exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 10.2s

Alternatives: 8

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, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[w_, l_] := N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[N[(w + w), $MachinePrecision]], $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. *-un-lft-identity 99.5%

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

   2. add-cube-cbrt 99.5%

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

   3. times-frac 99.5%

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

   4. cbrt-unprod 99.5%

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

   5. prod-exp 99.5%

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

5. Applied egg-rr 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

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: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{\sqrt[3]{e^{w + w}}} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

code[w_, l_] := N[(l / N[Power[N[Exp[N[(w + w), $MachinePrecision]], $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. *-un-lft-identity 99.5%

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

   2. add-cube-cbrt 99.5%

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

   3. times-frac 99.5%

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

   4. cbrt-unprod 99.5%

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

   5. prod-exp 99.5%

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

5. Applied egg-rr 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in w around 0 98.7%

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

9. Final simplification 98.7%

   \[\leadsto \frac{\ell}{\sqrt[3]{e^{w + w}}} \]

Alternative 4: 97.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{{\left(e^{w}\right)}^{0.6666666666666666}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ l (pow (exp w) 0.6666666666666666)))

C

double code(double w, double l) {
	return l / pow(exp(w), 0.6666666666666666);
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l / (exp(w) ** 0.6666666666666666d0)
end function

Java

public static double code(double w, double l) {
	return l / Math.pow(Math.exp(w), 0.6666666666666666);
}

Python

def code(w, l):
	return l / math.pow(math.exp(w), 0.6666666666666666)

Julia

function code(w, l)
	return Float64(l / (exp(w) ^ 0.6666666666666666))
end

MATLAB

function tmp = code(w, l)
	tmp = l / (exp(w) ^ 0.6666666666666666);
end

Wolfram

code[w_, l_] := N[(l / N[Power[N[Exp[w], $MachinePrecision], 0.6666666666666666], $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. *-un-lft-identity 99.5%

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

   2. add-cube-cbrt 99.5%

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

   3. times-frac 99.5%

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

   4. cbrt-unprod 99.5%

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

   5. prod-exp 99.5%

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

5. Applied egg-rr 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in w around 0 98.7%

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

   1. pow1/3 98.7%

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

   2. exp-sum 98.7%

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

   3. pow-prod-down 98.4%

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

   4. pow-prod-up 98.4%

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

   5. metadata-eval 98.4%

      \[\leadsto \frac{\ell}{{\left(e^{w}\right)}^{\color{blue}{0.6666666666666666}}} \]

10. Applied egg-rr 98.4%

    \[\leadsto \frac{\ell}{\color{blue}{{\left(e^{w}\right)}^{0.6666666666666666}}} \]

11. Final simplification 98.4%

    \[\leadsto \frac{\ell}{{\left(e^{w}\right)}^{0.6666666666666666}} \]

Alternative 5: 97.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -31:\\ \;\;\;\;\frac{\ell}{0}\\ \mathbf{elif}\;w \leq 0.0195:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -31.0)
   (/ l 0.0)
   (if (<= w 0.0195) (+ l (* w (- (* l (log l)) l))) (+ (+ l 1.0) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -31.0) {
		tmp = l / 0.0;
	} else if (w <= 0.0195) {
		tmp = l + (w * ((l * log(l)) - l));
	} else {
		tmp = (l + 1.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-31.0d0)) then
        tmp = l / 0.0d0
    else if (w <= 0.0195d0) then
        tmp = l + (w * ((l * log(l)) - l))
    else
        tmp = (l + 1.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -31.0) {
		tmp = l / 0.0;
	} else if (w <= 0.0195) {
		tmp = l + (w * ((l * Math.log(l)) - l));
	} else {
		tmp = (l + 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -31.0:
		tmp = l / 0.0
	elif w <= 0.0195:
		tmp = l + (w * ((l * math.log(l)) - l))
	else:
		tmp = (l + 1.0) + -1.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -31.0)
		tmp = Float64(l / 0.0);
	elseif (w <= 0.0195)
		tmp = Float64(l + Float64(w * Float64(Float64(l * log(l)) - l)));
	else
		tmp = Float64(Float64(l + 1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -31.0)
		tmp = l / 0.0;
	elseif (w <= 0.0195)
		tmp = l + (w * ((l * log(l)) - l));
	else
		tmp = (l + 1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -31.0], N[(l / 0.0), $MachinePrecision], If[LessEqual[w, 0.0195], N[(l + N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(l + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -31

   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. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. add-cube-cbrt 100.0%

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

      3. times-frac 100.0%

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

      4. cbrt-unprod 100.0%

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

      5. prod-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 100.0%

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

   9. Taylor expanded in w around 0 3.7%

      \[\leadsto \frac{\ell}{\color{blue}{1}} \]

   11. Simplified 100.0%

       \[\leadsto \frac{\ell}{\color{blue}{0}} \]

   if -31 < w < 0.0195

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-lft-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in w around 0 99.0%

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

   if 0.0195 < w

   1. Initial program 97.8%

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

      1. exp-neg 97.8%

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

      2. associate-*l/ 97.8%

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

      3. *-lft-identity 97.8%

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

   3. Simplified 97.8%

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

      1. expm1-log1p-u 97.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)\right)} \]

