exp-w (used to crash)

Percentage Accurate: 99.3% → 99.7%

Time: 22.5s

Alternatives: 14

Speedup: 0.6×

• 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: 99.3% 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: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{\left(e^{w}\right)}\\ t_1 := \sqrt[3]{e^{w}}\\ \mathbf{if}\;t_0 \cdot e^{-w} \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{t_0}{t_1}}{{t_1}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (pow l (exp w))) (t_1 (cbrt (exp w))))
   (if (<= (* t_0 (exp (- w))) 1e+305)
     (/ (/ t_0 t_1) (pow t_1 2.0))
     (exp (- (* (exp w) (log l)) w)))))

C

double code(double w, double l) {
	double t_0 = pow(l, exp(w));
	double t_1 = cbrt(exp(w));
	double tmp;
	if ((t_0 * exp(-w)) <= 1e+305) {
		tmp = (t_0 / t_1) / pow(t_1, 2.0);
	} else {
		tmp = exp(((exp(w) * log(l)) - w));
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double t_0 = Math.pow(l, Math.exp(w));
	double t_1 = Math.cbrt(Math.exp(w));
	double tmp;
	if ((t_0 * Math.exp(-w)) <= 1e+305) {
		tmp = (t_0 / t_1) / Math.pow(t_1, 2.0);
	} else {
		tmp = Math.exp(((Math.exp(w) * Math.log(l)) - w));
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = l ^ exp(w)
	t_1 = cbrt(exp(w))
	tmp = 0.0
	if (Float64(t_0 * exp(Float64(-w))) <= 1e+305)
		tmp = Float64(Float64(t_0 / t_1) / (t_1 ^ 2.0));
	else
		tmp = exp(Float64(Float64(exp(w) * log(l)) - w));
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[w], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Exp[(-w)], $MachinePrecision]), $MachinePrecision], 1e+305], N[(N[(t$95$0 / t$95$1), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Exp[w], $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.9999999999999994e304

   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. Step-by-step derivation

      1. *-un-lft-identity 99.7%

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

      2. add-cube-cbrt 99.7%

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

      3. times-frac 99.7%

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

      4. cbrt-unprod 99.7%

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

      5. prod-exp 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

   7. Simplified 99.7%

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

      1. exp-sum 99.7%

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

      2. cbrt-prod 99.7%

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

      3. pow2 99.7%

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

   9. Applied egg-rr 99.7%

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

   if 9.9999999999999994e304 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 93.2%

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

      1. exp-neg 93.2%

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

      2. associate-*l/ 93.2%

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

      3. *-lft-identity 93.2%

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

   3. Simplified 93.2%

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

      1. add-exp-log 93.2%

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

      2. log-div 93.2%

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

      3. log-pow 93.2%

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

      4. add-log-exp 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{e^{w} \cdot \log \ell - w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\ell}^{\left(e^{w}\right)} \cdot e^{-w} \leq 10^{+305}:\\ \;\;\;\;\frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(\sqrt[3]{e^{w}}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \end{array} \]

Alternative 2: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ \frac{\frac{{\ell}^{\left(e^{w}\right)}}{t_0}}{{\left(e^{\mathsf{log1p}\left(t_0\right)} + -1\right)}^{2}} \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (cbrt (exp w))))
   (/ (/ (pow l (exp w)) t_0) (pow (+ (exp (log1p t_0)) -1.0) 2.0))))

C

double code(double w, double l) {
	double t_0 = cbrt(exp(w));
	return (pow(l, exp(w)) / t_0) / pow((exp(log1p(t_0)) + -1.0), 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((Math.exp(Math.log1p(t_0)) + -1.0), 2.0);
}

Julia

function code(w, l)
	t_0 = cbrt(exp(w))
	return Float64(Float64((l ^ exp(w)) / t_0) / (Float64(exp(log1p(t_0)) + -1.0) ^ 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[N[(N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 98.2%

