exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 15.0s

Alternatives: 10

Speedup: N/A×

• could not determine a ground truth

  1. w = 2.00000000000000007e154
  2. l = -2

Specification

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* (exp (- w)) (pow l (exp w))))

C

double code(double w, double l) {
	return exp(-w) * pow(l, exp(w));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = exp(-w) * (l ** exp(w))
end function

Java

public static double code(double w, double l) {
	return Math.exp(-w) * Math.pow(l, Math.exp(w));
}

Python

def code(w, l):
	return math.exp(-w) * math.pow(l, math.exp(w))

Julia

function code(w, l)
	return Float64(exp(Float64(-w)) * (l ^ exp(w)))
end

MATLAB

function tmp = code(w, l)
	tmp = exp(-w) * (l ^ exp(w));
end

Wolfram

code[w_, l_] := N[(N[Exp[(-w)], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\sqrt[3]{e^{w}}}\\ \frac{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{t_0}}{{t_0}^{2}}}{{\left({t_0}^{3}\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (cbrt (cbrt (exp w)))))
   (/ (/ (/ (pow l (exp w)) t_0) (pow t_0 2.0)) (pow (pow t_0 3.0) 2.0))))

C

double code(double w, double l) {
	double t_0 = cbrt(cbrt(exp(w)));
	return ((pow(l, exp(w)) / t_0) / pow(t_0, 2.0)) / pow(pow(t_0, 3.0), 2.0);
}

Java

public static double code(double w, double l) {
	double t_0 = Math.cbrt(Math.cbrt(Math.exp(w)));
	return ((Math.pow(l, Math.exp(w)) / t_0) / Math.pow(t_0, 2.0)) / Math.pow(Math.pow(t_0, 3.0), 2.0);
}

Julia

function code(w, l)
	t_0 = cbrt(cbrt(exp(w)))
	return Float64(Float64(Float64((l ^ exp(w)) / t_0) / (t_0 ^ 2.0)) / ((t_0 ^ 3.0) ^ 2.0))
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Power[N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Taylor expanded in l around inf 94.1%

   \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]
5. Step-by-step derivation

   1. mul-1-neg 94.1%

      \[\leadsto \frac{e^{\color{blue}{-e^{w} \cdot \log \left(\frac{1}{\ell}\right)}}}{e^{w}} \]

   2. distribute-rgt-neg-in 94.1%

      \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \left(-\log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]

   3. log-rec 94.1%

      \[\leadsto \frac{e^{e^{w} \cdot \color{blue}{\log \left(\frac{1}{\frac{1}{\ell}}\right)}}}{e^{w}} \]

   4. remove-double-div 94.1%

      \[\leadsto \frac{e^{e^{w} \cdot \log \color{blue}{\ell}}}{e^{w}} \]

6. Simplified 94.1%

   \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell}}}{e^{w}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 94.1%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{e^{w} \cdot \log \ell}}}{e^{w}} \]

   2. metadata-eval 94.1%

      \[\leadsto \frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot e^{e^{w} \cdot \log \ell}}{e^{w}} \]

   3. add-cube-cbrt 94.1%

      \[\leadsto \frac{\left(3 \cdot 0.3333333333333333\right) \cdot e^{e^{w} \cdot \log \ell}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

   4. times-frac 94.1%

      \[\leadsto \color{blue}{\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}}} \]

   5. metadata-eval 94.1%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   6. pow2 94.1%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   7. *-commutative 94.1%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{e^{\color{blue}{\log \ell \cdot e^{w}}}}{\sqrt[3]{e^{w}}} \]

   8. exp-to-pow 99.2%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}} \]

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
9. Step-by-step derivation

   1. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]

   2. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

10. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]
11. Step-by-step derivation

    1. *-un-lft-identity 99.2%

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot {\ell}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    2. add-cube-cbrt 99.2%

       \[\leadsto \frac{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    3. unpow2 99.2%

       \[\leadsto \frac{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    4. cbrt-prod 99.2%

       \[\leadsto \frac{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{\color{blue}{\sqrt[3]{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    5. times-frac 99.2%

       \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    6. unpow2 99.2%

       \[\leadsto \frac{\frac{1}{\sqrt[3]{\color{blue}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    7. cbrt-prod 99.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{\sqrt[3]{\sqrt[3]{e^{w}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    8. pow2 99.2%

       \[\leadsto \frac{\frac{1}{\color{blue}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

