exp-w (used to crash)

Percentage Accurate: 99.3% → 99.3%

Time: 12.8s

Alternatives: 10

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: 99.3% 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: 99.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{e^{w}}\\ \frac{\frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{t_0}\right)}^{3}}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (sqrt (exp w))))
   (/ (/ (pow l (exp w)) (pow (cbrt t_0) 3.0)) t_0)))

C

double code(double w, double l) {
	double t_0 = sqrt(exp(w));
	return (pow(l, exp(w)) / pow(cbrt(t_0), 3.0)) / t_0;
}

Java

public static double code(double w, double l) {
	double t_0 = Math.sqrt(Math.exp(w));
	return (Math.pow(l, Math.exp(w)) / Math.pow(Math.cbrt(t_0), 3.0)) / t_0;
}

Julia

function code(w, l)
	t_0 = sqrt(exp(w))
	return Float64(Float64((l ^ exp(w)) / (cbrt(t_0) ^ 3.0)) / t_0)
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. exp-neg 99.7%

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

   3. div-inv 99.7%

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

   4. add-sqr-sqrt 99.7%

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

   5. associate-/r* 99.7%

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

3. Applied egg-rr 99.7%

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

   1. add-cube-cbrt 99.7%

      \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{\left(\sqrt[3]{\sqrt{e^{w}}} \cdot \sqrt[3]{\sqrt{e^{w}}}\right) \cdot \sqrt[3]{\sqrt{e^{w}}}}}}{\sqrt{e^{w}}} \]

   2. pow3 99.7%

      \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{{\left(\sqrt[3]{\sqrt{e^{w}}}\right)}^{3}}}}{\sqrt{e^{w}}} \]

5. Applied egg-rr 99.7%

   \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{\color{blue}{{\left(\sqrt[3]{\sqrt{e^{w}}}\right)}^{3}}}}{\sqrt{e^{w}}} \]

6. Final simplification 99.7%

   \[\leadsto \frac{\frac{{\ell}^{\left(e^{w}\right)}}{{\left(\sqrt[3]{\sqrt{e^{w}}}\right)}^{3}}}{\sqrt{e^{w}}} \]

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{e^{w}}\\ \frac{\frac{{\ell}^{\left(e^{w}\right)}}{t_0}}{t_0} \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (sqrt (exp w)))) (/ (/ (pow l (exp w)) t_0) t_0)))

C

double code(double w, double l) {
	double t_0 = sqrt(exp(w));
	return (pow(l, exp(w)) / t_0) / t_0;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    t_0 = sqrt(exp(w))
    code = ((l ** exp(w)) / t_0) / t_0
end function

Java

public static double code(double w, double l) {
	double t_0 = Math.sqrt(Math.exp(w));
	return (Math.pow(l, Math.exp(w)) / t_0) / t_0;
}

Python

def code(w, l):
	t_0 = math.sqrt(math.exp(w))
	return (math.pow(l, math.exp(w)) / t_0) / t_0

Julia

function code(w, l)
	t_0 = sqrt(exp(w))
	return Float64(Float64((l ^ exp(w)) / t_0) / t_0)
end

MATLAB

function tmp = code(w, l)
	t_0 = sqrt(exp(w));
	tmp = ((l ^ exp(w)) / t_0) / t_0;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Sqrt[N[Exp[w], $MachinePrecision]], $MachinePrecision]}, N[(N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]]

Derivation

1. Initial program 99.7%

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

   1. *-commutative 99.7%

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

   2. exp-neg 99.7%

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

   3. div-inv 99.7%

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

   4. add-sqr-sqrt 99.7%

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

   5. associate-/r* 99.7%

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

3. Applied egg-rr 99.7%

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

4. Final simplification 99.7%

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

Alternative 3: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] * N[((--1.0) / N[Exp[w], $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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

   1. frac-2neg 99.7%

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

   2. div-inv 99.7%

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

5. Applied egg-rr 99.7%

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

6. Taylor expanded in w around inf 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

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(Float64(-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. Final simplification 99.7%

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

Alternative 5: 99.3% 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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Final simplification 99.7%

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

Alternative 6: 97.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 1000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.7) (not (<= w 1000.0))) (exp (- w)) (* l (exp w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.7) || !(w <= 1000.0)) {
		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 <= 1000.0d0))) 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 <= 1000.0)) {
		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 <= 1000.0):
		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 <= 1000.0))
		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 <= 1000.0)))
		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, 1000.0]], $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 1e3 < w

