exp-w (used to crash)

Percentage Accurate: 99.5% → 99.7%

Time: 18.8s

Alternatives: 11

Speedup: 1.0×

• Could not converge on a ground truth

  1. w = -1.464133099440181e+278
  2. l = -1.5086660943530196e+67

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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\ell \cdot {\ell}^{\left(\mathsf{expm1}\left(w\right)\right)}}{e^{w}} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (/ (* l (pow l (expm1 w))) (exp w)))

C

double code(double w, double l) {
	return (l * pow(l, expm1(w))) / exp(w);
}

Java

public static double code(double w, double l) {
	return (l * Math.pow(l, Math.expm1(w))) / Math.exp(w);
}

Python

def code(w, l):
	return (l * math.pow(l, math.expm1(w))) / math.exp(w)

Julia

function code(w, l)
	return Float64(Float64(l * (l ^ expm1(w))) / exp(w))
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. exp-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. *-lft-identity 99.9%

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

   5. remove-double-neg 99.9%

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

3. Simplified 99.9%

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

   1. expm1-log1p-u 97.5%

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

6. Applied egg-rr 97.5%

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

   1. expm1-log1p-u 99.9%

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

   2. expm1-log1p-u 99.8%

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

   3. log1p-define 99.8%

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

   4. expm1-define 99.8%

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

   5. add-exp-log 99.8%

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

   6. associate--l+ 99.8%

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

   7. unpow-prod-up 99.8%

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

   8. pow1 99.8%

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

   9. expm1-define 100.0%

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

8. Applied egg-rr 100.0%

   \[\leadsto \frac{\color{blue}{\ell \cdot {\ell}^{\left(\mathsf{expm1}\left(w\right)\right)}}}{e^{w}} \]
9. Add Preprocessing

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. exp-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. *-lft-identity 99.9%

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

   5. remove-double-neg 99.9%

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

3. Simplified 99.9%

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

Alternative 3: 97.8% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. exp-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. *-lft-identity 99.9%

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

   5. remove-double-neg 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in w around 0 98.3%

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

6. Taylor expanded in l around 0 98.3%

   \[\leadsto \color{blue}{\frac{\ell}{e^{w}}} \]
7. Add Preprocessing

Alternative 4: 90.9% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2.4:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 2.4)
   (+ l (* l (* w (+ (* w (+ 0.5 (* w -0.16666666666666666))) -1.0))))
   (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 2.4) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -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 <= 2.4d0) then
        tmp = l + (l * (w * ((w * (0.5d0 + (w * (-0.16666666666666666d0)))) + (-1.0d0))))
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 2.4) {
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 2.4:
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)))
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 2.4)
		tmp = Float64(l + Float64(l * Float64(w * Float64(Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666))) + -1.0))));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 2.4)
		tmp = l + (l * (w * ((w * (0.5 + (w * -0.16666666666666666))) + -1.0)));
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 2.4], N[(l + N[(l * N[(w * N[(N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 2.39999999999999991

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 97.9%

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

   6. Taylor expanded in w around 0 86.2%

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

   7. Taylor expanded in l around 0 88.0%

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

   if 2.39999999999999991 < 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%

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

   6. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   8. Simplified 3.5%

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

   9. Taylor expanded in w around inf 3.5%

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

       1. mul-1-neg 3.5%

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

       2. *-commutative 3.5%

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

       3. distribute-rgt-neg-in 3.5%

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

   11. Simplified 3.5%

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

       1. expm1-log1p-u 3.5%

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

       2. expm1-undefine 89.1%

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

   13. Applied egg-rr 89.1%

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

       1. sub-neg 89.1%

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

       2. log1p-undefine 89.1%

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

       3. rem-exp-log 89.1%

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

       4. unsub-neg 89.1%

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

       5. metadata-eval 89.1%

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

   15. Simplified 89.1%

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

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 2.4:\\ \;\;\;\;\ell + \ell \cdot \left(w \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right) + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.4% accurate, 16.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 0.11:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot \left(w \cdot -0.16666666666666666\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 0.11)
   (+ l (* w (- (* w (* l (* w -0.16666666666666666))) l)))
   (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 0.11) {
		tmp = l + (w * ((w * (l * (w * -0.16666666666666666))) - l));
	} else {
		tmp = w * ((1.0 - l) + -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 <= 0.11d0) then
        tmp = l + (w * ((w * (l * (w * (-0.16666666666666666d0)))) - l))
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 0.11) {
		tmp = l + (w * ((w * (l * (w * -0.16666666666666666))) - l));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 0.11:
		tmp = l + (w * ((w * (l * (w * -0.16666666666666666))) - l))
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 0.11)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(l * Float64(w * -0.16666666666666666))) - l)));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 0.11)
		tmp = l + (w * ((w * (l * (w * -0.16666666666666666))) - l));
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 0.11], N[(l + N[(w * N[(N[(w * N[(l * N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 0.110000000000000001