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

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

   1. *-un-lft-identity 98.2%

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

   2. add-cube-cbrt 98.2%

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

   3. times-frac 98.2%

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

   4. cbrt-unprod 98.2%

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

   5. prod-exp 98.2%

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

5. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.2%

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

   2. *-lft-identity 98.2%

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

7. Simplified 98.2%

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

   1. exp-sum 98.2%

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

   2. cbrt-prod 98.2%

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

   3. pow2 98.2%

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

9. Applied egg-rr 98.2%

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

    1. expm1-log1p-u 98.2%

       \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{w}}\right)\right)\right)}}^{2}} \]

    2. expm1-udef 98.2%

       \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{e^{w}}\right)} - 1\right)}}^{2}} \]

11. Applied egg-rr 98.2%

    \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{e^{w}}\right)} - 1\right)}}^{2}} \]

12. Final simplification 98.2%

    \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}}}}{{\left(e^{\mathsf{log1p}\left(\sqrt[3]{e^{w}}\right)} + -1\right)}^{2}} \]

Alternative 3: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\ell}^{\left(e^{w}\right)}\\ \mathbf{if}\;t_0 \cdot e^{-w} \leq 10^{+305}:\\ \;\;\;\;\frac{t_0}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (pow l (exp w))))
   (if (<= (* t_0 (exp (- w))) 1e+305)
     (/ t_0 (exp w))
     (exp (- (* (exp w) (log l)) w)))))

C

double code(double w, double l) {
	double t_0 = pow(l, exp(w));
	double tmp;
	if ((t_0 * exp(-w)) <= 1e+305) {
		tmp = t_0 / exp(w);
	} else {
		tmp = exp(((exp(w) * log(l)) - w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = l ** exp(w)
    if ((t_0 * exp(-w)) <= 1d+305) then
        tmp = t_0 / exp(w)
    else
        tmp = exp(((exp(w) * log(l)) - w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.pow(l, Math.exp(w));
	double tmp;
	if ((t_0 * Math.exp(-w)) <= 1e+305) {
		tmp = t_0 / Math.exp(w);
	} else {
		tmp = Math.exp(((Math.exp(w) * Math.log(l)) - w));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = math.pow(l, math.exp(w))
	tmp = 0
	if (t_0 * math.exp(-w)) <= 1e+305:
		tmp = t_0 / math.exp(w)
	else:
		tmp = math.exp(((math.exp(w) * math.log(l)) - w))
	return tmp

Julia

function code(w, l)
	t_0 = l ^ exp(w)
	tmp = 0.0
	if (Float64(t_0 * exp(Float64(-w))) <= 1e+305)
		tmp = Float64(t_0 / exp(w));
	else
		tmp = exp(Float64(Float64(exp(w) * log(l)) - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = l ^ exp(w);
	tmp = 0.0;
	if ((t_0 * exp(-w)) <= 1e+305)
		tmp = t_0 / exp(w);
	else
		tmp = exp(((exp(w) * log(l)) - w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Exp[(-w)], $MachinePrecision]), $MachinePrecision], 1e+305], N[(t$95$0 / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Exp[w], $MachinePrecision] * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 9.9999999999999994e304

   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%

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

   if 9.9999999999999994e304 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 93.2%

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

      1. exp-neg 93.2%

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

      2. associate-*l/ 93.2%

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

      3. *-lft-identity 93.2%

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

   3. Simplified 93.2%

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

      1. add-exp-log 93.2%

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

      2. log-div 93.2%

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

      3. log-pow 93.2%

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

      4. add-log-exp 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{e^{w} \cdot \log \ell - w}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\ell}^{\left(e^{w}\right)} \cdot e^{-w} \leq 10^{+305}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{e^{w} \cdot \log \ell - w}\\ \end{array} \]

Alternative 4: 99.3% 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 98.2%

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

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

   1. *-un-lft-identity 98.2%

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

   2. add-cube-cbrt 98.2%

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

   3. times-frac 98.2%

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

   4. cbrt-unprod 98.2%

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

   5. prod-exp 98.2%

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

5. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.2%

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

   2. *-lft-identity 98.2%

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

7. Simplified 98.2%

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

8. Final simplification 98.2%

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

Alternative 5: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt{e^{w}}}}{e^{w \cdot 0.5}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/ (/ (pow l (exp w)) (sqrt (exp w))) (exp (* w 0.5))))

