exp-w (used to crash)

Percentage Accurate: 99.4% → 99.5%

Time: 14.3s

Alternatives: 14

Speedup: 1.4×

• 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: 99.4% 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, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (exp (- w))
   (/
    (pow l (exp w))
    (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-1.6d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** exp(w)) / (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -1.6) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.6:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, math.exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.6)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ exp(w)) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -1.6)
		tmp = exp(-w);
	else
		tmp = (l ^ exp(w)) / (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.6], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.6000000000000001

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around inf 100.0%

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

       1. exp-prod 100.0%

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

       2. log-E 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. exp-prod 100.0%

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

       5. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 98.9%

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

      1. exp-neg 98.9%

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

      2. remove-double-neg 98.9%

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

      3. associate-*l/ 98.9%

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

      4. *-lft-identity 98.9%

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

      5. remove-double-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in w around 0 99.2%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + w \cdot \left(0.5 + 0.16666666666666666 \cdot w\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 99.2%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + \color{blue}{w \cdot 0.16666666666666666}\right)\right)} \]

   7. Simplified 99.2%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{\sqrt[3]{e^{w}}}\right)}^{3}\\ \frac{{\ell}^{\left(e^{w}\right)}}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. exp-neg 99.2%

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

   2. remove-double-neg 99.2%

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

   3. associate-*l/ 99.2%

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

   4. *-lft-identity 99.2%

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

   5. remove-double-neg 99.2%

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

3. Simplified 99.2%

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

   1. add-cube-cbrt 99.2%

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

   2. pow3 99.2%

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

   3. add-sqr-sqrt 99.2%

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

   4. unpow-prod-down 99.2%

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

6. Applied egg-rr 99.2%

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

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{w}}\\ \frac{{\ell}^{\left(e^{w}\right)}}{t\_0 \cdot {\left({\left(\sqrt[3]{t\_0}\right)}^{2}\right)}^{3}} \end{array} \end{array} \]

FPCore

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

C

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

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.pow(Math.cbrt(t_0), 2.0), 3.0));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. exp-neg 99.2%

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

   2. remove-double-neg 99.2%

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

   3. associate-*l/ 99.2%

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

   4. *-lft-identity 99.2%

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

   5. remove-double-neg 99.2%

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

3. Simplified 99.2%

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

   1. add-cube-cbrt 99.2%

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

   2. pow3 99.2%

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

   3. add-cube-cbrt 99.2%

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

   4. unpow-prod-down 99.2%

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

   5. pow2 99.2%

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

   6. pow3 99.2%

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

   7. add-cube-cbrt 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w}} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{e^{w}}}\right)}^{2}\right)}^{3}} \]
8. Add Preprocessing

Alternative 4: 99.3% accurate, 0.8× speedup

Math

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

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 * 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 99.2%

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

   1. exp-neg 99.2%

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

   2. remove-double-neg 99.2%

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

   3. associate-*l/ 99.2%

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

   4. *-lft-identity 99.2%

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

   5. remove-double-neg 99.2%

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

3. Simplified 99.2%

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

   1. add-cbrt-cube 99.2%

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

   2. add-exp-log 99.2%

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

   3. pow3 99.2%

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

   4. log-pow 99.2%

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

   5. add-log-exp 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\sqrt[3]{e^{w \cdot 3}}} \]
8. Add Preprocessing

Alternative 5: 99.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return math.pow(l, math.exp(w)) / ((math.exp(w) + 1.0) + -1.0)

Julia

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

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / ((exp(w) + 1.0) + -1.0);
end

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. exp-neg 99.2%

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

   2. remove-double-neg 99.2%

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

   3. associate-*l/ 99.2%

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

   4. *-lft-identity 99.2%

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

   5. remove-double-neg 99.2%

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

3. Simplified 99.2%

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

   1. expm1-log1p-u 99.2%

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

   2. expm1-undefine 99.2%

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

   3. log1p-undefine 99.2%

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

   4. rem-exp-log 99.2%

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

6. Applied egg-rr 99.2%

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

7. Final simplification 99.2%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\left(e^{w} + 1\right) + -1} \]
8. Add Preprocessing

Alternative 6: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. exp-neg 99.2%

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

   2. remove-double-neg 99.2%

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

   3. associate-*l/ 99.2%

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

   4. *-lft-identity 99.2%

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

   5. remove-double-neg 99.2%

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

3. Simplified 99.2%

   \[\leadsto \color{blue}{\frac{{\ell}^{\left(e^{w}\right)}}{e^{w}}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 7: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3.6:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -3.6)
   (exp (- w))
   (/ (pow l (exp w)) (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -3.6) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-3.6d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** exp(w)) / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -3.6) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, Math.exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -3.6:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, math.exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -3.6)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ exp(w)) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -3.6)
		tmp = exp(-w);
	else
		tmp = (l ^ exp(w)) / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -3.6], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -3.60000000000000009

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around inf 100.0%

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

       1. exp-prod 100.0%

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

       2. log-E 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. exp-prod 100.0%

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

       5. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   if -3.60000000000000009 < w

