exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 13.9s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

5. Final simplification 99.3%

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

Alternative 2: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 410000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 410000.0))) (exp (- w)) (* l (exp w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 410000.0)) {
		tmp = exp(-w);
	} else {
		tmp = l * exp(w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.7d0)) .or. (.not. (w <= 410000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l * exp(w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 410000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l * Math.exp(w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.7) or not (w <= 410000.0):
		tmp = math.exp(-w)
	else:
		tmp = l * math.exp(w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.7) || !(w <= 410000.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l * exp(w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.7) || ~((w <= 410000.0)))
		tmp = exp(-w);
	else
		tmp = l * exp(w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.7], N[Not[LessEqual[w, 410000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l * N[Exp[w], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.69999999999999996 or 4.1e5 < 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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-div 100.0%

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

      3. log-pow 100.0%

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

      4. add-log-exp 100.0%

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

      5. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 99.5%

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

      1. neg-mul-1 99.5%

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

   9. Simplified 99.5%

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

   if -0.69999999999999996 < w < 4.1e5

   1. Initial program 98.7%

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

      1. exp-neg 98.7%

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

      2. associate-*l/ 98.7%

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

      3. *-lft-identity 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in w around 0 96.2%

      \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 91.9%

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

      2. expm1-udef 52.6%

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

      3. div-inv 52.6%

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

      4. exp-neg 52.6%

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

      5. add-sqr-sqrt 23.8%

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

      6. sqrt-unprod 52.6%

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

      7. sqr-neg 52.6%

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

      8. sqrt-unprod 28.8%

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

      9. add-sqr-sqrt 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. expm1-def 91.9%

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

      2. expm1-log1p 96.2%

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

      3. *-commutative 96.2%

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

   9. Simplified 96.2%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 410000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.66 \lor \neg \left(w \leq 350000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot w\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.66) (not (<= w 350000.0))) (exp (- w)) (+ l (* l w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.66) || !(w <= 350000.0)) {
		tmp = exp(-w);
	} else {
		tmp = l + (l * w);
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: tmp
    if ((w <= (-0.66d0)) .or. (.not. (w <= 350000.0d0))) then
        tmp = exp(-w)
    else
        tmp = l + (l * w)
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double tmp;
	if ((w <= -0.66) || !(w <= 350000.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l + (l * w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.66) or not (w <= 350000.0):
		tmp = math.exp(-w)
	else:
		tmp = l + (l * w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.66) || !(w <= 350000.0))
		tmp = exp(Float64(-w));
	else
		tmp = Float64(l + Float64(l * w));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if ((w <= -0.66) || ~((w <= 350000.0)))
		tmp = exp(-w);
	else
		tmp = l + (l * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.66], N[Not[LessEqual[w, 350000.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l + N[(l * w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.660000000000000031 or 3.5e5 < 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. associate-*l/ 100.0%

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

      3. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

      1. add-exp-log 100.0%

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

      2. log-div 100.0%

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

      3. log-pow 100.0%

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

      4. add-log-exp 100.0%

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

      5. fma-neg 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in w around inf 99.5%

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

      1. neg-mul-1 99.5%

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

   9. Simplified 99.5%

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

   if -0.660000000000000031 < w < 3.5e5

   1. Initial program 98.7%

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

      1. exp-neg 98.7%

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

      2. associate-*l/ 98.7%

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

      3. *-lft-identity 98.7%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in w around 0 96.2%

      \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 91.9%

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

      2. expm1-udef 52.6%

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

      3. div-inv 52.6%

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

      4. exp-neg 52.6%

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

      5. add-sqr-sqrt 23.8%

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

      6. sqrt-unprod 52.6%

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

      7. sqr-neg 52.6%

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

      8. sqrt-unprod 28.8%

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

      9. add-sqr-sqrt 52.6%

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

   7. Applied egg-rr 52.6%

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

      1. expm1-def 91.9%

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

      2. expm1-log1p 96.2%

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

      3. *-commutative 96.2%

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

   9. Simplified 96.2%

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

   10. Taylor expanded in w around 0 95.7%

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

       1. *-commutative 95.7%

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

   12. Simplified 95.7%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.66 \lor \neg \left(w \leq 350000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot w\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% 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 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in w around 0 97.5%

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

6. Final simplification 97.5%

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

Alternative 5: 63.3% 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 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in w around 0 97.5%

   \[\leadsto \frac{\color{blue}{\ell}}{e^{w}} \]
6. Step-by-step derivation

   1. expm1-log1p-u 95.1%

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

   2. expm1-udef 73.3%

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

   3. div-inv 73.3%

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

   4. exp-neg 73.3%

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

   5. add-sqr-sqrt 40.6%

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

   6. sqrt-unprod 56.8%

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

   7. sqr-neg 56.8%

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

   8. sqrt-unprod 16.2%

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

   9. add-sqr-sqrt 29.9%

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

7. Applied egg-rr 29.9%

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

   1. expm1-def 51.7%

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

   2. expm1-log1p 54.1%

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

   3. *-commutative 54.1%

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

9. Simplified 54.1%

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

10. Taylor expanded in w around 0 53.8%

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

    1. *-commutative 53.8%

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

12. Simplified 53.8%

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

    1. add-sqr-sqrt 28.4%

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

    2. sqrt-unprod 67.8%

       \[\leadsto \ell + \color{blue}{\sqrt{w \cdot w}} \cdot \ell \]

    3. sqr-neg 67.8%

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

    4. sqrt-unprod 32.0%

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

    5. add-sqr-sqrt 60.4%

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

    6. cancel-sign-sub-inv 60.4%

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

14. Applied egg-rr 60.4%

    \[\leadsto \color{blue}{\ell - w \cdot \ell} \]
15. Step-by-step derivation

    1. cancel-sign-sub-inv 60.4%

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

    2. *-lft-identity 60.4%

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

    3. distribute-rgt-in 60.4%

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

    4. sub-neg 60.4%

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

16. Simplified 60.4%

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

17. Final simplification 60.4%

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

Alternative 6: 63.3% 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 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

5. Taylor expanded in w around 0 97.5%

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

6. Taylor expanded in w around 0 60.4%

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

   1. mul-1-neg 60.4%

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

   2. unsub-neg 60.4%

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

   3. *-commutative 60.4%

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

8. Simplified 60.4%

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

9. Final simplification 60.4%

   \[\leadsto \ell - \ell \cdot w \]
10. Add Preprocessing

Alternative 7: 56.7% accurate, 305.0× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 l)

C

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

Fortran

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

Java

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

Python

def code(w, l):
	return l

Julia

function code(w, l)
	return l
end

MATLAB

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

Wolfram

code[w_, l_] := l

Derivation

1. Initial program 99.3%

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

   1. exp-neg 99.3%

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

   2. associate-*l/ 99.3%

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

   3. *-lft-identity 99.3%

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

3. Simplified 99.3%

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

   1. clear-num 99.1%

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

   2. inv-pow 99.1%

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

   3. div-inv 99.1%

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

   4. unpow-prod-down 99.1%

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

   5. inv-pow 99.1%

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

   6. rec-exp 99.1%

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

   7. pow-flip 99.1%

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

6. Applied egg-rr 99.1%

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

7. Taylor expanded in l around inf 99.2%

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

8. Taylor expanded in w around 0 55.2%

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

9. Final simplification 55.2%

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

Reproduce

herbie shell --seed 2024021 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))