normal distribution

Percentage Accurate: 99.4% → 99.4%

Time: 9.6s

Alternatives: 1

Speedup: 1.5×

Specification

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

Math

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

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: 99.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double u1, double u2) {
	return fma((cos((((double) M_PI) * (u2 + u2))) * 0.16666666666666666), sqrt((-2.0 * log(u1))), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(cos(Float64(pi * Float64(u2 + u2))) * 0.16666666666666666), sqrt(Float64(-2.0 * log(u1))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(Pi * N[(u2 + u2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sqrt[N[(-2.0 * N[Log[u1], $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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. accelerator-lowering-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. PI-lowering-PI.f64 N/A

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

   12. unpow1/2 N/A

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

   13. sqrt-lowering-sqrt.f64 N/A

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

   14. *-lowering-*.f64 N/A

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

   15. log-lowering-log.f64 99.3

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

4. Applied egg-rr 99.3%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. cos-2 N/A

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

   6. cos-sum N/A

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

   7. cos-lowering-cos.f64 N/A

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

   8. distribute-lft-out N/A

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

   9. *-lowering-*.f64 N/A

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

   10. PI-lowering-PI.f64 N/A

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

   11. +-lowering-+.f64 99.3

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

6. Applied egg-rr 99.3%

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

Reproduce

herbie shell --seed 2024208 
(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))