exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 18.7s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. w = -2.383798246865106e-16
  2. l = -1.690104436757282e-98

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

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

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 99.4%

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

Alternative 2: 97.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in w around 0

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

5. Simplified 98.0%

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

6. Taylor expanded in w around inf

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

   1. lower-*.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-neg.f64 98.0

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

8. Simplified 98.0%

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

9. Final simplification 98.0%

   \[\leadsto e^{-w} \cdot \ell \]
10. Add Preprocessing

Alternative 3: 83.6% accurate, 10.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \mathsf{fma}\left(w, -0.16666666666666666, 0.5\right), -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.25e-5)
   (* l (fma w (fma w (fma w -0.16666666666666666 0.5) -1.0) 1.0))
   (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.25e-5) {
		tmp = l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0);
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.25e-5)
		tmp = Float64(l * fma(w, fma(w, fma(w, -0.16666666666666666, 0.5), -1.0), 1.0));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.25e-5], N[(l * N[(w * N[(w * N[(w * -0.16666666666666666 + 0.5), $MachinePrecision] + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.25000000000000006e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 99.0

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

   8. Simplified 99.0%

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

   9. Taylor expanded in w around 0

      \[\leadsto \ell \cdot \color{blue}{\left(1 + w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \ell \cdot \color{blue}{\left(w \cdot \left(w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1\right) + 1\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\mathsf{fma}\left(w, w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) - 1, 1\right)} \]

       3. sub-neg N/A

          \[\leadsto \ell \cdot \mathsf{fma}\left(w, \color{blue}{w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

       4. metadata-eval N/A

          \[\leadsto \ell \cdot \mathsf{fma}\left(w, w \cdot \left(\frac{1}{2} + \frac{-1}{6} \cdot w\right) + \color{blue}{-1}, 1\right) \]

       5. lower-fma.f64 N/A

          \[\leadsto \ell \cdot \mathsf{fma}\left(w, \color{blue}{\mathsf{fma}\left(w, \frac{1}{2} + \frac{-1}{6} \cdot w, -1\right)}, 1\right) \]

       6. +-commutative N/A

          \[\leadsto \ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \color{blue}{\frac{-1}{6} \cdot w + \frac{1}{2}}, -1\right), 1\right) \]

       7. *-commutative N/A

          \[\leadsto \ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, \color{blue}{w \cdot \frac{-1}{6}} + \frac{1}{2}, -1\right), 1\right) \]

       8. lower-fma.f64 89.8

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

   11. Simplified 89.8%

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

   if 1.25000000000000006e-5 < w

   1. Initial program 97.1%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 92.4%

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

   6. Taylor expanded in w around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 3.2

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

   8. Simplified 3.2%

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

   9. Taylor expanded in w around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-/.f64 3.2

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

   11. Simplified 3.2%

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

   12. Taylor expanded in w around 0

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

       1. lower-/.f64 47.0

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

   14. Simplified 47.0%

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

Alternative 4: 80.0% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 9000000:\\ \;\;\;\;\ell \cdot \mathsf{fma}\left(w, \mathsf{fma}\left(w, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 9000000.0) (* l (fma w (fma w 0.5 -1.0) 1.0)) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 9000000.0) {
		tmp = l * fma(w, fma(w, 0.5, -1.0), 1.0);
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 9000000.0)
		tmp = Float64(l * fma(w, fma(w, 0.5, -1.0), 1.0));
	else
		tmp = Float64(w * Float64(l / w));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, 9000000.0], N[(l * N[(w * N[(w * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 9e6

   1. Initial program 99.3%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.7%

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

   6. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 97.7

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

   8. Simplified 97.7%

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

   9. Taylor expanded in w around 0

      \[\leadsto \ell \cdot \color{blue}{\left(1 + w \cdot \left(\frac{1}{2} \cdot w - 1\right)\right)} \]
   10. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \ell \cdot \color{blue}{\left(w \cdot \left(\frac{1}{2} \cdot w - 1\right) + 1\right)} \]

       2. lower-fma.f64 N/A

          \[\leadsto \ell \cdot \color{blue}{\mathsf{fma}\left(w, \frac{1}{2} \cdot w - 1, 1\right)} \]

       3. sub-neg N/A

          \[\leadsto \ell \cdot \mathsf{fma}\left(w, \color{blue}{\frac{1}{2} \cdot w + \left(\mathsf{neg}\left(1\right)\right)}, 1\right) \]

       4. *-commutative N/A

          \[\leadsto \ell \cdot \mathsf{fma}\left(w, \color{blue}{w \cdot \frac{1}{2}} + \left(\mathsf{neg}\left(1\right)\right), 1\right) \]

