exp-w (used to crash)

Percentage Accurate: 99.5% → 99.4%

Time: 21.4s

Alternatives: 21

Speedup: N/A×

• could not determine a ground truth

  1. w = 2.00000000000000007e154
  2. l = -2

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(w, l)
	return Float64((l ^ exp(w)) / Float64(exp(Float64(w * Float64(1.0 + Float64(1.0 / w)))) / exp(1)))
end

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / (exp((w * (1.0 + (1.0 / w)))) / 2.71828182845904523536);
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. remove-double-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-lft-identity 99.6%

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

   5. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

   1. expm1-log1p-u 75.7%

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

   2. expm1-undefine 75.7%

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

   3. exp-diff 75.7%

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

   4. log1p-undefine 75.7%

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

   5. rem-exp-log 99.6%

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

   6. exp-1-e 99.6%

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

6. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

8. Simplified 99.6%

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

9. Taylor expanded in w around inf 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(w, l)
	return Float64((l ^ exp(w)) / Float64(exp(Float64(w + 1.0)) / exp(1)))
end

MATLAB

function tmp = code(w, l)
	tmp = (l ^ exp(w)) / (exp((w + 1.0)) / 2.71828182845904523536);
end

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. remove-double-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-lft-identity 99.6%

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

   5. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

   1. expm1-log1p-u 75.7%

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

   2. expm1-undefine 75.7%

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

   3. exp-diff 75.7%

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

   4. log1p-undefine 75.7%

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

   5. rem-exp-log 99.6%

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

   6. exp-1-e 99.6%

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

6. Applied egg-rr 99.6%

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

   1. +-commutative 99.6%

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

8. Simplified 99.6%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. remove-double-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-lft-identity 99.6%

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

   5. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

Alternative 4: 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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

Alternative 5: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -4.1:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(1 + \mathsf{expm1}\left(w\right)\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -4.1)
   (exp (- w))
   (/ (pow l (+ 1.0 (expm1 w))) (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -4.1) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, (1.0 + expm1(w))) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -4.1) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, (1.0 + Math.expm1(w))) / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -4.1:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, (1.0 + math.expm1(w))) / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -4.1)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ Float64(1.0 + expm1(w))) / Float64(1.0 + Float64(w * Float64(1.0 + Float64(w * 0.5)))));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -4.1], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, N[(1.0 + N[(Exp[w] - 1), $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 < -4.0999999999999996

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -4.0999999999999996 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

      1. pow1 98.4%

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

      2. pow-to-exp 98.4%

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

      3. add-log-exp 98.4%

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

      4. log1p-expm1-u 98.4%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(w\right)\right)} \cdot 1}\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]

      5. log1p-undefine 98.4%

         \[\leadsto \frac{{\ell}^{\left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(w\right)\right)} \cdot 1}\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]

      6. exp-to-pow 98.4%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left({\left(1 + \mathsf{expm1}\left(w\right)\right)}^{1}\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]

      7. pow1 98.4%

         \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + \mathsf{expm1}\left(w\right)\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]

   9. Applied egg-rr 98.4%

      \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + \mathsf{expm1}\left(w\right)\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -3.8:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -3.8)
   (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 <= -3.8) {
		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 <= (-3.8d0)) 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 <= -3.8) {
		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 <= -3.8:
		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 <= -3.8)
		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 <= -3.8)
		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, -3.8], 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 < -3.7999999999999998

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -3.7999999999999998 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

Alternative 7: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\\ \mathbf{if}\;w \leq -1.6:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{t\_0}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* w (+ 1.0 (* w (+ 0.5 (* w 0.16666666666666666))))))))
   (if (<= w -1.6) (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 + (w * 0.16666666666666666)))));
	double tmp;
	if (w <= -1.6) {
		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 + (w * 0.16666666666666666d0)))))
    if (w <= (-1.6d0)) 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 + (w * 0.16666666666666666)))));
	double tmp;
	if (w <= -1.6) {
		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 + (w * 0.16666666666666666)))))
	tmp = 0
	if w <= -1.6:
		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 * Float64(0.5 + Float64(w * 0.16666666666666666))))))
	tmp = 0.0
	if (w <= -1.6)
		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 + (w * 0.16666666666666666)))));
	tmp = 0.0;
	if (w <= -1.6)
		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 * N[(0.5 + N[(w * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, -1.6], 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 < -1.6000000000000001

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.7%

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

      1. *-commutative 98.7%

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

   7. Simplified 98.7%

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

   8. Taylor expanded in w around 0 98.4%

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

      1. *-commutative 98.7%

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

   10. Simplified 98.4%

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

Alternative 8: 99.3% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.6)
   (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 <= -1.6) {
		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 <= (-1.6d0)) 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 <= -1.6) {
		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 <= -1.6:
		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 <= -1.6)
		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 <= -1.6)
		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, -1.6], 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 < -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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -1.6000000000000001 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in w around 0 98.3%

