exp-w (used to crash)

Percentage Accurate: 99.5% → 99.5%

Time: 33.4s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. w = -2.00000000000000007e154
  2. l = 0.0

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 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]

Derivation

1. Initial program 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
2. Add Preprocessing

3. Final simplification 99.6%

   \[\leadsto e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
4. 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 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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. Final simplification 99.6%

   \[\leadsto \frac{{\ell}^{\left(e^{w}\right)}}{e^{w}} \]
6. 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 99.6%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 98.0%

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

6. Final simplification 98.0%

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

Alternative 4: 83.6% accurate, 7.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell \cdot 0.5 - \ell\\ \mathbf{if}\;w \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(t\_0 + w \cdot \left(w \cdot \left(\ell \cdot -0.041666666666666664\right) - \left(t\_0 - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (- (* l 0.5) l)))
   (if (<= w 6.5e-18)
     (-
      l
      (*
       w
       (+
        l
        (*
         w
         (+
          t_0
          (*
           w
           (-
            (* w (* l -0.041666666666666664))
            (- t_0 (+ (* l -0.5) (* l 0.16666666666666666))))))))))
     (* w (/ l w)))))

C

double code(double w, double l) {
	double t_0 = (l * 0.5) - l;
	double tmp;
	if (w <= 6.5e-18) {
		tmp = l - (w * (l + (w * (t_0 + (w * ((w * (l * -0.041666666666666664)) - (t_0 - ((l * -0.5) + (l * 0.16666666666666666)))))))));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (l * 0.5d0) - l
    if (w <= 6.5d-18) then
        tmp = l - (w * (l + (w * (t_0 + (w * ((w * (l * (-0.041666666666666664d0))) - (t_0 - ((l * (-0.5d0)) + (l * 0.16666666666666666d0)))))))))
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = (l * 0.5) - l;
	double tmp;
	if (w <= 6.5e-18) {
		tmp = l - (w * (l + (w * (t_0 + (w * ((w * (l * -0.041666666666666664)) - (t_0 - ((l * -0.5) + (l * 0.16666666666666666)))))))));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = (l * 0.5) - l
	tmp = 0
	if w <= 6.5e-18:
		tmp = l - (w * (l + (w * (t_0 + (w * ((w * (l * -0.041666666666666664)) - (t_0 - ((l * -0.5) + (l * 0.16666666666666666)))))))))
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	t_0 = Float64(Float64(l * 0.5) - l)
	tmp = 0.0
	if (w <= 6.5e-18)
		tmp = Float64(l - Float64(w * Float64(l + Float64(w * Float64(t_0 + Float64(w * Float64(Float64(w * Float64(l * -0.041666666666666664)) - Float64(t_0 - Float64(Float64(l * -0.5) + Float64(l * 0.16666666666666666))))))))));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = (l * 0.5) - l;
	tmp = 0.0;
	if (w <= 6.5e-18)
		tmp = l - (w * (l + (w * (t_0 + (w * ((w * (l * -0.041666666666666664)) - (t_0 - ((l * -0.5) + (l * 0.16666666666666666)))))))));
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(l * 0.5), $MachinePrecision] - l), $MachinePrecision]}, If[LessEqual[w, 6.5e-18], N[(l - N[(w * N[(l + N[(w * N[(t$95$0 + N[(w * N[(N[(w * N[(l * -0.041666666666666664), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 - N[(N[(l * -0.5), $MachinePrecision] + N[(l * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 6.50000000000000008e-18

   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 98.8%

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

   6. Taylor expanded in w around 0 87.8%

      \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \left(-1 \cdot \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right) + \left(-0.5 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.16666666666666666 \cdot \ell + 0.041666666666666664 \cdot \ell\right)\right)\right)\right) - \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right)} \]

   7. Taylor expanded in l around 0 87.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(-0.041666666666666664 \cdot \ell\right)}\right) - \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]
   8. Step-by-step derivation

      1. *-commutative 87.8%

         \[\leadsto \ell + w \cdot \left(w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot -0.041666666666666664\right)}\right) - \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   9. Simplified 87.8%

      \[\leadsto \ell + w \cdot \left(w \cdot \left(w \cdot \left(-1 \cdot \left(w \cdot \color{blue}{\left(\ell \cdot -0.041666666666666664\right)}\right) - \left(-1 \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right) + \left(-0.5 \cdot \ell + 0.16666666666666666 \cdot \ell\right)\right)\right) - \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right) - \ell\right) \]