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-neg 100.0%

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

      3. div-inv 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate-/r* 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. pow-to-exp 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. div-exp 100.0%

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

      5. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.69999999999999996 < w < 1e3

   1. Initial program 99.5%

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

   2. Taylor expanded in w around 0 97.2%

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

      1. expm1-log1p-u 93.4%

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

      2. expm1-udef 48.1%

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

      3. add-sqr-sqrt 25.3%

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

      4. sqrt-unprod 48.1%

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

      5. sqr-neg 48.1%

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

      6. sqrt-unprod 22.8%

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

      7. add-sqr-sqrt 48.1%

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

   4. Applied egg-rr 48.1%

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

      1. expm1-def 93.4%

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

      2. expm1-log1p 97.2%

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

      3. *-commutative 97.2%

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

   6. Simplified 97.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.7 \lor \neg \left(w \leq 1000\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot e^{w}\\ \end{array} \]

Alternative 7: 97.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.72 \lor \neg \left(w \leq 640\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot w\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -0.72) (not (<= w 640.0))) (exp (- w)) (+ l (* l w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -0.72) || !(w <= 640.0)) {
		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 <= (-0.72d0)) .or. (.not. (w <= 640.0d0))) 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 <= -0.72) || !(w <= 640.0)) {
		tmp = Math.exp(-w);
	} else {
		tmp = l + (l * w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if (w <= -0.72) or not (w <= 640.0):
		tmp = math.exp(-w)
	else:
		tmp = l + (l * w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -0.72) || !(w <= 640.0))
		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 <= -0.72) || ~((w <= 640.0)))
		tmp = exp(-w);
	else
		tmp = l + (l * w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -0.72], N[Not[LessEqual[w, 640.0]], $MachinePrecision]], N[Exp[(-w)], $MachinePrecision], N[(l + N[(l * w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -0.71999999999999997 or 640 < w

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. exp-neg 100.0%

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

      3. div-inv 100.0%

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

      4. add-sqr-sqrt 100.0%

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

      5. associate-/r* 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. pow-to-exp 100.0%

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

      3. add-sqr-sqrt 100.0%

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

      4. div-exp 100.0%

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

      5. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

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

   6. Taylor expanded in w around inf 100.0%

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

      1. mul-1-neg 100.0%

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

   8. Simplified 100.0%

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

   if -0.71999999999999997 < w < 640

   1. Initial program 99.5%

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

   2. Taylor expanded in w around 0 97.2%

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

      1. expm1-log1p-u 93.4%

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

      2. expm1-udef 48.1%

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

      3. add-sqr-sqrt 25.3%

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

      4. sqrt-unprod 48.1%

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

      5. sqr-neg 48.1%

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

      6. sqrt-unprod 22.8%

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

      7. add-sqr-sqrt 48.1%

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

   4. Applied egg-rr 48.1%

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

      1. expm1-def 93.4%

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

      2. expm1-log1p 97.2%

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

      3. *-commutative 97.2%

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

   6. Simplified 97.2%

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

   7. Taylor expanded in w around 0 97.2%

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

      1. *-commutative 97.2%

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

   9. Simplified 97.2%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.72 \lor \neg \left(w \leq 640\right):\\ \;\;\;\;e^{-w}\\ \mathbf{else}:\\ \;\;\;\;\ell + \ell \cdot w\\ \end{array} \]

Alternative 8: 97.6% 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. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

3. Simplified 99.7%

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

4. Taylor expanded in w around 0 98.4%

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

5. Final simplification 98.4%

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

Alternative 9: 63.5% 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. Taylor expanded in w around 0 98.4%

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

3. Taylor expanded in w around 0 65.8%

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

   1. +-commutative 65.8%

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

   2. mul-1-neg 65.8%

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

   3. unsub-neg 65.8%

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

   4. *-commutative 65.8%

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

5. Simplified 65.8%

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

6. Final simplification 65.8%

   \[\leadsto \ell - \ell \cdot w \]

Alternative 10: 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 99.7%

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

   1. *-commutative 99.7%

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

   2. exp-neg 99.7%

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

   3. div-inv 99.7%

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

   4. add-sqr-sqrt 99.7%

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

   5. associate-/r* 99.7%

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

3. Applied egg-rr 99.7%

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

4. Taylor expanded in w around 0 59.2%

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

5. Final simplification 59.2%

   \[\leadsto \ell \]

Reproduce

herbie shell --seed 2023189 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))