12. Applied egg-rr 99.2%

    \[\leadsto \frac{\color{blue}{\frac{1}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
13. Step-by-step derivation

    1. associate-*l/ 99.2%

       \[\leadsto \frac{\color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    2. *-lft-identity 99.2%

       \[\leadsto \frac{\frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

14. Simplified 99.2%

    \[\leadsto \frac{\color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
15. Step-by-step derivation

    1. add-cube-cbrt 99.2%

       \[\leadsto \frac{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{e^{w}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)}}^{2}} \]

    2. pow3 99.2%

       \[\leadsto \frac{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{3}\right)}}^{2}} \]

16. Applied egg-rr 99.2%

    \[\leadsto \frac{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{3}\right)}}^{2}} \]

17. Final simplification 99.2%

    \[\leadsto \frac{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}{{\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{3}\right)}^{2}} \]

Alternative 2: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot -0.3333333333333333}}{{\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{3}\right)}^{2}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/
  (* (pow l (exp w)) (exp (* w -0.3333333333333333)))
  (pow (pow (cbrt (cbrt (exp w))) 3.0) 2.0)))

C

double code(double w, double l) {
	return (pow(l, exp(w)) * exp((w * -0.3333333333333333))) / pow(pow(cbrt(cbrt(exp(w))), 3.0), 2.0);
}

Java

public static double code(double w, double l) {
	return (Math.pow(l, Math.exp(w)) * Math.exp((w * -0.3333333333333333))) / Math.pow(Math.pow(Math.cbrt(Math.cbrt(Math.exp(w))), 3.0), 2.0);
}

Julia

function code(w, l)
	return Float64(Float64((l ^ exp(w)) * exp(Float64(w * -0.3333333333333333))) / ((cbrt(cbrt(exp(w))) ^ 3.0) ^ 2.0))
end

Wolfram

code[w_, l_] := N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[Exp[N[(w * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[N[Power[N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Taylor expanded in l around inf 94.1%

   \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]
5. Step-by-step derivation

   1. mul-1-neg 94.1%

      \[\leadsto \frac{e^{\color{blue}{-e^{w} \cdot \log \left(\frac{1}{\ell}\right)}}}{e^{w}} \]

   2. distribute-rgt-neg-in 94.1%

      \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \left(-\log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]

   3. log-rec 94.1%

      \[\leadsto \frac{e^{e^{w} \cdot \color{blue}{\log \left(\frac{1}{\frac{1}{\ell}}\right)}}}{e^{w}} \]

   4. remove-double-div 94.1%

      \[\leadsto \frac{e^{e^{w} \cdot \log \color{blue}{\ell}}}{e^{w}} \]

6. Simplified 94.1%

   \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell}}}{e^{w}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 94.1%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{e^{w} \cdot \log \ell}}}{e^{w}} \]

   2. metadata-eval 94.1%

      \[\leadsto \frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot e^{e^{w} \cdot \log \ell}}{e^{w}} \]

   3. add-cube-cbrt 94.1%

      \[\leadsto \frac{\left(3 \cdot 0.3333333333333333\right) \cdot e^{e^{w} \cdot \log \ell}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

   4. times-frac 94.1%

      \[\leadsto \color{blue}{\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}}} \]

   5. metadata-eval 94.1%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   6. pow2 94.1%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   7. *-commutative 94.1%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{e^{\color{blue}{\log \ell \cdot e^{w}}}}{\sqrt[3]{e^{w}}} \]

   8. exp-to-pow 99.2%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}} \]

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
9. Step-by-step derivation

   1. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]

   2. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

10. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]
11. Step-by-step derivation

    1. add-exp-log 94.1%

       \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    2. log-div 94.1%

       \[\leadsto \frac{e^{\color{blue}{\log \left({\ell}^{\left(e^{w}\right)}\right) - \log \left(\sqrt[3]{e^{w}}\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    3. log-pow 94.1%