   if 6.50000000000000008e-18 < w

   1. Initial program 98.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 13.8%

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

      1. neg-mul-1 13.8%

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

      2. +-commutative 13.8%

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

      3. sub-neg 13.8%

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

   5. Simplified 13.8%

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

   6. Taylor expanded in w around inf 13.8%

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

   7. Taylor expanded in w around 0 46.3%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\ell - w \cdot \left(\ell + w \cdot \left(\left(\ell \cdot 0.5 - \ell\right) + w \cdot \left(w \cdot \left(\ell \cdot -0.041666666666666664\right) - \left(\left(\ell \cdot 0.5 - \ell\right) - \left(\ell \cdot -0.5 + \ell \cdot 0.16666666666666666\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.7% accurate, 15.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 3e-17)
   (+ l (* w (- (* l (* w (+ 0.5 (* w -0.16666666666666666)))) l)))
   (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 3e-17) {
		tmp = l + (w * ((l * (w * (0.5 + (w * -0.16666666666666666)))) - l));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 3d-17) then
        tmp = l + (w * ((l * (w * (0.5d0 + (w * (-0.16666666666666666d0))))) - l))
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 3e-17) {
		tmp = l + (w * ((l * (w * (0.5 + (w * -0.16666666666666666)))) - l));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 3e-17:
		tmp = l + (w * ((l * (w * (0.5 + (w * -0.16666666666666666)))) - l))
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 3e-17)
		tmp = Float64(l + Float64(w * Float64(Float64(l * Float64(w * Float64(0.5 + Float64(w * -0.16666666666666666)))) - l)));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 3e-17)
		tmp = l + (w * ((l * (w * (0.5 + (w * -0.16666666666666666)))) - l));
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 3e-17], N[(l + N[(w * N[(N[(l * N[(w * N[(0.5 + N[(w * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 3.00000000000000006e-17

   1. Initial program 99.7%

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

      1. exp-neg 99.7%

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

      2. remove-double-neg 99.7%

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

      3. associate-*l/ 99.7%

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

      4. *-lft-identity 99.7%

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

      5. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in w around 0 98.8%

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

   6. Taylor expanded in w around 0 85.6%

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

   8. Simplified 85.6%

      \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(\left(-w\right) \cdot \mathsf{fma}\left(-1, \ell \cdot -0.5, \ell \cdot -0.3333333333333333\right) - \ell \cdot -0.5\right) - \ell\right)} \]

   9. Taylor expanded in l around 0 86.9%

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

       1. *-commutative 86.9%

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

   11. Simplified 86.9%

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

   if 3.00000000000000006e-17 < w

   1. Initial program 99.0%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 11.6%

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

      1. neg-mul-1 11.6%

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

      2. +-commutative 11.6%

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

      3. sub-neg 11.6%

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

   5. Simplified 11.6%

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

   6. Taylor expanded in w around inf 11.6%

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

   7. Taylor expanded in w around 0 45.0%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 3 \cdot 10^{-17}:\\ \;\;\;\;\ell + w \cdot \left(\ell \cdot \left(w \cdot \left(0.5 + w \cdot -0.16666666666666666\right)\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.1% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 6 \cdot 10^{-18}:\\ \;\;\;\;\ell + w \cdot \left(0.5 \cdot \left(w \cdot \ell\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 6e-18) (+ l (* w (- (* 0.5 (* w l)) l))) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 6e-18) {
		tmp = l + (w * ((0.5 * (w * l)) - l));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 6d-18) then
        tmp = l + (w * ((0.5d0 * (w * l)) - l))
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 6e-18) {
		tmp = l + (w * ((0.5 * (w * l)) - l));
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 6e-18:
		tmp = l + (w * ((0.5 * (w * l)) - l))
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 6e-18)
		tmp = Float64(l + Float64(w * Float64(Float64(0.5 * Float64(w * l)) - l)));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 6e-18)
		tmp = l + (w * ((0.5 * (w * l)) - l));
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 6e-18], N[(l + N[(w * N[(N[(0.5 * N[(w * l), $MachinePrecision]), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 5.99999999999999966e-18

   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 98.8%

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

   6. Taylor expanded in w around 0 85.6%

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

   8. Simplified 85.6%

      \[\leadsto \color{blue}{\ell + w \cdot \left(w \cdot \left(\left(-w\right) \cdot \mathsf{fma}\left(-1, \ell \cdot -0.5, \ell \cdot -0.3333333333333333\right) - \ell \cdot -0.5\right) - \ell\right)} \]