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

      1. *-commutative 98.7%

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

   10. Simplified 98.3%

       \[\leadsto \frac{{\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + w \cdot \left(0.5 + w \cdot 0.16666666666666666\right)\right)\right)}}}{1 + w \cdot \left(1 + w \cdot 0.5\right)} \]
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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -1.30000000000000004 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.4%

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

      1. *-commutative 98.4%

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

   7. Simplified 98.4%

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

   8. Taylor expanded in w around 0 98.0%

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

      1. *-commutative 98.4%

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

   10. Simplified 98.0%

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

Alternative 10: 99.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.0)
		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(w + 1.0));
	end
	return tmp
end

MATLAB

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

Wolfram

code[w_, l_] := If[LessEqual[w, -1.0], 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[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -1 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.1%

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

      1. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in w around 0 98.0%

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

      1. *-commutative 98.7%

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

   10. Simplified 98.0%

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

Alternative 11: 99.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -1 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.1%

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

      1. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in w around 0 98.0%

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

      1. *-commutative 98.4%

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

   10. Simplified 98.0%

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

Alternative 12: 98.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;e^{-w}\\ \mathbf{elif}\;w \leq 0.0115:\\ \;\;\;\;\ell\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(w + 1\right)}}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.68) (exp (- w)) (if (<= w 0.0115) l (/ (pow l (+ w 1.0)) w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = exp(-w);
	} else if (w <= 0.0115) {
		tmp = l;
	} else {
		tmp = pow(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 <= (-0.68d0)) then
        tmp = exp(-w)
    else if (w <= 0.0115d0) then
        tmp = l
    else
        tmp = (l ** (w + 1.0d0)) / w
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = Math.exp(-w);
	} else if (w <= 0.0115) {
		tmp = l;
	} else {
		tmp = Math.pow(l, (w + 1.0)) / w;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.68:
		tmp = math.exp(-w)
	elif w <= 0.0115:
		tmp = l
	else:
		tmp = math.pow(l, (w + 1.0)) / w
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.68)
		tmp = exp(Float64(-w));
	elseif (w <= 0.0115)
		tmp = l;
	else
		tmp = Float64((l ^ Float64(w + 1.0)) / w);
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.68)
		tmp = exp(-w);
	elseif (w <= 0.0115)
		tmp = l;
	else
		tmp = (l ^ (w + 1.0)) / w;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.68], N[Exp[(-w)], $MachinePrecision], If[LessEqual[w, 0.0115], l, N[(N[Power[l, N[(w + 1.0), $MachinePrecision]], $MachinePrecision] / w), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if w < -0.680000000000000049

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -0.680000000000000049 < w < 0.0115

   1. Initial program 99.3%

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

      1. exp-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. associate-*l/ 99.3%

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

      4. *-lft-identity 99.3%

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

      5. remove-double-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in w around 0 95.5%

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

   if 0.0115 < 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. Taylor expanded in w around 0 100.0%

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

      1. +-commutative 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around 0 100.0%

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

      1. +-commutative 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in w around inf 100.0%

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

       1. exp-to-pow 100.0%

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

       2. +-commutative 100.0%

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

   13. Simplified 100.0%

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

Alternative 13: 99.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.98:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{w} \cdot \frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.98) (exp (- w)) (* (pow l w) (/ l (+ w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.98) {
		tmp = exp(-w);
	} else {
		tmp = pow(l, w) * (l / (w + 1.0));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.98d0)) then
        tmp = exp(-w)
    else
        tmp = (l ** w) * (l / (w + 1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.98) {
		tmp = Math.exp(-w);
	} else {
		tmp = Math.pow(l, w) * (l / (w + 1.0));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.98:
		tmp = math.exp(-w)
	else:
		tmp = math.pow(l, w) * (l / (w + 1.0))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.98)
		tmp = exp(Float64(-w));
	else
		tmp = Float64((l ^ w) * Float64(l / Float64(w + 1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.98)
		tmp = exp(-w);
	else
		tmp = (l ^ w) * (l / (w + 1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.98], N[Exp[(-w)], $MachinePrecision], N[(N[Power[l, w], $MachinePrecision] * N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.97999999999999998

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -0.97999999999999998 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.1%

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

      1. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in w around 0 97.6%

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

      1. +-commutative 98.1%

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

   10. Simplified 97.6%

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

       1. pow-plus 97.8%

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

       2. *-un-lft-identity 97.8%

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

       3. times-frac 97.8%

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

   12. Applied egg-rr 97.8%

       \[\leadsto \color{blue}{\frac{{\ell}^{w}}{1} \cdot \frac{\ell}{w + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.98:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;{\ell}^{w} \cdot \frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 99.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if w < -1