C

double code(double w, double l) {
	return (pow(l, exp(w)) / sqrt(exp(w))) / exp((w * 0.5));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = ((l ** exp(w)) / sqrt(exp(w))) / exp((w * 0.5d0))
end function

Java

public static double code(double w, double l) {
	return (Math.pow(l, Math.exp(w)) / Math.sqrt(Math.exp(w))) / Math.exp((w * 0.5));
}

Python

def code(w, l):
	return (math.pow(l, math.exp(w)) / math.sqrt(math.exp(w))) / math.exp((w * 0.5))

Julia

function code(w, l)
	return Float64(Float64((l ^ exp(w)) / sqrt(exp(w))) / exp(Float64(w * 0.5)))
end

MATLAB

function tmp = code(w, l)
	tmp = ((l ^ exp(w)) / sqrt(exp(w))) / exp((w * 0.5));
end

Wolfram

code[w_, l_] := N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Exp[N[(w * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 98.2%

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

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

   1. *-un-lft-identity 98.2%

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

   2. add-sqr-sqrt 98.2%

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

   3. times-frac 98.2%

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

5. Applied egg-rr 98.2%

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

   1. associate-*l/ 98.2%

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

   2. *-lft-identity 98.2%

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

7. Simplified 98.2%

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

   1. pow1/2 98.2%

      \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt{e^{w}}}}{\color{blue}{{\left(e^{w}\right)}^{0.5}}} \]

   2. pow-exp 98.2%

      \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt{e^{w}}}}{\color{blue}{e^{w \cdot 0.5}}} \]

9. Applied egg-rr 98.2%

   \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt{e^{w}}}}{\color{blue}{e^{w \cdot 0.5}}} \]

10. Final simplification 98.2%

    \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\sqrt{e^{w}}}}{e^{w \cdot 0.5}} \]

Alternative 6: 99.3% 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 98.2%

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

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Final simplification 98.2%

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

Alternative 7: 99.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{elif}\;w \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell \cdot {\ell}^{w}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \ell \cdot \left(\left(w + 1\right) + 0.5 \cdot \left(w \cdot w\right)\right) - w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.3)
   (exp (- (log l) w))
   (if (<= w 1.25e-8)
     (/ (* l (pow l w)) (exp w))
     (exp (- (* (log l) (+ (+ w 1.0) (* 0.5 (* w w)))) w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = exp((log(l) - w));
	} else if (w <= 1.25e-8) {
		tmp = (l * pow(l, w)) / exp(w);
	} else {
		tmp = exp(((log(l) * ((w + 1.0) + (0.5 * (w * w)))) - w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.3d0)) then
        tmp = exp((log(l) - w))
    else if (w <= 1.25d-8) then
        tmp = (l * (l ** w)) / exp(w)
    else
        tmp = exp(((log(l) * ((w + 1.0d0) + (0.5d0 * (w * w)))) - w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = Math.exp((Math.log(l) - w));
	} else if (w <= 1.25e-8) {
		tmp = (l * Math.pow(l, w)) / Math.exp(w);
	} else {
		tmp = Math.exp(((Math.log(l) * ((w + 1.0) + (0.5 * (w * w)))) - w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.3:
		tmp = math.exp((math.log(l) - w))
	elif w <= 1.25e-8:
		tmp = (l * math.pow(l, w)) / math.exp(w)
	else:
		tmp = math.exp(((math.log(l) * ((w + 1.0) + (0.5 * (w * w)))) - w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.3)
		tmp = exp(Float64(log(l) - w));
	elseif (w <= 1.25e-8)
		tmp = Float64(Float64(l * (l ^ w)) / exp(w));
	else
		tmp = exp(Float64(Float64(log(l) * Float64(Float64(w + 1.0) + Float64(0.5 * Float64(w * w)))) - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.3)
		tmp = exp((log(l) - w));
	elseif (w <= 1.25e-8)
		tmp = (l * (l ^ w)) / exp(w);
	else
		tmp = exp(((log(l) * ((w + 1.0) + (0.5 * (w * w)))) - w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.3], N[Exp[N[(N[Log[l], $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 1.25e-8], N[(N[(l * N[Power[l, w], $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * N[(N[(w + 1.0), $MachinePrecision] + N[(0.5 * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -1.30000000000000004

