exp-w (used to crash)

Percentage Accurate: 99.5% → 99.4%

Time: 12.0s

Alternatives: 6

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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \left(\sqrt[3]{\frac{1}{{\left(e^{w}\right)}^{2}}} \cdot {\ell}^{\left(e^{w}\right)}\right) \cdot \sqrt[3]{e^{-w}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (* (* (cbrt (/ 1.0 (pow (exp w) 2.0))) (pow l (exp w))) (cbrt (exp (- w)))))

C

double code(double w, double l) {
	return (cbrt((1.0 / pow(exp(w), 2.0))) * pow(l, exp(w))) * cbrt(exp(-w));
}

Java

public static double code(double w, double l) {
	return (Math.cbrt((1.0 / Math.pow(Math.exp(w), 2.0))) * Math.pow(l, Math.exp(w))) * Math.cbrt(Math.exp(-w));
}

Julia

function code(w, l)
	return Float64(Float64(cbrt(Float64(1.0 / (exp(w) ^ 2.0))) * (l ^ exp(w))) * cbrt(exp(Float64(-w))))
end

Wolfram

code[w_, l_] := N[(N[(N[Power[N[(1.0 / N[Power[N[Exp[w], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[(-w)], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

   1. add-cbrt-cube 63.9%

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

   2. pow1/3 62.6%

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

   3. div-inv 62.6%

      \[\leadsto {\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot \color{blue}{\left({\ell}^{\left(e^{w}\right)} \cdot \frac{1}{e^{w}}\right)}\right)}^{0.3333333333333333} \]

   4. associate-*r* 62.6%

      \[\leadsto {\color{blue}{\left(\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot {\ell}^{\left(e^{w}\right)}\right) \cdot \frac{1}{e^{w}}\right)}}^{0.3333333333333333} \]

   5. unpow-prod-down 62.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{e^{w}}\right)}^{0.3333333333333333}} \]

   6. pow2 62.6%

      \[\leadsto {\left(\color{blue}{{\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)}^{2}} \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{e^{w}}\right)}^{0.3333333333333333} \]

   7. pow1/3 62.6%

      \[\leadsto {\left({\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)}^{2} \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\frac{1}{e^{w}}}} \]

   8. rec-exp 62.6%

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

5. Applied egg-rr 62.6%

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

   1. unpow1/3 63.9%

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

   2. *-commutative 63.9%

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

7. Simplified 63.9%

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

8. Taylor expanded in l around 0 99.5%

   \[\leadsto \color{blue}{\left({\left(\frac{1}{{\left(e^{w}\right)}^{2}}\right)}^{0.3333333333333333} \cdot {\ell}^{\left(e^{w}\right)}\right)} \cdot \sqrt[3]{e^{-w}} \]
9. Step-by-step derivation

   1. unpow1/3 99.5%

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

10. Simplified 99.5%

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

11. Final simplification 99.5%

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

Alternative 2: 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 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 97.4% accurate, 2.9× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 (* l (exp (* (- w) 0.3333333333333333))))

C

double code(double w, double l) {
	return l * exp((-w * 0.3333333333333333));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * exp((-w * 0.3333333333333333d0))
end function

Java

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

Python

def code(w, l):
	return l * math.exp((-w * 0.3333333333333333))

Julia

function code(w, l)
	return Float64(l * exp(Float64(Float64(-w) * 0.3333333333333333)))
end

MATLAB

function tmp = code(w, l)
	tmp = l * exp((-w * 0.3333333333333333));
end

Wolfram

code[w_, l_] := N[(l * N[Exp[N[((-w) * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

   1. add-cbrt-cube 63.9%

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

   2. pow1/3 62.6%

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

   3. div-inv 62.6%

      \[\leadsto {\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot \color{blue}{\left({\ell}^{\left(e^{w}\right)} \cdot \frac{1}{e^{w}}\right)}\right)}^{0.3333333333333333} \]

   4. associate-*r* 62.6%

      \[\leadsto {\color{blue}{\left(\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot {\ell}^{\left(e^{w}\right)}\right) \cdot \frac{1}{e^{w}}\right)}}^{0.3333333333333333} \]

   5. unpow-prod-down 62.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{e^{w}}\right)}^{0.3333333333333333}} \]

   6. pow2 62.6%

      \[\leadsto {\left(\color{blue}{{\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)}^{2}} \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{e^{w}}\right)}^{0.3333333333333333} \]

   7. pow1/3 62.6%

      \[\leadsto {\left({\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)}^{2} \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\frac{1}{e^{w}}}} \]

   8. rec-exp 62.6%

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

5. Applied egg-rr 62.6%

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

   1. unpow1/3 63.9%

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

   2. *-commutative 63.9%

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

7. Simplified 63.9%

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

8. Taylor expanded in l around 0 99.5%

   \[\leadsto \color{blue}{\left({\left(\frac{1}{{\left(e^{w}\right)}^{2}}\right)}^{0.3333333333333333} \cdot {\ell}^{\left(e^{w}\right)}\right)} \cdot \sqrt[3]{e^{-w}} \]
9. Step-by-step derivation

   1. unpow1/3 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in w around 0 99.3%

    \[\leadsto \color{blue}{\ell} \cdot \sqrt[3]{e^{-w}} \]
12. Step-by-step derivation

    1. exp-neg 99.3%

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

    2. cbrt-div 99.3%

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

    3. metadata-eval 99.3%

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

13. Applied egg-rr 99.3%

    \[\leadsto \ell \cdot \color{blue}{\frac{1}{\sqrt[3]{e^{w}}}} \]
14. Step-by-step derivation