   9. Taylor expanded in w around 0 79.8%

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

       1. *-commutative 79.8%

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

   11. Simplified 79.8%

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

   if 5.99999999999999966e-18 < w

   1. Initial program 98.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 13.8%

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

      1. neg-mul-1 13.8%

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

      2. +-commutative 13.8%

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

      3. sub-neg 13.8%

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

   5. Simplified 13.8%

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

   6. Taylor expanded in w around inf 13.8%

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

   7. Taylor expanded in w around 0 46.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 6 \cdot 10^{-18}:\\ \;\;\;\;\ell + w \cdot \left(0.5 \cdot \left(w \cdot \ell\right) - \ell\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.1% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 6.7 \cdot 10^{-18}:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 6.7e-18) (* l (- 1.0 w)) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 6.7e-18) {
		tmp = l * (1.0 - w);
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 6.7d-18) then
        tmp = l * (1.0d0 - w)
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 6.7e-18) {
		tmp = l * (1.0 - w);
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 6.7e-18:
		tmp = l * (1.0 - w)
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 6.7e-18)
		tmp = Float64(l * Float64(1.0 - w));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 6.7e-18)
		tmp = l * (1.0 - w);
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 6.7e-18], N[(l * N[(1.0 - w), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 6.6999999999999998e-18

   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 98.8%

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

   6. Taylor expanded in w around 0 70.9%

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

      1. mul-1-neg 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. *-commutative 70.9%

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

   8. Simplified 70.9%

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

   9. Taylor expanded in l around 0 70.9%

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

       1. neg-mul-1 70.9%

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

       2. unsub-neg 70.9%

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

   11. Simplified 70.9%

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

   if 6.6999999999999998e-18 < w

   1. Initial program 98.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 13.8%

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

      1. neg-mul-1 13.8%

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

      2. +-commutative 13.8%

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

      3. sub-neg 13.8%

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

   5. Simplified 13.8%

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

   6. Taylor expanded in w around inf 13.8%

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

   7. Taylor expanded in w around 0 46.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 6.7 \cdot 10^{-18}:\\ \;\;\;\;\ell \cdot \left(1 - w\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.1% accurate, 30.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 6.5e-18) (- l (* w l)) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 6.5e-18) {
		tmp = l - (w * l);
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if (w <= 6.5d-18) then
        tmp = l - (w * l)
    else
        tmp = w * (l / w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if (w <= 6.5e-18) {
		tmp = l - (w * l);
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 6.5e-18:
		tmp = l - (w * l)
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 6.5e-18)
		tmp = Float64(l - Float64(w * l));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= 6.5e-18)
		tmp = l - (w * l);
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 6.5e-18], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 6.50000000000000008e-18

   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 98.8%

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

   6. Taylor expanded in w around 0 70.9%

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

      1. mul-1-neg 70.9%

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

      2. distribute-lft-neg-out 70.9%

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

      3. *-commutative 70.9%

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

   8. Simplified 70.9%

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

   if 6.50000000000000008e-18 < w

   1. Initial program 98.9%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in w around 0 13.8%

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

      1. neg-mul-1 13.8%

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

      2. +-commutative 13.8%

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

      3. sub-neg 13.8%

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

   5. Simplified 13.8%

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

   6. Taylor expanded in w around inf 13.8%

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

   7. Taylor expanded in w around 0 46.3%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 6.5 \cdot 10^{-18}:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.7% accurate, 33.8× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 w around 0 100.0%

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

   6. Taylor expanded in w around 0 24.6%

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

      1. mul-1-neg 24.6%

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

      2. distribute-lft-neg-out 24.6%

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

      3. *-commutative 24.6%

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

   8. Simplified 24.6%

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

   9. Taylor expanded in w around inf 24.6%

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

       1. *-commutative 24.6%

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

       2. neg-mul-1 24.6%

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

       3. distribute-rgt-neg-in 24.6%

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

   11. Simplified 24.6%

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

   if -1 < 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 97.1%

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

   6. Taylor expanded in w around 0 79.0%

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

4. Final simplification 62.0%

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

Alternative 10: 63.4% accurate, 61.0× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 - w\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (- 1.0 w)))

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l * (1.0 - w)

Julia

function code(w, l)
	return Float64(l * Float64(1.0 - w))
end

MATLAB

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

Wolfram

code[w_, l_] := N[(l * N[(1.0 - 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. Taylor expanded in w around 0 98.0%

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

6. Taylor expanded in w around 0 61.7%

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

   1. mul-1-neg 61.7%

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

   2. distribute-lft-neg-out 61.7%

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

   3. *-commutative 61.7%

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

8. Simplified 61.7%

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

9. Taylor expanded in l around 0 61.7%

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

    1. neg-mul-1 61.7%

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

    2. unsub-neg 61.7%

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

11. Simplified 61.7%

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

12. Final simplification 61.7%

    \[\leadsto \ell \cdot \left(1 - w\right) \]
13. Add Preprocessing

Alternative 11: 56.8% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 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 98.0%

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

6. Taylor expanded in w around 0 55.3%

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

7. Final simplification 55.3%

   \[\leadsto \ell \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))