       \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell} - \log \left(\sqrt[3]{e^{w}}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    4. pow1/3 94.1%

       \[\leadsto \frac{e^{e^{w} \cdot \log \ell - \log \color{blue}{\left({\left(e^{w}\right)}^{0.3333333333333333}\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    5. log-pow 94.1%

       \[\leadsto \frac{e^{e^{w} \cdot \log \ell - \color{blue}{0.3333333333333333 \cdot \log \left(e^{w}\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    6. add-log-exp 94.1%

       \[\leadsto \frac{e^{e^{w} \cdot \log \ell - 0.3333333333333333 \cdot \color{blue}{w}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

12. Applied egg-rr 94.1%

    \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell - 0.3333333333333333 \cdot w}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
13. Step-by-step derivation

    1. sub-neg 94.1%

       \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + \left(-0.3333333333333333 \cdot w\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    2. exp-sum 94.1%

       \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell} \cdot e^{-0.3333333333333333 \cdot w}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    3. *-commutative 94.1%

       \[\leadsto \frac{e^{\color{blue}{\log \ell \cdot e^{w}}} \cdot e^{-0.3333333333333333 \cdot w}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    4. exp-to-pow 99.2%

       \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}} \cdot e^{-0.3333333333333333 \cdot w}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    5. *-commutative 99.2%

       \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{-\color{blue}{w \cdot 0.3333333333333333}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    6. distribute-rgt-neg-in 99.2%

       \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{\color{blue}{w \cdot \left(-0.3333333333333333\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    7. metadata-eval 99.2%

       \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot \color{blue}{-0.3333333333333333}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

14. Simplified 99.2%

    \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot -0.3333333333333333}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
15. Step-by-step derivation

    1. add-cube-cbrt 99.2%

       \[\leadsto \frac{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}{{\color{blue}{\left(\left(\sqrt[3]{\sqrt[3]{e^{w}}} \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{w}}}\right)}}^{2}} \]

    2. pow3 99.2%

       \[\leadsto \frac{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}}}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{3}\right)}}^{2}} \]

16. Applied egg-rr 99.2%

    \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot -0.3333333333333333}}{{\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{3}\right)}}^{2}} \]

17. Final simplification 99.2%

    \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot -0.3333333333333333}}{{\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{3}\right)}^{2}} \]

Alternative 3: 99.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ \frac{\frac{{\ell}^{\left(e^{w}\right)}}{t_0}}{{t_0}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (cbrt (exp w)))) (/ (/ (pow l (exp w)) t_0) (pow t_0 2.0))))

C

double code(double w, double l) {
	double t_0 = cbrt(exp(w));
	return (pow(l, exp(w)) / t_0) / pow(t_0, 2.0);
}

Java

public static double code(double w, double l) {
	double t_0 = Math.cbrt(Math.exp(w));
	return (Math.pow(l, Math.exp(w)) / t_0) / Math.pow(t_0, 2.0);
}

Julia

function code(w, l)
	t_0 = cbrt(exp(w))
	return Float64(Float64((l ^ exp(w)) / t_0) / (t_0 ^ 2.0))
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Taylor expanded in l around inf 94.1%

   \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]
5. Step-by-step derivation

   1. mul-1-neg 94.1%

      \[\leadsto \frac{e^{\color{blue}{-e^{w} \cdot \log \left(\frac{1}{\ell}\right)}}}{e^{w}} \]

   2. distribute-rgt-neg-in 94.1%

      \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \left(-\log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]

   3. log-rec 94.1%

      \[\leadsto \frac{e^{e^{w} \cdot \color{blue}{\log \left(\frac{1}{\frac{1}{\ell}}\right)}}}{e^{w}} \]

   4. remove-double-div 94.1%

      \[\leadsto \frac{e^{e^{w} \cdot \log \color{blue}{\ell}}}{e^{w}} \]

6. Simplified 94.1%

   \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell}}}{e^{w}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 94.1%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{e^{w} \cdot \log \ell}}}{e^{w}} \]

   2. metadata-eval 94.1%

      \[\leadsto \frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot e^{e^{w} \cdot \log \ell}}{e^{w}} \]

   3. add-cube-cbrt 94.1%

      \[\leadsto \frac{\left(3 \cdot 0.3333333333333333\right) \cdot e^{e^{w} \cdot \log \ell}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

   4. times-frac 94.1%

      \[\leadsto \color{blue}{\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}}} \]

   5. metadata-eval 94.1%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   6. pow2 94.1%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   7. *-commutative 94.1%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{e^{\color{blue}{\log \ell \cdot e^{w}}}}{\sqrt[3]{e^{w}}} \]

