normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 8.6s

Alternatives: 2

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double u1, double u2) {
	return (((1.0 / 6.0) * pow((-2.0 * log(u1)), 0.5)) * cos(((2.0 * ((double) M_PI)) * u2))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (((1.0 / 6.0) * Math.pow((-2.0 * Math.log(u1)), 0.5)) * Math.cos(((2.0 * Math.PI) * u2))) + 0.5;
}

Python

def code(u1, u2):
	return (((1.0 / 6.0) * math.pow((-2.0 * math.log(u1)), 0.5)) * math.cos(((2.0 * math.pi) * u2))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(Float64(Float64(1.0 / 6.0) * (Float64(-2.0 * log(u1)) ^ 0.5)) * cos(Float64(Float64(2.0 * pi) * u2))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (((1.0 / 6.0) * ((-2.0 * log(u1)) ^ 0.5)) * cos(((2.0 * pi) * u2))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(1.0 / 6.0), $MachinePrecision] * N[Power[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Alternative 1: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double u1, double u2) {
	return (cos(((2.0 * ((double) M_PI)) * u2)) * sqrt((log(u1) * -0.05555555555555555))) + 0.5;
}

Java

public static double code(double u1, double u2) {
	return (Math.cos(((2.0 * Math.PI) * u2)) * Math.sqrt((Math.log(u1) * -0.05555555555555555))) + 0.5;
}

Python

def code(u1, u2):
	return (math.cos(((2.0 * math.pi) * u2)) * math.sqrt((math.log(u1) * -0.05555555555555555))) + 0.5

Julia

function code(u1, u2)
	return Float64(Float64(cos(Float64(Float64(2.0 * pi) * u2)) * sqrt(Float64(log(u1) * -0.05555555555555555))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = (cos(((2.0 * pi) * u2)) * sqrt((log(u1) * -0.05555555555555555))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(N[(2.0 * Pi), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.3%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.3%

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

   3. pow1/2 99.3%

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

   4. *-commutative 99.3%

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

   5. pow1/2 99.3%

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

   6. *-commutative 99.3%

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

   7. swap-sqr 99.4%

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

   8. add-sqr-sqrt 99.6%

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

   9. metadata-eval 99.6%

      \[\leadsto \sqrt{\left(-2 \cdot \log u1\right) \cdot \left(\color{blue}{0.16666666666666666} \cdot \frac{1}{6}\right)} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   10. metadata-eval 99.6%

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

   11. metadata-eval 99.6%

       \[\leadsto \sqrt{\left(-2 \cdot \log u1\right) \cdot \color{blue}{0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

3. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\sqrt{\left(-2 \cdot \log u1\right) \cdot 0.027777777777777776}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
4. Step-by-step derivation

   1. *-commutative 99.6%

      \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot 0.027777777777777776} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   2. associate-*l* 99.6%

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

   3. metadata-eval 99.6%

      \[\leadsto \sqrt{\log u1 \cdot \color{blue}{-0.05555555555555555}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\sqrt{\log u1 \cdot -0.05555555555555555}} \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]

6. Final simplification 99.6%

   \[\leadsto \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) \cdot \sqrt{\log u1 \cdot -0.05555555555555555} + 0.5 \]

Alternative 2: 98.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

2. Taylor expanded in u2 around 0 97.0%

   \[\leadsto \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \color{blue}{1} + 0.5 \]
3. Step-by-step derivation

   1. *-rgt-identity 97.0%

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}} + 0.5 \]

   2. metadata-eval 97.0%

      \[\leadsto \color{blue}{0.16666666666666666} \cdot {\left(-2 \cdot \log u1\right)}^{0.5} + 0.5 \]

   3. pow1/2 97.0%

      \[\leadsto 0.16666666666666666 \cdot \color{blue}{\sqrt{-2 \cdot \log u1}} + 0.5 \]

   4. add-cube-cbrt 95.9%

      \[\leadsto \color{blue}{\left(\sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \cdot \sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}\right) \cdot \sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}} + 0.5 \]

   5. associate-*r* 95.9%

      \[\leadsto \color{blue}{\sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \cdot \left(\sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \cdot \sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}\right)} + 0.5 \]

   6. fma-def 95.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}, \sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}} \cdot \sqrt[3]{0.16666666666666666 \cdot \sqrt{-2 \cdot \log u1}}, 0.5\right)} \]

4. Applied egg-rr 96.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}}, \sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}, 0.5\right)} \]
5. Step-by-step derivation

   1. fma-udef 96.6%

      \[\leadsto \color{blue}{\sqrt{\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}} \cdot \sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} + 0.5} \]

   2. unpow1/2 96.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{0.5}} \cdot \sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776} + 0.5 \]

   3. pow-plus 96.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{\left(0.5 + 1\right)}} + 0.5 \]

   4. metadata-eval 96.6%

      \[\leadsto {\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{\color{blue}{1.5}} + 0.5 \]

   5. metadata-eval 96.6%

      \[\leadsto {\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{\color{blue}{\left(\frac{3}{2}\right)}} + 0.5 \]

   6. associate-*l* 96.6%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\log u1 \cdot \left(-2 \cdot 0.027777777777777776\right)}}\right)}^{\left(\frac{3}{2}\right)} + 0.5 \]

   7. metadata-eval 96.6%

      \[\leadsto {\left(\sqrt[3]{\log u1 \cdot \color{blue}{-0.05555555555555555}}\right)}^{\left(\frac{3}{2}\right)} + 0.5 \]

   8. +-commutative 96.6%

      \[\leadsto \color{blue}{0.5 + {\left(\sqrt[3]{\log u1 \cdot -0.05555555555555555}\right)}^{\left(\frac{3}{2}\right)}} \]

   9. metadata-eval 96.6%

      \[\leadsto 0.5 + {\left(\sqrt[3]{\log u1 \cdot \color{blue}{\left(-2 \cdot 0.027777777777777776\right)}}\right)}^{\left(\frac{3}{2}\right)} \]

   10. associate-*l* 96.6%

       \[\leadsto 0.5 + {\left(\sqrt[3]{\color{blue}{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}}\right)}^{\left(\frac{3}{2}\right)} \]

   11. metadata-eval 96.6%

       \[\leadsto 0.5 + {\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{\color{blue}{1.5}} \]

   12. sqr-pow 96.6%

       \[\leadsto 0.5 + \color{blue}{{\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{\left(\frac{1.5}{2}\right)} \cdot {\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{\left(\frac{1.5}{2}\right)}} \]

   13. fabs-sqr 96.6%

       \[\leadsto 0.5 + \color{blue}{\left|{\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{\left(\frac{1.5}{2}\right)} \cdot {\left(\sqrt[3]{\left(\log u1 \cdot -2\right) \cdot 0.027777777777777776}\right)}^{\left(\frac{1.5}{2}\right)}\right|} \]

6. Simplified 97.3%

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

7. Final simplification 97.3%

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

Reproduce

herbie shell --seed 2023188 
(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))