    1. unpow1/3 99.3%

       \[\leadsto \ell \cdot \frac{1}{\color{blue}{{\left(e^{w}\right)}^{0.3333333333333333}}} \]

    2. exp-prod 99.3%

       \[\leadsto \ell \cdot \frac{1}{\color{blue}{e^{w \cdot 0.3333333333333333}}} \]

    3. *-commutative 99.3%

       \[\leadsto \ell \cdot \frac{1}{e^{\color{blue}{0.3333333333333333 \cdot w}}} \]

    4. rec-exp 99.3%

       \[\leadsto \ell \cdot \color{blue}{e^{-0.3333333333333333 \cdot w}} \]

    5. *-commutative 99.3%

       \[\leadsto \ell \cdot e^{-\color{blue}{w \cdot 0.3333333333333333}} \]

15. Simplified 99.3%

    \[\leadsto \ell \cdot \color{blue}{e^{-w \cdot 0.3333333333333333}} \]

16. Final simplification 99.3%

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

Alternative 4: 64.3% accurate, 43.6× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 + w \cdot -0.3333333333333333\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (+ 1.0 (* w -0.3333333333333333))))

C

double code(double w, double l) {
	return l * (1.0 + (w * -0.3333333333333333));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 + (w * (-0.3333333333333333d0)))
end function

Java

public static double code(double w, double l) {
	return l * (1.0 + (w * -0.3333333333333333));
}

Python

def code(w, l):
	return l * (1.0 + (w * -0.3333333333333333))

Julia

function code(w, l)
	return Float64(l * Float64(1.0 + Float64(w * -0.3333333333333333)))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 + (w * -0.3333333333333333));
end

Wolfram

code[w_, l_] := N[(l * N[(1.0 + N[(w * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

   1. add-cbrt-cube 63.9%

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

   2. pow1/3 62.6%

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

   3. div-inv 62.6%

      \[\leadsto {\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot \color{blue}{\left({\ell}^{\left(e^{w}\right)} \cdot \frac{1}{e^{w}}\right)}\right)}^{0.3333333333333333} \]

   4. associate-*r* 62.6%

      \[\leadsto {\color{blue}{\left(\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot {\ell}^{\left(e^{w}\right)}\right) \cdot \frac{1}{e^{w}}\right)}}^{0.3333333333333333} \]

   5. unpow-prod-down 62.6%

      \[\leadsto \color{blue}{{\left(\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \cdot \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right) \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{e^{w}}\right)}^{0.3333333333333333}} \]

   6. pow2 62.6%

      \[\leadsto {\left(\color{blue}{{\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)}^{2}} \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot {\left(\frac{1}{e^{w}}\right)}^{0.3333333333333333} \]

   7. pow1/3 62.6%

      \[\leadsto {\left({\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)}^{2} \cdot {\ell}^{\left(e^{w}\right)}\right)}^{0.3333333333333333} \cdot \color{blue}{\sqrt[3]{\frac{1}{e^{w}}}} \]

   8. rec-exp 62.6%

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

5. Applied egg-rr 62.6%

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

   1. unpow1/3 63.9%

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

   2. *-commutative 63.9%

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

7. Simplified 63.9%

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

8. Taylor expanded in l around 0 99.5%

   \[\leadsto \color{blue}{\left({\left(\frac{1}{{\left(e^{w}\right)}^{2}}\right)}^{0.3333333333333333} \cdot {\ell}^{\left(e^{w}\right)}\right)} \cdot \sqrt[3]{e^{-w}} \]
9. Step-by-step derivation

   1. unpow1/3 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in w around 0 99.3%

    \[\leadsto \color{blue}{\ell} \cdot \sqrt[3]{e^{-w}} \]

12. Taylor expanded in w around 0 65.4%

    \[\leadsto \ell \cdot \color{blue}{\left(1 + -0.3333333333333333 \cdot w\right)} \]
13. Step-by-step derivation

    1. *-commutative 65.4%

       \[\leadsto \ell \cdot \left(1 + \color{blue}{w \cdot -0.3333333333333333}\right) \]

14. Simplified 65.4%

    \[\leadsto \ell \cdot \color{blue}{\left(1 + w \cdot -0.3333333333333333\right)} \]

15. Final simplification 65.4%

    \[\leadsto \ell \cdot \left(1 + w \cdot -0.3333333333333333\right) \]

Alternative 5: 64.6% accurate, 50.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -10:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -10.0) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -10.0) {
		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 <= (-10.0d0)) 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 <= -10.0) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -10.0:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -10.0)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -10.0)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -10.0], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -10

   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 1.7%

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

   5. Taylor expanded in w around inf 1.7%

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

      1. add-log-exp 1.6%

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

      2. *-commutative 1.6%

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

      3. exp-to-pow 1.6%

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

   7. Applied egg-rr 1.6%

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

   8. Taylor expanded in l around inf 30.9%

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

      1. neg-mul-1 30.9%

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

   10. Simplified 30.9%

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

   if -10 < w

   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 78.5%

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

4. Final simplification 65.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -10:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 6: 57.5% 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 99.5%

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

   1. exp-neg 99.5%

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

   2. associate-*l/ 99.5%

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

   3. *-lft-identity 99.5%

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

3. Simplified 99.5%

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

4. Taylor expanded in w around 0 58.6%

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

5. Final simplification 58.6%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023339 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))