   8. exp-to-pow 99.2%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}} \]

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
9. Step-by-step derivation

   1. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]

   2. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

10. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]

11. Final simplification 99.2%

    \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

Alternative 4: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot -0.3333333333333333}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/
  (* (pow l (exp w)) (exp (* w -0.3333333333333333)))
  (pow (cbrt (exp w)) 2.0)))

C

double code(double w, double l) {
	return (pow(l, exp(w)) * exp((w * -0.3333333333333333))) / pow(cbrt(exp(w)), 2.0);
}

Java

public static double code(double w, double l) {
	return (Math.pow(l, Math.exp(w)) * Math.exp((w * -0.3333333333333333))) / Math.pow(Math.cbrt(Math.exp(w)), 2.0);
}

Julia

function code(w, l)
	return Float64(Float64((l ^ exp(w)) * exp(Float64(w * -0.3333333333333333))) / (cbrt(exp(w)) ^ 2.0))
end

Wolfram

code[w_, l_] := N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[Exp[N[(w * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Power[N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Taylor expanded in l around inf 94.1%

   \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]
5. Step-by-step derivation

   1. mul-1-neg 94.1%

      \[\leadsto \frac{e^{\color{blue}{-e^{w} \cdot \log \left(\frac{1}{\ell}\right)}}}{e^{w}} \]

   2. distribute-rgt-neg-in 94.1%

      \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \left(-\log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]

   3. log-rec 94.1%

      \[\leadsto \frac{e^{e^{w} \cdot \color{blue}{\log \left(\frac{1}{\frac{1}{\ell}}\right)}}}{e^{w}} \]

   4. remove-double-div 94.1%

      \[\leadsto \frac{e^{e^{w} \cdot \log \color{blue}{\ell}}}{e^{w}} \]

6. Simplified 94.1%

   \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell}}}{e^{w}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 94.1%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{e^{w} \cdot \log \ell}}}{e^{w}} \]

   2. metadata-eval 94.1%

      \[\leadsto \frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot e^{e^{w} \cdot \log \ell}}{e^{w}} \]

   3. add-cube-cbrt 94.1%

      \[\leadsto \frac{\left(3 \cdot 0.3333333333333333\right) \cdot e^{e^{w} \cdot \log \ell}}{\color{blue}{\left(\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}\right) \cdot \sqrt[3]{e^{w}}}} \]

   4. times-frac 94.1%

      \[\leadsto \color{blue}{\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}}} \]

   5. metadata-eval 94.1%

      \[\leadsto \frac{\color{blue}{1}}{\sqrt[3]{e^{w}} \cdot \sqrt[3]{e^{w}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   6. pow2 94.1%

      \[\leadsto \frac{1}{\color{blue}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \cdot \frac{e^{e^{w} \cdot \log \ell}}{\sqrt[3]{e^{w}}} \]

   7. *-commutative 94.1%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{e^{\color{blue}{\log \ell \cdot e^{w}}}}{\sqrt[3]{e^{w}}} \]

   8. exp-to-pow 99.2%

      \[\leadsto \frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{\sqrt[3]{e^{w}}} \]

8. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\frac{1}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}} \]
9. Step-by-step derivation

   1. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]

   2. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

10. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}} \]
11. Step-by-step derivation

    1. add-exp-log 94.1%

       \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    2. log-div 94.1%

       \[\leadsto \frac{e^{\color{blue}{\log \left({\ell}^{\left(e^{w}\right)}\right) - \log \left(\sqrt[3]{e^{w}}\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    3. log-pow 94.1%

       \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell} - \log \left(\sqrt[3]{e^{w}}\right)}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    4. pow1/3 94.1%

       \[\leadsto \frac{e^{e^{w} \cdot \log \ell - \log \color{blue}{\left({\left(e^{w}\right)}^{0.3333333333333333}\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    5. log-pow 94.1%

       \[\leadsto \frac{e^{e^{w} \cdot \log \ell - \color{blue}{0.3333333333333333 \cdot \log \left(e^{w}\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    6. add-log-exp 94.1%

       \[\leadsto \frac{e^{e^{w} \cdot \log \ell - 0.3333333333333333 \cdot \color{blue}{w}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

