exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 12.7s

Alternatives: 7

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

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. add-cube-cbrt 99.6%

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

   3. times-frac 99.6%

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

   4. cbrt-unprod 99.6%

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

   5. prod-exp 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 2: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{w \cdot -0.3333333333333333}\\ {\ell}^{\left(e^{w}\right)} \cdot \left(t_0 \cdot {t_0}^{2}\right) \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (* w -0.3333333333333333))))
   (* (pow l (exp w)) (* t_0 (pow t_0 2.0)))))

C

double code(double w, double l) {
	double t_0 = exp((w * -0.3333333333333333));
	return pow(l, exp(w)) * (t_0 * pow(t_0, 2.0));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = exp((w * (-0.3333333333333333d0)))
    code = (l ** exp(w)) * (t_0 * (t_0 ** 2.0d0))
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.exp((w * -0.3333333333333333));
	return Math.pow(l, Math.exp(w)) * (t_0 * Math.pow(t_0, 2.0));
}

Python

def code(w, l):
	t_0 = math.exp((w * -0.3333333333333333))
	return math.pow(l, math.exp(w)) * (t_0 * math.pow(t_0, 2.0))

Julia

function code(w, l)
	t_0 = exp(Float64(w * -0.3333333333333333))
	return Float64((l ^ exp(w)) * Float64(t_0 * (t_0 ^ 2.0)))
end

MATLAB

function tmp = code(w, l)
	t_0 = exp((w * -0.3333333333333333));
	tmp = (l ^ exp(w)) * (t_0 * (t_0 ^ 2.0));
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[N[(w * -0.3333333333333333), $MachinePrecision]], $MachinePrecision]}, N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[(t$95$0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in l around 0 94.3%

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

   1. pow-to-exp 99.6%

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

   2. add-cbrt-cube 99.6%

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

   3. exp-sum 99.6%

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

   4. cbrt-prod 99.6%

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

   5. associate-/l/ 99.6%

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

   6. div-inv 99.6%

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

   7. div-inv 99.6%

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

   8. associate-*l* 99.6%

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

   9. add-exp-log 99.6%

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

   10. rec-exp 99.6%

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

   11. pow1/3 99.6%

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

   12. log-pow 99.6%

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

   13. add-log-exp 99.6%

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

6. Applied egg-rr 99.6%

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

7. Taylor expanded in w around -inf 99.6%

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

   1. *-commutative 99.6%

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

9. Simplified 99.6%

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

10. Taylor expanded in w around -inf 99.6%

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

11. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. add-cube-cbrt 99.6%

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

   3. times-frac 99.6%

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

   4. cbrt-unprod 99.6%

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

   5. prod-exp 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in w around inf 99.6%

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

   1. metadata-eval 99.6%

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

   2. distribute-lft-neg-in 99.6%

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

   3. unpow1/3 99.6%

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

   4. prod-exp 99.6%

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

   5. distribute-lft-neg-in 99.6%

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

   6. metadata-eval 99.6%

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

   7. rec-exp 99.6%

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

   8. distribute-lft1-in 99.6%

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

   9. metadata-eval 99.6%

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

8. Simplified 99.6%

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

9. Final simplification 99.6%

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

Alternative 4: 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 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 97.8% 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 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

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 6: 63.8% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \ell - w \cdot \ell \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in w around 0 97.3%

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

5. Taylor expanded in w around 0 67.2%

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

   1. +-commutative 67.2%

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

   2. mul-1-neg 67.2%

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

   3. unsub-neg 67.2%

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

   4. *-commutative 67.2%

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

7. Simplified 67.2%

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

8. Final simplification 67.2%

   \[\leadsto \ell - w \cdot \ell \]

Alternative 7: 57.3% 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 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

   1. *-un-lft-identity 99.6%

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

   2. add-cube-cbrt 99.6%

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

   3. times-frac 99.6%

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

   4. cbrt-unprod 99.6%

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

   5. prod-exp 99.6%

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

5. Applied egg-rr 99.6%

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

6. Taylor expanded in w around 0 61.7%

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

7. Final simplification 61.7%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023203 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))