval 99.5

      \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot \color{blue}{0.16666666666666666}, \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right), 0.5\right) \]

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666, \cos \color{blue}{\left(u2 \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)}, 0.5\right) \]

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.5

       \[\leadsto \mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666, \cos \left(u2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)}\right), 0.5\right) \]

6. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2} \cdot 0.16666666666666666, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.5\right)} \]
7. Add Preprocessing

Alternative 3: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} + 0.5 \end{array} \]

FPCore

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

C

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

Python

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

Julia

function code(u1, u2)
	return Float64(sqrt(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[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\sqrt{{\left(\log u1 \cdot -2\right)}^{1} \cdot 0.027777777777777776}} + 0.5 \]
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. *-commutative N/A

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

   8. lift-log.f64 N/A

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

   9. log-pow-rev N/A

      \[\leadsto \sqrt{\color{blue}{\log \left({u1}^{\frac{-1}{18}}\right)}} + \frac{1}{2} \]

   10. lower-log.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\log \left({u1}^{\frac{-1}{18}}\right)}} + \frac{1}{2} \]

   11. lower-pow.f64 98.8

       \[\leadsto \sqrt{\log \color{blue}{\left({u1}^{-0.05555555555555555}\right)}} + 0.5 \]

6. Applied rewrites 98.8%

   \[\leadsto \sqrt{\color{blue}{\log \left({u1}^{-0.05555555555555555}\right)}} + 0.5 \]
7. Add Preprocessing

Alternative 4: 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 99.5%

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

4. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\sqrt{{\left(\log u1 \cdot -2\right)}^{1} \cdot 0.027777777777777776}} + 0.5 \]
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. lower-*.f64 98.7

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

6. Applied rewrites 98.7%

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

Reproduce

herbie shell --seed 2024326 
(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))