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-div 99.9%

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

      3. log-pow 99.9%

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

      4. add-log-exp 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in w around 0 98.3%

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

   if -1.30000000000000004 < w < 1.2499999999999999e-8

   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. Step-by-step derivation

      1. add-exp-log 91.2%

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

      2. log-div 91.2%

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

      3. log-pow 91.3%

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

      4. add-log-exp 91.3%

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

   5. Applied egg-rr 91.3%

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

   6. Taylor expanded in w around 0 90.4%

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

      1. distribute-lft1-in 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in w around inf 90.4%

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

       1. exp-diff 90.4%

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

       2. *-commutative 90.4%

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

       3. exp-to-pow 98.7%

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

       4. +-commutative 98.7%

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

   11. Simplified 98.7%

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

       1. pow-plus 99.0%

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

   13. Applied egg-rr 99.0%

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

   if 1.2499999999999999e-8 < w

   1. Initial program 90.8%

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

      1. exp-neg 90.8%

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

      2. associate-*l/ 90.8%

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

      3. *-lft-identity 90.8%

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

   3. Simplified 90.8%

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

      1. add-exp-log 90.7%

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

      2. log-div 90.7%

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

      3. log-pow 90.7%

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

      4. add-log-exp 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in w around 0 99.7%

      \[\leadsto e^{\color{blue}{\left(0.5 \cdot \left({w}^{2} \cdot \log \ell\right) + \left(w \cdot \log \ell + \log \ell\right)\right)} - w} \]
   7. Step-by-step derivation

      1. +-commutative 99.7%

         \[\leadsto e^{\color{blue}{\left(\left(w \cdot \log \ell + \log \ell\right) + 0.5 \cdot \left({w}^{2} \cdot \log \ell\right)\right)} - w} \]

      2. distribute-lft1-in 99.8%

         \[\leadsto e^{\left(\color{blue}{\left(w + 1\right) \cdot \log \ell} + 0.5 \cdot \left({w}^{2} \cdot \log \ell\right)\right) - w} \]

      3. associate-*r* 99.8%

         \[\leadsto e^{\left(\left(w + 1\right) \cdot \log \ell + \color{blue}{\left(0.5 \cdot {w}^{2}\right) \cdot \log \ell}\right) - w} \]

      4. distribute-rgt-out 99.8%

         \[\leadsto e^{\color{blue}{\log \ell \cdot \left(\left(w + 1\right) + 0.5 \cdot {w}^{2}\right)} - w} \]

      5. unpow2 99.8%

         \[\leadsto e^{\log \ell \cdot \left(\left(w + 1\right) + 0.5 \cdot \color{blue}{\left(w \cdot w\right)}\right) - w} \]

   8. Simplified 99.8%

      \[\leadsto e^{\color{blue}{\log \ell \cdot \left(\left(w + 1\right) + 0.5 \cdot \left(w \cdot w\right)\right)} - w} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{elif}\;w \leq 1.25 \cdot 10^{-8}:\\ \;\;\;\;\frac{\ell \cdot {\ell}^{w}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \ell \cdot \left(\left(w + 1\right) + 0.5 \cdot \left(w \cdot w\right)\right) - w}\\ \end{array} \]