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -1 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.1%

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

      1. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in w around 0 97.6%

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

      1. +-commutative 98.1%

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

   10. Simplified 97.6%

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

       1. pow-plus 97.8%

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

   12. Applied egg-rr 97.8%

       \[\leadsto \frac{\color{blue}{{\ell}^{w} \cdot \ell}}{w + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1:\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot {\ell}^{w}}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 560\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.68) (not (<= w 560.0))) (exp (- w)) l))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.68) || !(w <= 560.0)) {
		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 <= (-0.68d0)) .or. (.not. (w <= 560.0d0))) 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 <= -0.68) || !(w <= 560.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.68) or not (w <= 560.0):
		tmp = math.exp(-w)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.68) || !(w <= 560.0))
		tmp = exp(Float64(-w));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.68) || ~((w <= 560.0)))
		tmp = exp(-w);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.68], N[Not[LessEqual[w, 560.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.680000000000000049 or 560 < 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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   if -0.680000000000000049 < w < 560

   1. Initial program 99.3%

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

      1. exp-neg 99.3%

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

      2. remove-double-neg 99.3%

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

      3. associate-*l/ 99.3%

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

      4. *-lft-identity 99.3%

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

      5. remove-double-neg 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in w around 0 95.0%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68 \lor \neg \left(w \leq 560\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 85.1% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2.9e+20)
   (+ 1.0 (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0)))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2.9e+20) {
		tmp = 1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0));
	} else {
		tmp = l / (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 <= (-2.9d+20)) then
        tmp = 1.0d0 + (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0)))
    else
        tmp = l / (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 <= -2.9e+20) {
		tmp = 1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -2.9e+20:
		tmp = 1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2.9e+20)
		tmp = Float64(1.0 + Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0)));
	else
		tmp = Float64(l / 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 <= -2.9e+20)
		tmp = 1.0 + (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2.9e+20], N[(1.0 + N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / 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 < -2.9e20

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in w around 0 76.7%

       \[\leadsto \color{blue}{1 + w \cdot \left(w \cdot \left(0.5 + -0.16666666666666666 \cdot w\right) - 1\right)} \]

   if -2.9e20 < w

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 96.0%

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

      1. *-commutative 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in w around 0 95.9%

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

      1. *-commutative 96.2%

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

   10. Simplified 95.9%

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

   11. Taylor expanded in w around 0 87.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -2.9 \cdot 10^{+20}:\\ \;\;\;\;1 + w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 80.7% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;1 + w \cdot \left(w \cdot 0.5 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.8e+20)
   (+ 1.0 (* w (+ (* w 0.5) -1.0)))
   (/ l (+ 1.0 (* w (+ 1.0 (* w 0.5)))))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.8e+20) {
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	} else {
		tmp = l / (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 <= (-1.8d+20)) then
        tmp = 1.0d0 + (w * ((w * 0.5d0) + (-1.0d0)))
    else
        tmp = l / (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 <= -1.8e+20) {
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	} else {
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -1.8e+20:
		tmp = 1.0 + (w * ((w * 0.5) + -1.0))
	else:
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))))
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.8e+20)
		tmp = Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0)));
	else
		tmp = Float64(l / 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 <= -1.8e+20)
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	else
		tmp = l / (1.0 + (w * (1.0 + (w * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.8e+20], N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / 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 < -1.8e20

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in w around 0 51.2%

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

   if -1.8e20 < w

   1. Initial program 99.5%

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

      1. exp-neg 99.5%

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

      2. remove-double-neg 99.5%

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

      3. associate-*l/ 99.5%

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

      4. *-lft-identity 99.5%

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

      5. remove-double-neg 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in w around 0 96.0%

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

      1. *-commutative 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in w around 0 95.9%

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

      1. *-commutative 96.2%

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

   10. Simplified 95.9%

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

   11. Taylor expanded in w around 0 87.1%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -1.8 \cdot 10^{+20}:\\ \;\;\;\;1 + w \cdot \left(w \cdot 0.5 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{1 + w \cdot \left(1 + w \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 76.9% accurate, 21.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;1 + w \cdot \left(w \cdot 0.5 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -0.68) (+ 1.0 (* w (+ (* w 0.5) -1.0))) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= (-0.68d0)) then
        tmp = 1.0d0 + (w * ((w * 0.5d0) + (-1.0d0)))
    else
        tmp = l / (w + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -0.68) {
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.68:
		tmp = 1.0 + (w * ((w * 0.5) + -1.0))
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.68)
		tmp = Float64(1.0 + Float64(w * Float64(Float64(w * 0.5) + -1.0)));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.68)
		tmp = 1.0 + (w * ((w * 0.5) + -1.0));
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.68], N[(1.0 + N[(w * N[(N[(w * 0.5), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.680000000000000049