   1. Initial program 98.9%

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

      1. exp-neg 98.9%

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

      2. remove-double-neg 98.9%

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

      3. associate-*l/ 98.9%

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

      4. *-lft-identity 98.9%

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

      5. remove-double-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in w around 0 99.0%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + 0.5 \cdot w\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + w \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4500:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4500.0)
   (exp (- w))
   (/
    (pow l (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666)))))))
    (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4500.0) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-4500.0d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** (1.0d0 + (w * (1.0d0 + (w * (0.5d0 + (w * 0.16666666666666666d0))))))) / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -4500.0) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -4500.0:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4500.0)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * Float64(0.5 + Float64(w * 0.16666666666666666))))))) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -4500.0)
		tmp = exp(-w);
	else
		tmp = (l ^ (1.0 + (w * (1.0 + (w * (0.5 + (w * 0.16666666666666666))))))) / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4500.0], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[(1.0 + N[(w * N[(1.0 + N[(w * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4500

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around inf 100.0%

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

       1. exp-prod 100.0%

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

       2. log-E 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. exp-prod 100.0%

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

       5. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   if -4500 < w

   1. Initial program 98.9%

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

      1. exp-neg 98.9%

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

      2. remove-double-neg 98.9%

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

      3. associate-*l/ 98.9%

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

      4. *-lft-identity 98.9%

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

      5. remove-double-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in w around 0 98.5%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + 0.5 \cdot w\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in w around 0 99.0%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + 0.16666666666666666 \cdot w\right)\right)\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]
   9. Step-by-step derivation

      1. *-commutative 98.6%

         \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot \left(0.5 + \color{blue}{w \cdot 0.16666666666666666}\right)\right)} \]

   10. Simplified 99.0%

       \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + w \cdot \left(1 + w \cdot 0.5\right)\\ \mathbf{if}\;w \leq -1.3:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* w (+ 1.0 (* w 0.5))))))
   (if (<= w -1.3) (exp (- w)) (/ (pow l t_0) t_0))))

C

double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * 0.5)));
	double tmp;
	if (w <= -1.3) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, t_0) / 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 = 1.0d0 + (w * (1.0d0 + (w * 0.5d0)))
    if (w <= (-1.3d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** t_0) / t_0
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = 1.0 + (w * (1.0 + (w * 0.5)));
	double tmp;
	if (w <= -1.3) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, t_0) / t_0;
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = 1.0 + (w * (1.0 + (w * 0.5)))
	tmp = 0
	if w <= -1.3:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, t_0) / t_0
	return tmp

Julia

function code(w, l)
	t_0 = Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5))))
	tmp = 0.0
	if (w <= -1.3)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ t_0) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = 1.0 + (w * (1.0 + (w * 0.5)));
	tmp = 0.0;
	if (w <= -1.3)
		tmp = exp(-w);
	else
		tmp = (l ^ t_0) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.3], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, t$95$0], $MachinePrecision] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < -1.30000000000000004

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around inf 100.0%

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

       1. exp-prod 100.0%

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

       2. log-E 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. exp-prod 100.0%

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

       5. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   if -1.30000000000000004 < w

   1. Initial program 98.9%

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

      1. exp-neg 98.9%

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

      2. remove-double-neg 98.9%

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

      3. associate-*l/ 98.9%

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

      4. *-lft-identity 98.9%

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

      5. remove-double-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in w around 0 99.0%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + 0.5 \cdot w\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in w around 0 98.8%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + 0.5 \cdot w\right)\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]
   9. Step-by-step derivation

      1. *-commutative 99.0%

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

   10. Simplified 98.8%

       \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot 0.5\right)\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 98.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.38 \lor \neg \left(w \leq 170000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.38) (not (<= w 170000.0)))
   (exp (- w))
   (+ l (* w (- (* l (log l)) l)))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.38) || !(w <= 170000.0)) {
		tmp = exp(-w);
	} else {
		tmp = l + (w * ((l * log(l)) - l));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.38d0)) .or. (.not. (w <= 170000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l + (w * ((l * log(l)) - l))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.38) || !(w <= 170000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l + (w * ((l * Math.log(l)) - l));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.38) or not (w <= 170000.0):
		tmp = math.exp(-w)
	else:
		tmp = l + (w * ((l * math.log(l)) - l))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.38) || !(w <= 170000.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l + Float64(w * Float64(Float64(l * log(l)) - l)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.38) || ~((w <= 170000.0)))
		tmp = exp(-w);
	else
		tmp = l + (w * ((l * log(l)) - l));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.38], N[Not[LessEqual[w, 170000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l + N[(w * N[(N[(l * N[Log[l], $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.38 or 1.7e5 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around inf 100.0%

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

       1. exp-prod 100.0%

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

       2. log-E 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. exp-prod 100.0%

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

       5. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   if -0.38 < w < 1.7e5

   1. Initial program 98.6%

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

      1. exp-neg 98.6%

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

      2. remove-double-neg 98.6%

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

      3. associate-*l/ 98.6%

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

      4. *-lft-identity 98.6%

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

      5. remove-double-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in w around 0 95.5%