Alternative 8: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{elif}\;w \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\ell \cdot {\ell}^{w}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \ell \cdot \left(w + 1\right) - w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.3)
   (exp (- (log l) w))
   (if (<= w 1.7e-7)
     (/ (* l (pow l w)) (exp w))
     (exp (- (* (log l) (+ w 1.0)) w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = exp((log(l) - w));
	} else if (w <= 1.7e-7) {
		tmp = (l * pow(l, w)) / exp(w);
	} else {
		tmp = exp(((log(l) * (w + 1.0)) - w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.3d0)) then
        tmp = exp((log(l) - w))
    else if (w <= 1.7d-7) then
        tmp = (l * (l ** w)) / exp(w)
    else
        tmp = exp(((log(l) * (w + 1.0d0)) - w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.3) {
		tmp = Math.exp((Math.log(l) - w));
	} else if (w <= 1.7e-7) {
		tmp = (l * Math.pow(l, w)) / Math.exp(w);
	} else {
		tmp = Math.exp(((Math.log(l) * (w + 1.0)) - w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.3:
		tmp = math.exp((math.log(l) - w))
	elif w <= 1.7e-7:
		tmp = (l * math.pow(l, w)) / math.exp(w)
	else:
		tmp = math.exp(((math.log(l) * (w + 1.0)) - w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.3)
		tmp = exp(Float64(log(l) - w));
	elseif (w <= 1.7e-7)
		tmp = Float64(Float64(l * (l ^ w)) / exp(w));
	else
		tmp = exp(Float64(Float64(log(l) * Float64(w + 1.0)) - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.3)
		tmp = exp((log(l) - w));
	elseif (w <= 1.7e-7)
		tmp = (l * (l ^ w)) / exp(w);
	else
		tmp = exp(((log(l) * (w + 1.0)) - w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.3], N[Exp[N[(N[Log[l], $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 1.7e-7], N[(N[(l * N[Power[l, w], $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(N[Log[l], $MachinePrecision] * N[(w + 1.0), $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -1.30000000000000004

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-div 99.9%

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

      3. log-pow 99.9%

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

      4. add-log-exp 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in w around 0 98.3%

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

   if -1.30000000000000004 < w < 1.69999999999999987e-7

   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. Step-by-step derivation

      1. add-exp-log 91.2%

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

      2. log-div 91.2%

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

      3. log-pow 91.3%

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

      4. add-log-exp 91.3%

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

   5. Applied egg-rr 91.3%

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

   6. Taylor expanded in w around 0 90.4%

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

      1. distribute-lft1-in 90.4%

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

   8. Simplified 90.4%

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

   9. Taylor expanded in w around inf 90.4%

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

       1. exp-diff 90.4%

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

       2. *-commutative 90.4%

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

       3. exp-to-pow 98.7%

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

       4. +-commutative 98.7%

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

   11. Simplified 98.7%

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

       1. pow-plus 99.0%

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

   13. Applied egg-rr 99.0%

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

   if 1.69999999999999987e-7 < w

   1. Initial program 90.8%

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

      1. exp-neg 90.8%

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

      2. associate-*l/ 90.8%

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

      3. *-lft-identity 90.8%

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

   3. Simplified 90.8%

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

      1. add-exp-log 90.7%

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

      2. log-div 90.7%

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

      3. log-pow 90.7%

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

      4. add-log-exp 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in w around 0 99.1%

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

      1. distribute-lft1-in 99.1%

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

   8. Simplified 99.1%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{elif}\;w \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\frac{\ell \cdot {\ell}^{w}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \ell \cdot \left(w + 1\right) - w}\\ \end{array} \]

Alternative 9: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.42:\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{elif}\;w \leq 10:\\ \;\;\;\;\frac{\ell \cdot {\ell}^{w}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{w \cdot \log \ell - w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.42)
   (exp (- (log l) w))
   (if (<= w 10.0) (/ (* l (pow l w)) (exp w)) (exp (- (* w (log l)) w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.42) {
		tmp = exp((log(l) - w));
	} else if (w <= 10.0) {
		tmp = (l * pow(l, w)) / exp(w);
	} else {
		tmp = exp(((w * log(l)) - w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.42d0)) then
        tmp = exp((log(l) - w))
    else if (w <= 10.0d0) then
        tmp = (l * (l ** w)) / exp(w)
    else
        tmp = exp(((w * log(l)) - w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.42) {
		tmp = Math.exp((Math.log(l) - w));
	} else if (w <= 10.0) {
		tmp = (l * Math.pow(l, w)) / Math.exp(w);
	} else {
		tmp = Math.exp(((w * Math.log(l)) - w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.42:
		tmp = math.exp((math.log(l) - w))
	elif w <= 10.0:
		tmp = (l * math.pow(l, w)) / math.exp(w)
	else:
		tmp = math.exp(((w * math.log(l)) - w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.42)
		tmp = exp(Float64(log(l) - w));
	elseif (w <= 10.0)
		tmp = Float64(Float64(l * (l ^ w)) / exp(w));
	else
		tmp = exp(Float64(Float64(w * log(l)) - w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.42)
		tmp = exp((log(l) - w));
	elseif (w <= 10.0)
		tmp = (l * (l ^ w)) / exp(w);
	else
		tmp = exp(((w * log(l)) - w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.42], N[Exp[N[(N[Log[l], $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 10.0], N[(N[(l * N[Power[l, w], $MachinePrecision]), $MachinePrecision] / N[Exp[w], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(w * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -1.4199999999999999