   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. Taylor expanded in l around inf 100.0%

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

      1. mul-1-neg 100.0%

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

      2. distribute-rgt-neg-in 100.0%

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

      3. log-rec 100.0%

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

      4. remove-double-neg 100.0%

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

      5. +-rgt-identity 100.0%

         \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

      6. exp-diff 100.0%

         \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

      7. +-rgt-identity 100.0%

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

      8. remove-double-neg 100.0%

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

      9. distribute-lft-neg-in 100.0%

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

      10. distribute-rgt-neg-out 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in w around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
   9. Step-by-step derivation

      1. neg-mul-1 100.0%

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

   10. Simplified 100.0%

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

   11. Taylor expanded in w around 0 47.3%

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

   if -0.680000000000000049 < w

   1. Initial program 99.4%

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

      1. exp-neg 99.4%

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

      2. remove-double-neg 99.4%

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

      3. associate-*l/ 99.4%

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

      4. *-lft-identity 99.4%

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

      5. remove-double-neg 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in w around 0 98.1%

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

      1. +-commutative 98.1%

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

   7. Simplified 98.1%

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

   8. Taylor expanded in w around 0 98.0%

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

      1. *-commutative 98.7%

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

   10. Simplified 98.0%

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

   11. Taylor expanded in w around 0 86.3%

       \[\leadsto \frac{\color{blue}{\ell}}{w + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.68:\\ \;\;\;\;1 + w \cdot \left(w \cdot 0.5 + -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 62.7% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell}{w + 1} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ l (+ w 1.0)))

C

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

Fortran

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

Java

public static double code(double w, double l) {
	return l / (w + 1.0);
}

Python

def code(w, l):
	return l / (w + 1.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. remove-double-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-lft-identity 99.6%

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

   5. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in w around 0 75.0%

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

   1. +-commutative 75.0%

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

7. Simplified 75.0%

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

8. Taylor expanded in w around 0 74.9%

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

   1. *-commutative 75.5%

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

10. Simplified 74.9%

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

11. Taylor expanded in w around 0 66.0%

    \[\leadsto \frac{\color{blue}{\ell}}{w + 1} \]
12. Add Preprocessing

Alternative 20: 57.0% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. remove-double-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-lft-identity 99.6%

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

   5. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in w around 0 59.3%

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

Alternative 21: 4.4% accurate, 305.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (w l) :precision binary64 1.0)

C

double code(double w, double l) {
	return 1.0;
}

Fortran

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

Java

public static double code(double w, double l) {
	return 1.0;
}

Python

def code(w, l):
	return 1.0

Julia

function code(w, l)
	return 1.0
end

MATLAB

function tmp = code(w, l)
	tmp = 1.0;
end

Wolfram

code[w_, l_] := 1.0

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. remove-double-neg 99.6%

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

   3. associate-*l/ 99.6%

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

   4. *-lft-identity 99.6%

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

   5. remove-double-neg 99.6%

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

3. Simplified 99.6%

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

5. Taylor expanded in l around inf 94.5%

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

   1. mul-1-neg 94.5%

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

   2. distribute-rgt-neg-in 94.5%

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

   3. log-rec 94.5%

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

   4. remove-double-neg 94.5%

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

   5. +-rgt-identity 94.5%

      \[\leadsto \frac{e^{\color{blue}{e^{w} \cdot \log \ell + 0}}}{e^{w}} \]

   6. exp-diff 94.5%

      \[\leadsto \color{blue}{e^{\left(e^{w} \cdot \log \ell + 0\right) - w}} \]

   7. +-rgt-identity 94.5%

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

   8. remove-double-neg 94.5%

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

   9. distribute-lft-neg-in 94.5%

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

   10. distribute-rgt-neg-out 94.5%

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

7. Simplified 94.5%

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

8. Taylor expanded in w around inf 42.7%

   \[\leadsto e^{\color{blue}{-1 \cdot w}} \]
9. Step-by-step derivation

   1. neg-mul-1 42.7%

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

10. Simplified 42.7%

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

11. Taylor expanded in w around 0 4.9%

    \[\leadsto \color{blue}{1 + -1 \cdot w} \]
12. Step-by-step derivation

    1. neg-mul-1 4.9%

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

    2. sub-neg 4.9%

       \[\leadsto \color{blue}{1 - w} \]

13. Simplified 4.9%

    \[\leadsto \color{blue}{1 - w} \]

14. Taylor expanded in w around 0 4.5%

    \[\leadsto \color{blue}{1} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))