12. Applied egg-rr 94.1%

    \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell - 0.3333333333333333 \cdot w}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]
13. Step-by-step derivation

    1. sub-neg 94.1%

       \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + \left(-0.3333333333333333 \cdot w\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    2. exp-sum 94.1%

       \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell} \cdot e^{-0.3333333333333333 \cdot w}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    3. *-commutative 94.1%

       \[\leadsto \frac{e^{\color{blue}{\log \ell \cdot e^{w}}} \cdot e^{-0.3333333333333333 \cdot w}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    4. exp-to-pow 99.2%

       \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}} \cdot e^{-0.3333333333333333 \cdot w}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    5. *-commutative 99.2%

       \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{-\color{blue}{w \cdot 0.3333333333333333}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    6. distribute-rgt-neg-in 99.2%

       \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{\color{blue}{w \cdot \left(-0.3333333333333333\right)}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

    7. metadata-eval 99.2%

       \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot \color{blue}{-0.3333333333333333}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

14. Simplified 99.2%

    \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot -0.3333333333333333}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

15. Final simplification 99.2%

    \[\leadsto \frac{{\ell}^{\left(e^{w}\right)} \cdot e^{w \cdot -0.3333333333333333}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ (pow l (exp w)) (exp w)))

C

double code(double w, double l) {
	return pow(l, exp(w)) / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = (l ** exp(w)) / exp(w)
end function

Java

public static double code(double w, double l) {
	return Math.pow(l, Math.exp(w)) / Math.exp(w);
}

Python

def code(w, l):
	return math.pow(l, math.exp(w)) / math.exp(w)

Julia

function code(w, l)
	return Float64((l ^ exp(w)) / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / exp(w);
end

Wolfram

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Final simplification 99.2%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]

Alternative 6: 97.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 650\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.68) (not (<= w 650.0))) (exp (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 650.0)) {
		tmp = exp(-w);
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.68d0)) .or. (.not. (w <= 650.0d0))) then
        tmp = exp(-w)
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 650.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.68) or not (w <= 650.0):
		tmp = math.exp(-w)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.68) || !(w <= 650.0))
		tmp = exp(Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.68) || ~((w <= 650.0)))
		tmp = exp(-w);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.68], N[Not[LessEqual[w, 650.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.680000000000000049 or 650 < w

   1. Initial program 98.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 98.9%

         \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      2. associate-*l/ 98.9%

         \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

      3. *-lft-identity 98.9%

         \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   4. Taylor expanded in l around inf 98.9%

      \[\leadsto \frac{\color{blue}{e^{-1 \cdot \left(e^{w} \cdot \log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]
   5. Step-by-step derivation

      1. mul-1-neg 98.9%

         \[\leadsto \frac{e^{\color{blue}{-e^{w} \cdot \log \left(\frac{1}{\ell}\right)}}}{e^{w}} \]

      2. distribute-rgt-neg-in 98.9%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \left(-\log \left(\frac{1}{\ell}\right)\right)}}}{e^{w}} \]

      3. log-rec 98.9%

         \[\leadsto \frac{e^{e^{w} \cdot \color{blue}{\log \left(\frac{1}{\frac{1}{\ell}}\right)}}}{e^{w}} \]

      4. remove-double-div 98.9%

         \[\leadsto \frac{e^{e^{w} \cdot \log \color{blue}{\ell}}}{e^{w}} \]

   6. Simplified 98.9%

      \[\leadsto \frac{\color{blue}{e^{e^{w} \cdot \log \ell}}}{e^{w}} \]
   7. Step-by-step derivation

      1. div-exp 99.9%

         \[\leadsto \color{blue}{e^{e^{w} \cdot \log \ell - w}} \]

      2. fma-neg 99.9%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(e^{w}, \log \ell, -w\right)}} \]

   8. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(e^{w}, \log \ell, -w\right)}} \]

   9. Taylor expanded in w around inf 98.1%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   10. Step-by-step derivation

       1. mul-1-neg 98.1%

          \[\leadsto e^{\color{blue}{-w}} \]

   11. Simplified 98.1%

       \[\leadsto e^{\color{blue}{-w}} \]

   if -0.680000000000000049 < w < 650

   1. Initial program 99.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.4%

         \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      2. associate-*l/ 99.4%

         \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

      3. *-lft-identity 99.4%

         \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   4. Taylor expanded in w around 0 96.8%