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 98.3%

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

   6. Taylor expanded in w around 0 86.6%

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

   7. Taylor expanded in w around inf 86.6%

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

   9. Simplified 86.6%

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

   if 0.110000000000000001 < 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 98.0%

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

   6. Taylor expanded in w around 0 3.4%

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

      1. mul-1-neg 3.4%

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

      2. unsub-neg 3.4%

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

   8. Simplified 3.4%

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

   9. Taylor expanded in w around inf 3.4%

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

       1. mul-1-neg 3.4%

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

       2. *-commutative 3.4%

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

       3. distribute-rgt-neg-in 3.4%

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

   11. Simplified 3.4%

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

       1. expm1-log1p-u 3.4%

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

       2. expm1-undefine 87.2%

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

   13. Applied egg-rr 87.2%

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

       1. sub-neg 87.2%

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

       2. log1p-undefine 87.2%

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

       3. rem-exp-log 87.2%

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

       4. unsub-neg 87.2%

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

       5. metadata-eval 87.2%

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

   15. Simplified 87.2%

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

Alternative 6: 84.0% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 2.4:\\ \;\;\;\;\ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 2.4) (+ l (* w (- (* w (* l 0.5)) l))) (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 2.4) {
		tmp = l + (w * ((w * (l * 0.5)) - l));
	} else {
		tmp = w * ((1.0 - l) + -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 <= 2.4d0) then
        tmp = l + (w * ((w * (l * 0.5d0)) - l))
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 2.4) {
		tmp = l + (w * ((w * (l * 0.5)) - l));
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 2.4:
		tmp = l + (w * ((w * (l * 0.5)) - l))
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 2.4)
		tmp = Float64(l + Float64(w * Float64(Float64(w * Float64(l * 0.5)) - l)));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 2.4)
		tmp = l + (w * ((w * (l * 0.5)) - l));
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 2.4], N[(l + N[(w * N[(N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 2.39999999999999991

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 97.9%

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

   6. Taylor expanded in w around 0 86.2%

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

   7. Taylor expanded in w around 0 81.3%

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

      1. neg-mul-1 81.3%

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

      2. +-commutative 81.3%

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

      3. sub-neg 81.3%

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

      4. mul-1-neg 81.3%

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

      5. distribute-rgt-out 81.3%

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

      6. metadata-eval 81.3%

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

      7. distribute-rgt-neg-out 81.3%

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

      8. distribute-rgt-neg-in 81.3%

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

      9. metadata-eval 81.3%

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

   9. Simplified 81.3%

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

   if 2.39999999999999991 < 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%

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

   6. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   8. Simplified 3.5%

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

   9. Taylor expanded in w around inf 3.5%

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

       1. mul-1-neg 3.5%

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

       2. *-commutative 3.5%

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

       3. distribute-rgt-neg-in 3.5%

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

   11. Simplified 3.5%

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

       1. expm1-log1p-u 3.5%

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

       2. expm1-undefine 89.1%

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

   13. Applied egg-rr 89.1%

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

       1. sub-neg 89.1%

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

       2. log1p-undefine 89.1%

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

       3. rem-exp-log 89.1%

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

       4. unsub-neg 89.1%

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

       5. metadata-eval 89.1%

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

   15. Simplified 89.1%

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

Alternative 7: 76.7% accurate, 25.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 30500000000:\\ \;\;\;\;\ell - \ell \cdot w\\ \mathbf{else}:\\ \;\;\;\;w \cdot \left(\left(1 - \ell\right) + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 30500000000.0) (- l (* l w)) (* w (+ (- 1.0 l) -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 30500000000.0) {
		tmp = l - (l * w);
	} else {
		tmp = w * ((1.0 - l) + -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 <= 30500000000.0d0) then
        tmp = l - (l * w)
    else
        tmp = w * ((1.0d0 - l) + (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 30500000000.0) {
		tmp = l - (l * w);
	} else {
		tmp = w * ((1.0 - l) + -1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 30500000000.0:
		tmp = l - (l * w)
	else:
		tmp = w * ((1.0 - l) + -1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 30500000000.0)
		tmp = Float64(l - Float64(l * w));
	else
		tmp = Float64(w * Float64(Float64(1.0 - l) + -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 30500000000.0)
		tmp = l - (l * w);
	else
		tmp = w * ((1.0 - l) + -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 30500000000.0], N[(l - N[(l * w), $MachinePrecision]), $MachinePrecision], N[(w * N[(N[(1.0 - l), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3.05e10

