normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 9.3s

Alternatives: 8

Speedup: 1.4×

Specification

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

Math

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

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: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(N[(Pi * u2), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
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. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-log.f64 98.8

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 2: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[Cos[N[(N[(Pi * 2.0), $MachinePrecision] * u2), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Final simplification 99.4%

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

Alternative 3: 98.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(u2 \cdot u2\right) \cdot -0.3333333333333333\\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), t\_0 \cdot t\_0, -0.027777777777777776\right)}{\mathsf{fma}\left(\pi \cdot \pi, t\_0, -0.16666666666666666\right)}, \sqrt{\log u1 \cdot -2}, 0.5\right) \end{array} \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (let* ((t_0 (* (* u2 u2) -0.3333333333333333)))
   (fma
    (/
     (fma (* (* PI PI) (* PI PI)) (* t_0 t_0) -0.027777777777777776)
     (fma (* PI PI) t_0 -0.16666666666666666))
    (sqrt (* (log u1) -2.0))
    0.5)))

C

double code(double u1, double u2) {
	double t_0 = (u2 * u2) * -0.3333333333333333;
	return fma((fma(((((double) M_PI) * ((double) M_PI)) * (((double) M_PI) * ((double) M_PI))), (t_0 * t_0), -0.027777777777777776) / fma((((double) M_PI) * ((double) M_PI)), t_0, -0.16666666666666666)), sqrt((log(u1) * -2.0)), 0.5);
}

Julia

function code(u1, u2)
	t_0 = Float64(Float64(u2 * u2) * -0.3333333333333333)
	return fma(Float64(fma(Float64(Float64(pi * pi) * Float64(pi * pi)), Float64(t_0 * t_0), -0.027777777777777776) / fma(Float64(pi * pi), t_0, -0.16666666666666666)), sqrt(Float64(log(u1) * -2.0)), 0.5)
end

Wolfram

code[u1_, u2_] := Block[{t$95$0 = N[(N[(u2 * u2), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]}, N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -0.027777777777777776), $MachinePrecision] / N[(N[(Pi * Pi), $MachinePrecision] * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]]

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} + \frac{-1}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{6}}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + \frac{1}{6}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right)}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot \frac{-1}{3}}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot \frac{-1}{3}}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-1}{3}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-1}{3}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   11. lower-PI.f64 98.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, \pi \cdot \color{blue}{\pi}, 0.16666666666666666\right), \sqrt{\log u1 \cdot -2}, 0.5\right) \]

7. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, \pi \cdot \pi, 0.16666666666666666\right)}, \sqrt{\log u1 \cdot -2}, 0.5\right) \]
8. Step-by-step derivation

9. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(-0.3333333333333333 \cdot \left(u2 \cdot u2\right)\right) \cdot \left(-0.3333333333333333 \cdot \left(u2 \cdot u2\right)\right), -0.027777777777777776\right)}{\color{blue}{\mathsf{fma}\left(\pi \cdot \pi, -0.3333333333333333 \cdot \left(u2 \cdot u2\right), -0.16666666666666666\right)}}, \sqrt{\log u1 \cdot -2}, 0.5\right) \]

10. Final simplification 98.9%

    \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot \left(\pi \cdot \pi\right), \left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333\right) \cdot \left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333\right), -0.027777777777777776\right)}{\mathsf{fma}\left(\pi \cdot \pi, \left(u2 \cdot u2\right) \cdot -0.3333333333333333, -0.16666666666666666\right)}, \sqrt{\log u1 \cdot -2}, 0.5\right) \]
11. Add Preprocessing

Alternative 4: 98.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot -2, u2 \cdot u2, 1\right) \cdot 0.16666666666666666, \sqrt{\log u1 \cdot -2}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (fma (* (* PI PI) -2.0) (* u2 u2) 1.0) 0.16666666666666666)
  (sqrt (* (log u1) -2.0))
  0.5))