   5. Applied egg-rr 97.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)\right)} \]

   6. Taylor expanded in w around 0 5.0%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-udef 89.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1} \]

      2. log1p-udef 89.3%

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

      3. add-exp-log 89.3%

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

      4. +-commutative 89.3%

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

   8. Applied egg-rr 89.3%

      \[\leadsto \color{blue}{\left(\ell + 1\right) - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -31:\\ \;\;\;\;\frac{\ell}{0}\\ \mathbf{elif}\;w \leq 0.0195:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + 1\right) + -1\\ \end{array} \]

Alternative 6: 96.9% accurate, 33.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -440:\\ \;\;\;\;\frac{\ell}{0}\\ \mathbf{elif}\;w \leq 4.7 \cdot 10^{-19}:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + 1\right) + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -440.0) (/ l 0.0) (if (<= w 4.7e-19) l (+ (+ l 1.0) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -440.0) {
		tmp = l / 0.0;
	} else if (w <= 4.7e-19) {
		tmp = l;
	} else {
		tmp = (l + 1.0) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-440.0d0)) then
        tmp = l / 0.0d0
    else if (w <= 4.7d-19) then
        tmp = l
    else
        tmp = (l + 1.0d0) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -440.0) {
		tmp = l / 0.0;
	} else if (w <= 4.7e-19) {
		tmp = l;
	} else {
		tmp = (l + 1.0) + -1.0;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -440.0:
		tmp = l / 0.0
	elif w <= 4.7e-19:
		tmp = l
	else:
		tmp = (l + 1.0) + -1.0
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -440.0)
		tmp = Float64(l / 0.0);
	elseif (w <= 4.7e-19)
		tmp = l;
	else
		tmp = Float64(Float64(l + 1.0) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -440.0)
		tmp = l / 0.0;
	elseif (w <= 4.7e-19)
		tmp = l;
	else
		tmp = (l + 1.0) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -440.0], N[(l / 0.0), $MachinePrecision], If[LessEqual[w, 4.7e-19], l, N[(N[(l + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -440

   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. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. add-cube-cbrt 100.0%

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

      3. times-frac 100.0%

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

      4. cbrt-unprod 100.0%

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

      5. prod-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 100.0%

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

   9. Taylor expanded in w around 0 3.7%

      \[\leadsto \frac{\ell}{\color{blue}{1}} \]

   11. Simplified 100.0%

       \[\leadsto \frac{\ell}{\color{blue}{0}} \]

   if -440 < w < 4.7e-19

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. associate-*l/ 99.8%

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

      3. *-lft-identity 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in w around 0 98.8%

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

   if 4.7e-19 < w

   1. Initial program 97.5%

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

      1. exp-neg 97.5%

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

      2. associate-*l/ 97.5%

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

      3. *-lft-identity 97.5%

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

   3. Simplified 97.5%

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

      1. expm1-log1p-u 97.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)\right)} \]

   5. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\right)\right)} \]

   6. Taylor expanded in w around 0 7.6%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\ell}\right)\right) \]
   7. Step-by-step derivation

      1. expm1-udef 88.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\ell\right)} - 1} \]

      2. log1p-udef 88.2%

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

      3. add-exp-log 88.4%

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

      4. +-commutative 88.4%

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

   8. Applied egg-rr 88.4%

      \[\leadsto \color{blue}{\left(\ell + 1\right) - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -440:\\ \;\;\;\;\frac{\ell}{0}\\ \mathbf{elif}\;w \leq 4.7 \cdot 10^{-19}:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\left(\ell + 1\right) + -1\\ \end{array} \]

Alternative 7: 84.2% accurate, 60.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -310:\\ \;\;\;\;\frac{\ell}{0}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -310.0) (/ l 0.0) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -310.0) {
		tmp = l / 0.0;
	} 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 <= (-310.0d0)) then
        tmp = l / 0.0d0
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -310.0) {
		tmp = l / 0.0;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -310.0:
		tmp = l / 0.0
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -310.0)
		tmp = Float64(l / 0.0);
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -310.0)
		tmp = l / 0.0;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -310.0], N[(l / 0.0), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -310

   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. Step-by-step derivation

      1. *-un-lft-identity 100.0%

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

      2. add-cube-cbrt 100.0%

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

      3. times-frac 100.0%

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

      4. cbrt-unprod 100.0%

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

      5. prod-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. associate-*l/ 100.0%

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

      2. *-lft-identity 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 100.0%

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

   9. Taylor expanded in w around 0 3.7%

      \[\leadsto \frac{\ell}{\color{blue}{1}} \]

   11. Simplified 100.0%

       \[\leadsto \frac{\ell}{\color{blue}{0}} \]

   if -310 < w

   1. Initial program 99.3%

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

      1. exp-neg 99.3%

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

      2. associate-*l/ 99.3%

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

      3. *-lft-identity 99.3%

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

   3. Simplified 99.3%

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

   4. Taylor expanded in w around 0 76.2%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -310:\\ \;\;\;\;\frac{\ell}{0}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]

Alternative 8: 56.7% 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 99.5%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 57.2%

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

5. Final simplification 57.2%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023230 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))