      \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   5. Taylor expanded in w around 0 96.8%

      \[\leadsto \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 650\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 7: 97.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ l (exp w)))

C

double code(double w, double l) {
	return l / exp(w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l / exp(w)
end function

Java

public static double code(double w, double l) {
	return l / Math.exp(w);
}

Python

def code(w, l):
	return l / math.exp(w)

Julia

function code(w, l)
	return Float64(l / exp(w))
end

MATLAB

function tmp = code(w, l)
	tmp = l / exp(w);
end

Wolfram

code[w_, l_] := N[(l / N[Exp[w], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Taylor expanded in w around 0 97.3%

   \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

5. Final simplification 97.3%

   \[\leadsto \frac{\ell}{e^{w}} \]

Alternative 8: 65.0% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -1.0) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.0d0)) then
        tmp = w * -l
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.0) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.0:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.0)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -1

   1. Initial program 100.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 100.0%

         \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      2. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

      3. *-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   4. Taylor expanded in w around 0 100.0%

      \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   5. Taylor expanded in w around 0 24.6%

      \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 24.6%

         \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

      2. unsub-neg 24.6%

         \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   7. Simplified 24.6%

      \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

   8. Taylor expanded in w around inf 24.6%

      \[\leadsto \color{blue}{-1 \cdot \left(\ell \cdot w\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 24.6%

         \[\leadsto \color{blue}{-\ell \cdot w} \]

      2. *-commutative 24.6%

         \[\leadsto -\color{blue}{w \cdot \ell} \]

      3. distribute-rgt-neg-in 24.6%

         \[\leadsto \color{blue}{w \cdot \left(-\ell\right)} \]

   10. Simplified 24.6%

       \[\leadsto \color{blue}{w \cdot \left(-\ell\right)} \]

   if -1 < w

   1. Initial program 99.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Step-by-step derivation

      1. exp-neg 99.0%

         \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

      2. associate-*l/ 99.0%

         \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

      3. *-lft-identity 99.0%

         \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

   3. Simplified 99.0%

      \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   4. Taylor expanded in w around 0 96.5%

      \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

   5. Taylor expanded in w around 0 76.9%

      \[\leadsto \color{blue}{\ell} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 9: 64.7% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 - w\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (- 1.0 w)))

C

double code(double w, double l) {
	return l * (1.0 - w);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 - w)
end function

Java

public static double code(double w, double l) {
	return l * (1.0 - w);
}

Python

def code(w, l):
	return l * (1.0 - w)

Julia

function code(w, l)
	return Float64(l * Float64(1.0 - w))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 - w);
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Taylor expanded in w around 0 97.3%

   \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

5. Taylor expanded in w around 0 63.9%

   \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
6. Step-by-step derivation

   1. mul-1-neg 63.9%

      \[\leadsto \ell + \color{blue}{\left(-\ell \cdot w\right)} \]

   2. unsub-neg 63.9%

      \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

7. Simplified 63.9%

   \[\leadsto \color{blue}{\ell - \ell \cdot w} \]

8. Taylor expanded in l around 0 63.9%

   \[\leadsto \color{blue}{\ell \cdot \left(1 - w\right)} \]

9. Final simplification 63.9%

   \[\leadsto \ell \cdot \left(1 - w\right) \]

Alternative 10: 57.7% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ \ell \end{array} \]

FPCore

(FPCore (w l) :precision binary64 l)

C

double code(double w, double l) {
	return l;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l
end function

Java

public static double code(double w, double l) {
	return l;
}

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

function tmp = code(w, l)
	tmp = l;
end

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.2%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Step-by-step derivation

   1. exp-neg 99.2%

      \[\leadsto \color{blue}{\frac{1}{e^{w}}} \cdot {\ell}^{\left(e^{w}\right)} \]

   2. associate-*l/ 99.2%

      \[\leadsto \color{blue}{\frac{1 \cdot {\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

   3. *-lft-identity 99.2%

      \[\leadsto \frac{\color{blue}{{\ell}^{\left(e^{w}\right)}}}{e^{w}} \]

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]

4. Taylor expanded in w around 0 97.3%

   \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]

5. Taylor expanded in w around 0 59.2%

   \[\leadsto \color{blue}{\ell} \]

6. Final simplification 59.2%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023300 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))