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-div 99.9%

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

      3. log-pow 99.9%

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

      4. add-log-exp 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in w around 0 98.3%

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

   if -1.4199999999999999 < w < 10

   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. add-exp-log 91.2%

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

      2. log-div 91.2%

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

      3. log-pow 91.2%

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

      4. add-log-exp 91.2%

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

   5. Applied egg-rr 91.2%

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

   6. Taylor expanded in w around 0 90.2%

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

      1. distribute-lft1-in 90.2%

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

   8. Simplified 90.2%

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

   9. Taylor expanded in w around inf 90.2%

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

       1. exp-diff 90.2%

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

       2. *-commutative 90.2%

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

       3. exp-to-pow 98.5%

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

       4. +-commutative 98.5%

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

   11. Simplified 98.5%

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

       1. pow-plus 98.8%

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

   13. Applied egg-rr 98.8%

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

   if 10 < w

   1. Initial program 90.7%

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

      1. exp-neg 90.7%

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

      2. associate-*l/ 90.7%

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

      3. *-lft-identity 90.7%

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

   3. Simplified 90.7%

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

      1. add-exp-log 90.7%

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

      2. log-div 90.7%

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

      3. log-pow 90.7%

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

      4. add-log-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in w around 0 100.0%

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

      1. distribute-lft1-in 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in w around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. sub-neg 100.0%

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

       3. metadata-eval 100.0%

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

       4. distribute-rgt-in 100.0%

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

       5. neg-mul-1 100.0%

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

       6. sub-neg 100.0%

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

   11. Simplified 100.0%

       \[\leadsto e^{\color{blue}{\log \ell \cdot w - w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.42:\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{elif}\;w \leq 10:\\ \;\;\;\;\frac{\ell \cdot {\ell}^{w}}{e^{w}}\\ \mathbf{else}:\\ \;\;\;\;e^{w \cdot \log \ell - w}\\ \end{array} \]

Alternative 10: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := w \cdot \log \ell - w\\ \mathbf{if}\;w \leq -0.005:\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{elif}\;w \leq 0.98:\\ \;\;\;\;\ell + \ell \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;e^{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (- (* w (log l)) w)))
   (if (<= w -0.005)
     (exp (- (log l) w))
     (if (<= w 0.98) (+ l (* l t_0)) (exp t_0)))))

C

double code(double w, double l) {
	double t_0 = (w * log(l)) - w;
	double tmp;
	if (w <= -0.005) {
		tmp = exp((log(l) - w));
	} else if (w <= 0.98) {
		tmp = l + (l * t_0);
	} else {
		tmp = exp(t_0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (w * log(l)) - w
    if (w <= (-0.005d0)) then
        tmp = exp((log(l) - w))
    else if (w <= 0.98d0) then
        tmp = l + (l * t_0)
    else
        tmp = exp(t_0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = (w * Math.log(l)) - w;
	double tmp;
	if (w <= -0.005) {
		tmp = Math.exp((Math.log(l) - w));
	} else if (w <= 0.98) {
		tmp = l + (l * t_0);
	} else {
		tmp = Math.exp(t_0);
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = (w * math.log(l)) - w
	tmp = 0
	if w <= -0.005:
		tmp = math.exp((math.log(l) - w))
	elif w <= 0.98:
		tmp = l + (l * t_0)
	else:
		tmp = math.exp(t_0)
	return tmp