   1. Initial program 99.9%

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

      1. exp-neg 99.9%

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

      2. remove-double-neg 99.9%

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

      3. associate-*l/ 99.9%

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

      4. *-lft-identity 99.9%

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

      5. remove-double-neg 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in w around 0 97.9%

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

   6. Taylor expanded in w around 0 72.7%

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

      1. mul-1-neg 72.7%

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

      2. unsub-neg 72.7%

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

   8. Simplified 72.7%

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

   if 3.05e10 < 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%

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

   6. Taylor expanded in w around 0 3.5%

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

      1. mul-1-neg 3.5%

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

      2. unsub-neg 3.5%

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

   8. Simplified 3.5%

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

   9. Taylor expanded in w around inf 3.5%

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

       1. mul-1-neg 3.5%

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

       2. *-commutative 3.5%

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

       3. distribute-rgt-neg-in 3.5%

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

   11. Simplified 3.5%

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

       1. expm1-log1p-u 3.5%

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

       2. expm1-undefine 91.1%

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

   13. Applied egg-rr 91.1%

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

       1. sub-neg 91.1%

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

       2. log1p-undefine 91.1%

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

       3. rem-exp-log 91.1%

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

       4. unsub-neg 91.1%

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

       5. metadata-eval 91.1%

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

   15. Simplified 91.1%

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

Alternative 8: 71.0% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.6:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{w + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.6) (* w (- l)) (/ l (+ w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.6) {
		tmp = w * -l;
	} 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 <= (-0.6d0)) then
        tmp = w * -l
    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 <= -0.6) {
		tmp = w * -l;
	} else {
		tmp = l / (w + 1.0);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.6:
		tmp = w * -l
	else:
		tmp = l / (w + 1.0)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.6)
		tmp = Float64(w * Float64(-l));
	else
		tmp = Float64(l / Float64(w + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.6)
		tmp = w * -l;
	else
		tmp = l / (w + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.6], N[(w * (-l)), $MachinePrecision], N[(l / N[(w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.599999999999999978

   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 98.8%

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

   6. Taylor expanded in w around 0 34.1%

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

      1. mul-1-neg 34.1%

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

      2. unsub-neg 34.1%

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

   8. Simplified 34.1%

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

   9. Taylor expanded in w around inf 34.1%

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

       1. mul-1-neg 34.1%

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

       2. *-commutative 34.1%

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

       3. distribute-rgt-neg-in 34.1%

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

   11. Simplified 34.1%

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

   if -0.599999999999999978 < w

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 98.0%

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

   6. Taylor expanded in l around 0 98.0%

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

   7. Taylor expanded in w around 0 84.1%

      \[\leadsto \frac{\ell}{\color{blue}{1 + w}} \]
   8. Step-by-step derivation

      1. +-commutative 84.1%

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

   9. Simplified 84.1%

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

Alternative 9: 64.5% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -150:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -150.0) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -150.0) {
		tmp = w * -l;
	} 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 <= (-150.0d0)) then
        tmp = w * -l
    else
        tmp = l
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= -150.0) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -150.0:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -150.0)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -150.0)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -150.0], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -150

   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%

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

   6. Taylor expanded in w around 0 34.5%

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

      1. mul-1-neg 34.5%

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

      2. unsub-neg 34.5%

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

   8. Simplified 34.5%

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

   9. Taylor expanded in w around inf 34.5%

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

       1. mul-1-neg 34.5%

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

       2. *-commutative 34.5%

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

       3. distribute-rgt-neg-in 34.5%

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

   11. Simplified 34.5%

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

   if -150 < w

   1. Initial program 99.8%

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

      1. exp-neg 99.8%

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

      2. remove-double-neg 99.8%

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

      3. associate-*l/ 99.8%

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

      4. *-lft-identity 99.8%

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

      5. remove-double-neg 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in w around 0 73.3%

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

Alternative 10: 64.2% accurate, 61.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. exp-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. *-lft-identity 99.9%

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

   5. remove-double-neg 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in w around 0 98.3%

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

6. Taylor expanded in w around 0 60.8%

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

   1. mul-1-neg 60.8%

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

   2. unsub-neg 60.8%

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

8. Simplified 60.8%

   \[\leadsto \color{blue}{\ell - \ell \cdot w} \]
9. Add Preprocessing

Alternative 11: 57.5% 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 99.9%

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

   1. exp-neg 99.9%

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

   2. remove-double-neg 99.9%

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

   3. associate-*l/ 99.9%

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

   4. *-lft-identity 99.9%

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

   5. remove-double-neg 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in w around 0 51.7%

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

Reproduce

herbie shell --seed 2024185 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))