C

double code(double u1, double u2) {
	return fma((fma(((((double) M_PI) * ((double) M_PI)) * -2.0), (u2 * u2), 1.0) * 0.16666666666666666), sqrt((log(u1) * -2.0)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(fma(Float64(Float64(pi * pi) * -2.0), Float64(u2 * u2), 1.0) * 0.16666666666666666), sqrt(Float64(log(u1) * -2.0)), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(N[(Pi * Pi), $MachinePrecision] * -2.0), $MachinePrecision] * N[(u2 * u2), $MachinePrecision] + 1.0), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. rem-square-sqrt N/A

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

   3. unpow2 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. rem-square-sqrt N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-PI.f64 N/A

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

   16. lower-PI.f64 N/A

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

   17. unpow2 N/A

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

   18. rem-square-sqrt N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 98.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot -2, \color{blue}{u2 \cdot u2}, 1\right) \cdot 0.16666666666666666, \sqrt{\log u1 \cdot -2}, 0.5\right) \]

7. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(\pi \cdot \pi\right) \cdot -2, u2 \cdot u2, 1\right)} \cdot 0.16666666666666666, \sqrt{\log u1 \cdot -2}, 0.5\right) \]
8. Add Preprocessing

Alternative 5: 98.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, \pi \cdot \pi, 0.16666666666666666\right), \sqrt{\log u1 \cdot -2}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (fma (* (* u2 u2) -0.3333333333333333) (* PI PI) 0.16666666666666666)
  (sqrt (* (log u1) -2.0))
  0.5))

C

double code(double u1, double u2) {
	return fma(fma(((u2 * u2) * -0.3333333333333333), (((double) M_PI) * ((double) M_PI)), 0.16666666666666666), sqrt((log(u1) * -2.0)), 0.5);
}

Julia

function code(u1, u2)
	return fma(fma(Float64(Float64(u2 * u2) * -0.3333333333333333), Float64(pi * pi), 0.16666666666666666), sqrt(Float64(log(u1) * -2.0)), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[(N[(u2 * u2), $MachinePrecision] * -0.3333333333333333), $MachinePrecision] * N[(Pi * Pi), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} + \frac{-1}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{6}}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{-1}{3} \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + \frac{1}{6}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{3} \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right)}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot \frac{-1}{3}}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot \frac{-1}{3}}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-1}{3}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot \frac{-1}{3}, {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   10. lower-PI.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot \frac{-1}{3}, \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), \frac{1}{6}\right), \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]

   11. lower-PI.f64 98.9

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, \pi \cdot \color{blue}{\pi}, 0.16666666666666666\right), \sqrt{\log u1 \cdot -2}, 0.5\right) \]

7. Applied rewrites 98.9%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -0.3333333333333333, \pi \cdot \pi, 0.16666666666666666\right)}, \sqrt{\log u1 \cdot -2}, 0.5\right) \]
8. Add Preprocessing

Alternative 6: 98.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
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. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-log.f64 98.8

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{2}, \sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}}, \frac{1}{2}\right) \]
9. Step-by-step derivation

10. Applied rewrites 98.7%

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

Alternative 7: 98.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(0.16666666666666666 * N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \pi\right) \cdot u2\right) + 0.5 \]
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. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-log.f64 98.8

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.8

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

4. Applied rewrites 98.8%

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

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.4%

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

10. Taylor expanded in u2 around 0

    \[\leadsto \mathsf{fma}\left(\frac{1}{6}, \sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}}, \frac{1}{2}\right) \]
11. Step-by-step derivation

12. Applied rewrites 98.5%

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

Alternative 8: 98.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[u1_, u2_] := N[(0.16666666666666666 * N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

4. Applied rewrites 99.4%

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

5. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6}}, \sqrt{\log u1 \cdot -2}, \frac{1}{2}\right) \]
6. Step-by-step derivation

7. Applied rewrites 98.5%

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

Reproduce

herbie shell --seed 2024240 
(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))