exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 29.1s

Alternatives: 10

Speedup: N/A×

• could not determine a ground truth

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% 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 99.7%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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. Final simplification 99.7%

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

Alternative 2: 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 99.7%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 97.9%

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

6. Final simplification 97.9%

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

Alternative 3: 74.4% accurate, 17.9× speedup

Math

\[\begin{array}{l} \\ \ell + w \cdot \left(w \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot w\right) + \ell \cdot 0.5\right) - \ell\right) \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (+ l (* w (- (* w (+ (* -0.16666666666666666 (* l w)) (* l 0.5))) l))))

C

double code(double w, double l) {
	return l + (w * ((w * ((-0.16666666666666666 * (l * w)) + (l * 0.5))) - l));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (w * ((w * (((-0.16666666666666666d0) * (l * w)) + (l * 0.5d0))) - l))
end function

Java

public static double code(double w, double l) {
	return l + (w * ((w * ((-0.16666666666666666 * (l * w)) + (l * 0.5))) - l));
}

Python

def code(w, l):
	return l + (w * ((w * ((-0.16666666666666666 * (l * w)) + (l * 0.5))) - l))

Julia

function code(w, l)
	return Float64(l + Float64(w * Float64(Float64(w * Float64(Float64(-0.16666666666666666 * Float64(l * w)) + Float64(l * 0.5))) - l)))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (w * ((w * ((-0.16666666666666666 * (l * w)) + (l * 0.5))) - l));
end

Wolfram

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

Derivation

1. Initial program 99.7%

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

3. Taylor expanded in w around 0 97.9%

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

4. Taylor expanded in w around 0 73.4%

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

5. Final simplification 73.4%

   \[\leadsto \ell + w \cdot \left(w \cdot \left(-0.16666666666666666 \cdot \left(\ell \cdot w\right) + \ell \cdot 0.5\right) - \ell\right) \]
6. Add Preprocessing

Alternative 4: 69.9% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \ell + w \cdot \left(\ell + \left(\ell \cdot w\right) \cdot 0.5\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (+ l (* w (+ l (* (* l w) 0.5)))))

C

double code(double w, double l) {
	return l + (w * (l + ((l * w) * 0.5)));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (w * (l + ((l * w) * 0.5d0)))
end function

Java

public static double code(double w, double l) {
	return l + (w * (l + ((l * w) * 0.5)));
}

Python

def code(w, l):
	return l + (w * (l + ((l * w) * 0.5)))

Julia

function code(w, l)
	return Float64(l + Float64(w * Float64(l + Float64(Float64(l * w) * 0.5))))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (w * (l + ((l * w) * 0.5)));
end

Wolfram

code[w_, l_] := N[(l + N[(w * N[(l + N[(N[(l * w), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 97.9%

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

   1. clear-num 97.8%

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

   2. associate-/r/ 97.9%

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

   3. exp-neg 97.9%

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

   4. add-sqr-sqrt 59.5%

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

   5. sqrt-unprod 83.3%

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

   6. sqr-neg 83.3%

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

   7. sqrt-unprod 23.8%

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

   8. add-sqr-sqrt 53.3%

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

7. Applied egg-rr 53.3%

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

8. Taylor expanded in w around 0 67.2%

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

   1. *-commutative 67.2%

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

10. Simplified 67.2%

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

11. Final simplification 67.2%

    \[\leadsto \ell + w \cdot \left(\ell + \left(\ell \cdot w\right) \cdot 0.5\right) \]
12. Add Preprocessing

Alternative 5: 73.0% accurate, 27.7× speedup

Math

\[\begin{array}{l} \\ \ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (* l (+ 1.0 (* w (+ -1.0 (* w 0.5))))))

C

double code(double w, double l) {
	return l * (1.0 + (w * (-1.0 + (w * 0.5))));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l * (1.0d0 + (w * ((-1.0d0) + (w * 0.5d0))))
end function

Java

public static double code(double w, double l) {
	return l * (1.0 + (w * (-1.0 + (w * 0.5))));
}

Python

def code(w, l):
	return l * (1.0 + (w * (-1.0 + (w * 0.5))))

Julia

function code(w, l)
	return Float64(l * Float64(1.0 + Float64(w * Float64(-1.0 + Float64(w * 0.5)))))
end