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

4. Final simplification 97.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.38 \lor \neg \left(w \leq 170000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \log \ell - \ell\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 98.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4500:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(w + 1\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4500.0)
   (exp (- w))
   (/ (pow l (+ w 1.0)) (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4500.0) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, (w + 1.0)) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-4500.0d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** (w + 1.0d0)) / (1.0d0 + (w * (1.0d0 + (w * 0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -4500.0) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, (w + 1.0)) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -4500.0:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, (w + 1.0)) / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4500.0)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ Float64(w + 1.0)) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -4500.0)
		tmp = exp(-w);
	else
		tmp = (l ^ (w + 1.0)) / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4500.0], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision] / N[(1.0 + N[(w * N[(1.0 + N[(w * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -4500

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around inf 100.0%

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

       1. exp-prod 100.0%

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

       2. log-E 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. exp-prod 100.0%

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

       5. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   if -4500 < w

   1. Initial program 98.9%

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

      1. exp-neg 98.9%

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

      2. remove-double-neg 98.9%

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

      3. associate-*l/ 98.9%

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

      4. *-lft-identity 98.9%

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

      5. remove-double-neg 98.9%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in w around 0 98.5%

      \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{1 + w \cdot \left(1 + 0.5 \cdot w\right)}} \]
   6. Step-by-step derivation

      1. *-commutative 98.5%

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

   7. Simplified 98.5%

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

   8. Taylor expanded in w around 0 98.2%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]
   9. Step-by-step derivation

      1. +-commutative 98.2%

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

   10. Simplified 98.2%

       \[\leadsto \frac{{\ell}^{\color{blue}{\left(w + 1\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 97.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 170000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 170000.0))) (exp (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 170000.0)) {
		tmp = exp(-w);
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.7d0)) .or. (.not. (w <= 170000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 170000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 170000.0):
		tmp = math.exp(-w)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 170000.0))
		tmp = exp(Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 170000.0)))
		tmp = exp(-w);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 170000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 1.7e5 < w

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around inf 100.0%

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

       1. exp-prod 100.0%

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

       2. log-E 100.0%

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

       3. *-rgt-identity 100.0%

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

       4. exp-prod 100.0%

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

       5. mul-1-neg 100.0%

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

   12. Simplified 100.0%

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

   if -0.69999999999999996 < w < 1.7e5

   1. Initial program 98.6%

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

      1. exp-neg 98.6%

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

      2. remove-double-neg 98.6%

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

      3. associate-*l/ 98.6%

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

      4. *-lft-identity 98.6%

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

      5. remove-double-neg 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in w around 0 94.5%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 170000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.3% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3.6 \cdot 10^{+55}:\\ \;\;\;\;1 + w \cdot \left(w \cdot 0.5 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -3.6e+55) (+ 1.0 (* w (+ (* w 0.5) -1.0))) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -3.6e+55) {
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	} else {
		tmp = l;
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-3.6d+55)) then
        tmp = 1.0d0 + (w * ((w * 0.5d0) + (-1.0d0)))
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -3.6e+55) {
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -3.6e+55:
		tmp = 1.0 + (w * ((w * 0.5) + -1.0))
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -3.6e+55)
		tmp = Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0)));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -3.6e+55)
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -3.6e+55], N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -3.59999999999999987e55

   1. Initial program 100.0%

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

      1. exp-neg 100.0%

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

      2. remove-double-neg 100.0%

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

      3. associate-*l/ 100.0%

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

      4. *-lft-identity 100.0%

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

      5. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. pow1 100.0%

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

      3. log-pow 100.0%

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

      4. exp-prod 100.0%

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

      5. exp-1-e 100.0%

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

      6. log-div 100.0%

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

      7. log-pow 100.0%

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

      8. add-log-exp 100.0%

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

      9. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in w around 0 62.6%

       \[\leadsto \color{blue}{1 + w \cdot \left(-1 \cdot \log e + 0.5 \cdot \left(w \cdot {\log e}^{2}\right)\right)} \]
   11. Step-by-step derivation

       1. log-E 62.6%

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

       2. metadata-eval 62.6%

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

       3. associate-*r* 62.6%

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

       4. *-commutative 62.6%

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

       5. log-E 62.6%

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

       6. metadata-eval 62.6%

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

       7. *-rgt-identity 62.6%

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

   12. Simplified 62.6%

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

   if -3.59999999999999987e55 < w

   1. Initial program 99.0%

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

      1. exp-neg 99.0%

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

      2. remove-double-neg 99.0%

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

      3. associate-*l/ 99.0%

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

      4. *-lft-identity 99.0%

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

      5. remove-double-neg 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in w around 0 69.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -3.6 \cdot 10^{+55}:\\ \;\;\;\;1 + w \cdot \left(w \cdot 0.5 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 57.7% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.2%

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

   1. exp-neg 99.2%

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

   2. remove-double-neg 99.2%

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

   3. associate-*l/ 99.2%

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

   4. *-lft-identity 99.2%

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

   5. remove-double-neg 99.2%

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

3. Simplified 99.2%

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

5. Taylor expanded in w around 0 55.6%

   \[\leadsto \color{blue}{\ell} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024132 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))