       5. metadata-eval N/A

          \[\leadsto \ell \cdot \mathsf{fma}\left(w, w \cdot \frac{1}{2} + \color{blue}{-1}, 1\right) \]

       6. lower-fma.f64 82.2

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

   11. Simplified 82.2%

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

   if 9e6 < w

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in w around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 3.3

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

   8. Simplified 3.3%

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

   9. Taylor expanded in w around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-/.f64 3.3

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

   11. Simplified 3.3%

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

   12. Taylor expanded in w around 0

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

       1. lower-/.f64 50.6

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

   14. Simplified 50.6%

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

Alternative 5: 69.7% accurate, 13.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w 1.25e-5) (- l (* w l)) (* w (/ l w))))

C

double code(double w, double l) {
	double tmp;
	if (w <= 1.25e-5) {
		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 <= 1.25d-5) 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 <= 1.25e-5) {
		tmp = l - (w * l);
	} else {
		tmp = w * (l / w);
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= 1.25e-5:
		tmp = l - (w * l)
	else:
		tmp = w * (l / w)
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= 1.25e-5)
		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 <= 1.25e-5)
		tmp = l - (w * l);
	else
		tmp = w * (l / w);
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, 1.25e-5], N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision], N[(w * N[(l / w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < 1.25000000000000006e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 99.0%

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

   6. Taylor expanded in w around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 71.8

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

   8. Simplified 71.8%

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

   if 1.25000000000000006e-5 < w

   1. Initial program 97.1%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 92.4%

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

   6. Taylor expanded in w around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 3.2

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

   8. Simplified 3.2%

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

   9. Taylor expanded in w around inf

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

       1. lower-*.f64 N/A

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

       2. lower--.f64 N/A

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

       3. lower-/.f64 3.2

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

   11. Simplified 3.2%

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

   12. Taylor expanded in w around 0

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

       1. lower-/.f64 47.0

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

   14. Simplified 47.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\ell - w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;w \cdot \frac{\ell}{w}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 22.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -0.39:\\ \;\;\;\;-w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \end{array} \]

FPCore

(FPCore (w l) :precision binary64 (if (<= w -0.39) (- (* w l)) l))

C

double code(double w, double l) {
	double tmp;
	if (w <= -0.39) {
		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 <= (-0.39d0)) 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 <= -0.39) {
		tmp = -(w * l);
	} else {
		tmp = l;
	}
	return tmp;
}

Python

def code(w, l):
	tmp = 0
	if w <= -0.39:
		tmp = -(w * l)
	else:
		tmp = l
	return tmp

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -0.39)
		tmp = Float64(-Float64(w * l));
	else
		tmp = l;
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	tmp = 0.0;
	if (w <= -0.39)
		tmp = -(w * l);
	else
		tmp = l;
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := If[LessEqual[w, -0.39], (-N[(w * l), $MachinePrecision]), l]

Derivation

1. Split input into 2 regimes

2. if w < -0.39000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in w around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 22.2

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

   8. Simplified 22.2%

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

   9. Taylor expanded in w around inf

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

       1. mul-1-neg N/A

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

       2. distribute-rgt-neg-in N/A

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

       3. neg-mul-1 N/A

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

       4. lower-*.f64 N/A

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

       5. neg-mul-1 N/A

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

       6. lower-neg.f64 22.2

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

   11. Simplified 22.2%

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

   if -0.39000000000000001 < w

   1. Initial program 99.1%

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

   3. Taylor expanded in w around 0

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

   5. Simplified 97.2%

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

   6. Taylor expanded in w around inf

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 97.2

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

   8. Simplified 97.2%

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

   9. Taylor expanded in w around 0

      \[\leadsto \ell \cdot \color{blue}{1} \]
   10. Step-by-step derivation

   11. Simplified 78.8%

       \[\leadsto \ell \cdot \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -0.39:\\ \;\;\;\;-w \cdot \ell\\ \mathbf{else}:\\ \;\;\;\;\ell\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 34.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in w around 0

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

5. Simplified 98.0%

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

6. Taylor expanded in w around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 61.7

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

8. Simplified 61.7%

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

9. Final simplification 61.7%

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

Alternative 8: 56.4% accurate, 309.0× 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 99.4%

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

3. Taylor expanded in w around 0

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

5. Simplified 98.0%

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

6. Taylor expanded in w around inf

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

   1. lower-*.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-neg.f64 98.0

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

8. Simplified 98.0%

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

9. Taylor expanded in w around 0

   \[\leadsto \ell \cdot \color{blue}{1} \]
10. Step-by-step derivation

11. Simplified 56.5%

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

12. Final simplification 56.5%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))