MATLAB

function tmp = code(w, l)
	tmp = l * (1.0 + (w * (-1.0 + (w * 0.5))));
end

Wolfram

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

Derivation

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 97.9%

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

6. Taylor expanded in w around 0 67.2%

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

   1. mul-1-neg 67.2%

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

   2. distribute-rgt-out 67.2%

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

   3. metadata-eval 67.2%

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

8. Simplified 67.2%

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

9. Taylor expanded in l around 0 72.3%

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

    1. metadata-eval 72.3%

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

    2. cancel-sign-sub-inv 72.3%

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

    3. associate-*r* 72.3%

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

    4. neg-mul-1 72.3%

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

    5. *-commutative 72.3%

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

11. Simplified 72.3%

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

12. Final simplification 72.3%

    \[\leadsto \ell \cdot \left(1 + w \cdot \left(-1 + w \cdot 0.5\right)\right) \]
13. Add Preprocessing

Alternative 6: 63.0% accurate, 33.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.4:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.4) (* w (- l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.4) {
		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 <= (-0.4d0)) 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 <= -0.4) {
		tmp = w * -l;
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.4:
		tmp = w * -l
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.4)
		tmp = Float64(w * Float64(-l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.4)
		tmp = w * -l;
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.4], N[(w * (-l)), $MachinePrecision], l]

Derivation

1. Split input into 2 regimes

2. if w < -0.40000000000000002

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0 98.8%

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

   4. Taylor expanded in w around 0 23.8%

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

      1. mul-1-neg 23.8%

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

      2. unsub-neg 23.8%

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

      3. *-rgt-identity 23.8%

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

      4. distribute-lft-out-- 23.8%

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

   6. Simplified 23.8%

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

   7. Taylor expanded in w around inf 23.8%

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

      1. mul-1-neg 23.8%

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

      2. distribute-rgt-neg-out 23.8%

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

   9. Simplified 23.8%

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

   if -0.40000000000000002 < w

   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 97.5%

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

   6. Taylor expanded in w around 0 77.2%

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

4. Final simplification 60.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.4:\\ \;\;\;\;w \cdot \left(-\ell\right)\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.9% accurate, 33.9× speedup

Math

\[\begin{array}{l} \\ \ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (+ l (* w (* w (* l 0.5)))))

C

double code(double w, double l) {
	return l + (w * (w * (l * 0.5)));
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    code = l + (w * (w * (l * 0.5d0)))
end function

Java

public static double code(double w, double l) {
	return l + (w * (w * (l * 0.5)));
}

Python

def code(w, l):
	return l + (w * (w * (l * 0.5)))

Julia

function code(w, l)
	return Float64(l + Float64(w * Float64(w * Float64(l * 0.5))))
end

MATLAB

function tmp = code(w, l)
	tmp = l + (w * (w * (l * 0.5)));
end

Wolfram

code[w_, l_] := N[(l + N[(w * N[(w * N[(l * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 97.9%

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

6. Taylor expanded in w around 0 67.2%

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

   1. mul-1-neg 67.2%

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

   2. distribute-rgt-out 67.2%

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

   3. metadata-eval 67.2%

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

8. Simplified 67.2%

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

9. Taylor expanded in w around inf 67.2%

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

    1. *-commutative 67.2%

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

    2. *-commutative 67.2%

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

    3. associate-*r* 67.2%

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

11. Simplified 67.2%

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

12. Final simplification 67.2%

    \[\leadsto \ell + w \cdot \left(w \cdot \left(\ell \cdot 0.5\right)\right) \]
13. Add Preprocessing

Alternative 8: 62.7% 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 99.7%

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

3. Taylor expanded in w around 0 97.9%

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

4. Taylor expanded in w around 0 60.4%

   \[\leadsto \color{blue}{\ell + -1 \cdot \left(\ell \cdot w\right)} \]
5. 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. *-rgt-identity 60.4%

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

   4. distribute-lft-out-- 60.4%

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

6. Simplified 60.4%

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

7. Final simplification 60.4%

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

Alternative 9: 62.7% 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 99.7%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 97.9%

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

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%

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

8. Simplified 60.4%

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

9. Final simplification 60.4%

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

Alternative 10: 56.3% 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 99.7%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 97.9%

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

6. Taylor expanded in w around 0 54.6%

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

7. Final simplification 54.6%

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

Reproduce

herbie shell --seed 2024074 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))