Julia

function code(w, l)
	t_0 = Float64(Float64(w * log(l)) - w)
	tmp = 0.0
	if (w <= -0.005)
		tmp = exp(Float64(log(l) - w));
	elseif (w <= 0.98)
		tmp = Float64(l + Float64(l * t_0));
	else
		tmp = exp(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = (w * log(l)) - w;
	tmp = 0.0;
	if (w <= -0.005)
		tmp = exp((log(l) - w));
	elseif (w <= 0.98)
		tmp = l + (l * t_0);
	else
		tmp = exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(w * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]}, If[LessEqual[w, -0.005], N[Exp[N[(N[Log[l], $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision], If[LessEqual[w, 0.98], N[(l + N[(l * t$95$0), $MachinePrecision]), $MachinePrecision], N[Exp[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if w < -0.0050000000000000001

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. associate-*l/ 99.9%

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

      3. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

      1. add-exp-log 99.9%

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

      2. log-div 99.9%

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

      3. log-pow 99.9%

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

      4. add-log-exp 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in w around 0 96.8%

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

   if -0.0050000000000000001 < w < 0.97999999999999998

   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 99.2%

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

      1. sub-neg 99.2%

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

      2. distribute-rgt-in 99.2%

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in l around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. neg-mul-1 99.2%

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

      4. sub-neg 99.2%

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

   9. Simplified 99.2%

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

   if 0.97999999999999998 < w

   1. Initial program 90.7%

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

      1. exp-neg 90.7%

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

      2. associate-*l/ 90.7%

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

      3. *-lft-identity 90.7%

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

   3. Simplified 90.7%

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

      1. add-exp-log 90.7%

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

      2. log-div 90.7%

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

      3. log-pow 90.7%

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

      4. add-log-exp 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in w around 0 100.0%

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

      1. distribute-lft1-in 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in w around inf 100.0%

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

       1. *-commutative 100.0%

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

       2. sub-neg 100.0%

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

       3. metadata-eval 100.0%

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

       4. distribute-rgt-in 100.0%

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

       5. neg-mul-1 100.0%

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

       6. sub-neg 100.0%

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

   11. Simplified 100.0%

       \[\leadsto e^{\color{blue}{\log \ell \cdot w - w}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.005:\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{elif}\;w \leq 0.98:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \log \ell - w\right)\\ \mathbf{else}:\\ \;\;\;\;e^{w \cdot \log \ell - w}\\ \end{array} \]

Alternative 11: 98.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.00295 \lor \neg \left(w \leq 1450\right):\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \log \ell - w\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.00295) (not (<= w 1450.0)))
   (exp (- (log l) w))
   (+ l (* l (- (* w (log l)) w)))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.00295) || !(w <= 1450.0)) {
		tmp = exp((log(l) - w));
	} else {
		tmp = l + (l * ((w * log(l)) - w));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.00295d0)) .or. (.not. (w <= 1450.0d0))) then
        tmp = exp((log(l) - w))
    else
        tmp = l + (l * ((w * log(l)) - w))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.00295) || !(w <= 1450.0)) {
		tmp = Math.exp((Math.log(l) - w));
	} else {
		tmp = l + (l * ((w * Math.log(l)) - w));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.00295) or not (w <= 1450.0):
		tmp = math.exp((math.log(l) - w))
	else:
		tmp = l + (l * ((w * math.log(l)) - w))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.00295) || !(w <= 1450.0))
		tmp = exp(Float64(log(l) - w));
	else
		tmp = Float64(l + Float64(l * Float64(Float64(w * log(l)) - w)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.00295) || ~((w <= 1450.0)))
		tmp = exp((log(l) - w));
	else
		tmp = l + (l * ((w * log(l)) - w));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.00295], N[Not[LessEqual[w, 1450.0]], $MachinePrecision]], N[Exp[N[(N[Log[l], $MachinePrecision] - w), $MachinePrecision]], $MachinePrecision], N[(l + N[(l * N[(N[(w * N[Log[l], $MachinePrecision]), $MachinePrecision] - w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.00294999999999999993 or 1450 < w

   1. Initial program 97.9%

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

      1. exp-neg 97.9%

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

      2. associate-*l/ 97.9%

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

      3. *-lft-identity 97.9%

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

   3. Simplified 97.9%

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

      1. add-exp-log 97.9%

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

      2. log-div 97.9%

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

      3. log-pow 97.9%

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

      4. add-log-exp 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in w around 0 96.1%

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

   if -0.00294999999999999993 < w < 1450

   1. Initial program 98.4%

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

      1. exp-neg 98.4%

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

      2. associate-*l/ 98.4%

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

      3. *-lft-identity 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in w around 0 98.6%

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

      1. sub-neg 98.6%

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

      2. distribute-rgt-in 98.0%

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

   6. Applied egg-rr 98.0%

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

   7. Taylor expanded in l around 0 98.6%

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

      1. *-commutative 98.6%

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

      2. *-commutative 98.6%

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

      3. neg-mul-1 98.6%

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

      4. sub-neg 98.6%

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

   9. Simplified 98.6%

      \[\leadsto \ell + \color{blue}{\ell \cdot \left(\log \ell \cdot w - w\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.00295 \lor \neg \left(w \leq 1450\right):\\ \;\;\;\;e^{\log \ell - w}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \log \ell - w\right)\\ \end{array} \]

Alternative 12: 63.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l - ((l * (w * math.log(l))) + (l * w))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in w around 0 61.8%

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

   1. sub-neg 61.8%

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

   2. distribute-rgt-in 61.4%

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

6. Applied egg-rr 61.4%

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

7. Taylor expanded in l around inf 61.4%

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

   1. add-sqr-sqrt 41.3%

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

   2. sqrt-unprod 58.3%

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

   3. sqr-neg 58.3%

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

   4. mul-1-neg 58.3%

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

   5. mul-1-neg 58.3%

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

   6. sqrt-unprod 33.0%

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

   7. add-sqr-sqrt 64.9%

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

   8. expm1-log1p-u 50.6%

      \[\leadsto \ell + \left(-1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-1 \cdot \left(\log \left(\frac{1}{\ell}\right) \cdot \left(\ell \cdot w\right)\right)\right)\right)} + \left(-\ell\right) \cdot w\right) \]

   9. expm1-udef 50.8%

      \[\leadsto \ell + \left(-1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(-1 \cdot \left(\log \left(\frac{1}{\ell}\right) \cdot \left(\ell \cdot w\right)\right)\right)} - 1\right)} + \left(-\ell\right) \cdot w\right) \]

9. Applied egg-rr 50.8%

   \[\leadsto \ell + \left(-1 \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\log \ell \cdot \left(w \cdot \ell\right)\right)} - 1\right)} + \left(-\ell\right) \cdot w\right) \]
10. Step-by-step derivation

    1. expm1-def 50.6%

       \[\leadsto \ell + \left(-1 \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \ell \cdot \left(w \cdot \ell\right)\right)\right)} + \left(-\ell\right) \cdot w\right) \]

    2. expm1-log1p 64.9%

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

    3. *-commutative 64.9%

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

    4. *-commutative 64.9%

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

    5. associate-*r* 64.9%

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

11. Simplified 64.9%

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

12. Final simplification 64.9%

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

Alternative 13: 56.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l + (l * ((w * math.log(l)) - w))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.2%

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

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in w around 0 61.8%

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

   1. sub-neg 61.8%

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

   2. distribute-rgt-in 61.4%

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

6. Applied egg-rr 61.4%

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

7. Taylor expanded in l around 0 61.8%

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

   1. *-commutative 61.8%

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

   2. *-commutative 61.8%

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

   3. neg-mul-1 61.8%

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

   4. sub-neg 61.8%

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

9. Simplified 61.8%

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

10. Final simplification 61.8%

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

Alternative 14: 56.8% 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 98.2%

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

   1. exp-neg 98.2%

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

   2. associate-*l/ 98.2%

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

   3. *-lft-identity 98.2%

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

3. Simplified 98.2%

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

4. Taylor expanded in w around 0 61.8%

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

5. Final simplification 61.8%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023279 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))