exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 12.2s

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: 99.4% 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 99.6%

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

2. Final simplification 99.6%

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

Alternative 2: 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 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 3: 97.5% 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 99.6%

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

2. Taylor expanded in w around 0 98.1%

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

3. Final simplification 98.1%

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

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 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in w around 0 98.1%

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

5. Final simplification 98.1%

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

Alternative 5: 75.5% accurate, 9.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \ell - \left(\ell + \ell\right)\\ \mathbf{if}\;w \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;\left(\ell - w \cdot \ell\right) + 0.5 \cdot \left(\ell \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell + \left(w \cdot w\right) \cdot \left(t_0 \cdot \left(\left(\ell + \ell\right) - \ell\right)\right)}{\ell + w \cdot t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (- l (+ l l))))
   (if (<= w 3.2e-40)
     (+ (- l (* w l)) (* 0.5 (* l (* w w))))
     (/ (+ (* l l) (* (* w w) (* t_0 (- (+ l l) l)))) (+ l (* w t_0))))))

C

double code(double w, double l) {
	double t_0 = l - (l + l);
	double tmp;
	if (w <= 3.2e-40) {
		tmp = (l - (w * l)) + (0.5 * (l * (w * w)));
	} else {
		tmp = ((l * l) + ((w * w) * (t_0 * ((l + l) - l)))) / (l + (w * t_0));
	}
	return tmp;
}

Fortran

real(8) function code(w, l)
    real(8), intent (in) :: w
    real(8), intent (in) :: l
    real(8) :: t_0
    real(8) :: tmp
    t_0 = l - (l + l)
    if (w <= 3.2d-40) then
        tmp = (l - (w * l)) + (0.5d0 * (l * (w * w)))
    else
        tmp = ((l * l) + ((w * w) * (t_0 * ((l + l) - l)))) / (l + (w * t_0))
    end if
    code = tmp
end function

Java

public static double code(double w, double l) {
	double t_0 = l - (l + l);
	double tmp;
	if (w <= 3.2e-40) {
		tmp = (l - (w * l)) + (0.5 * (l * (w * w)));
	} else {
		tmp = ((l * l) + ((w * w) * (t_0 * ((l + l) - l)))) / (l + (w * t_0));
	}
	return tmp;
}

Python

def code(w, l):
	t_0 = l - (l + l)
	tmp = 0
	if w <= 3.2e-40:
		tmp = (l - (w * l)) + (0.5 * (l * (w * w)))
	else:
		tmp = ((l * l) + ((w * w) * (t_0 * ((l + l) - l)))) / (l + (w * t_0))
	return tmp

Julia

function code(w, l)
	t_0 = Float64(l - Float64(l + l))
	tmp = 0.0
	if (w <= 3.2e-40)
		tmp = Float64(Float64(l - Float64(w * l)) + Float64(0.5 * Float64(l * Float64(w * w))));
	else
		tmp = Float64(Float64(Float64(l * l) + Float64(Float64(w * w) * Float64(t_0 * Float64(Float64(l + l) - l)))) / Float64(l + Float64(w * t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(w, l)
	t_0 = l - (l + l);
	tmp = 0.0;
	if (w <= 3.2e-40)
		tmp = (l - (w * l)) + (0.5 * (l * (w * w)));
	else
		tmp = ((l * l) + ((w * w) * (t_0 * ((l + l) - l)))) / (l + (w * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(l - N[(l + l), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[w, 3.2e-40], N[(N[(l - N[(w * l), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(l * N[(w * w), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l * l), $MachinePrecision] + N[(N[(w * w), $MachinePrecision] * N[(t$95$0 * N[(N[(l + l), $MachinePrecision] - l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l + N[(w * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if w < 3.20000000000000002e-40

   1. Initial program 99.6%

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

   2. Taylor expanded in w around 0 97.9%

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

   3. Taylor expanded in w around 0 86.6%

      \[\leadsto \color{blue}{\ell + \left(-1 \cdot \left(\ell \cdot w\right) + 0.5 \cdot \left(\ell \cdot {w}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. neg-mul-1 86.6%

         \[\leadsto \ell + \left(\color{blue}{\left(-\ell \cdot w\right)} + 0.5 \cdot \left(\ell \cdot {w}^{2}\right)\right) \]

      2. associate-+r+ 86.6%

         \[\leadsto \color{blue}{\left(\ell + \left(-\ell \cdot w\right)\right) + 0.5 \cdot \left(\ell \cdot {w}^{2}\right)} \]

      3. sub-neg 86.6%

         \[\leadsto \color{blue}{\left(\ell - \ell \cdot w\right)} + 0.5 \cdot \left(\ell \cdot {w}^{2}\right) \]

      4. *-commutative 86.6%

         \[\leadsto \left(\ell - \color{blue}{w \cdot \ell}\right) + 0.5 \cdot \left(\ell \cdot {w}^{2}\right) \]

      5. unpow2 86.6%

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

   5. Simplified 86.6%

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

   if 3.20000000000000002e-40 < w

   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 98.8%

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

   5. Taylor expanded in w around 0 10.8%

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

      1. mul-1-neg 10.8%

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

      2. unsub-neg 10.8%

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

   7. Simplified 10.8%

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

      1. *-un-lft-identity 10.8%

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

      2. prod-diff 10.8%

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

      3. *-commutative 10.8%

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

      4. fma-neg 10.8%

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

      5. *-un-lft-identity 10.8%

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

      6. add-sqr-sqrt 0.0%

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

      7. sqrt-unprod 10.3%

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

      8. sqr-neg 10.3%

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

      9. sqrt-unprod 10.8%

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

      10. add-sqr-sqrt 10.8%

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

      11. *-commutative 10.8%

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

   9. Applied egg-rr 10.8%

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

       1. *-commutative 10.8%

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

       2. fma-udef 10.8%

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

       3. *-commutative 10.8%

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

       4. distribute-lft-out 10.8%

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

   11. Simplified 10.8%

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

       1. associate-+l- 10.8%

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

       2. flip-- 17.1%

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

       3. distribute-lft-out-- 17.1%

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

       4. distribute-lft-out-- 17.1%

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

       5. distribute-lft-out-- 17.1%

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

   13. Applied egg-rr 17.1%

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

       1. swap-sqr 28.9%

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

   15. Simplified 28.9%

       \[\leadsto \color{blue}{\frac{\ell \cdot \ell - \left(w \cdot w\right) \cdot \left(\left(\ell - \left(\ell + \ell\right)\right) \cdot \left(\ell - \left(\ell + \ell\right)\right)\right)}{\ell + w \cdot \left(\ell - \left(\ell + \ell\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq 3.2 \cdot 10^{-40}:\\ \;\;\;\;\left(\ell - w \cdot \ell\right) + 0.5 \cdot \left(\ell \cdot \left(w \cdot w\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell \cdot \ell + \left(w \cdot w\right) \cdot \left(\left(\ell - \left(\ell + \ell\right)\right) \cdot \left(\left(\ell + \ell\right) - \ell\right)\right)}{\ell + w \cdot \left(\ell - \left(\ell + \ell\right)\right)}\\ \end{array} \]

Alternative 6: 73.8% accurate, 23.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in w around 0 98.1%

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

3. Taylor expanded in w around 0 73.1%

   \[\leadsto \color{blue}{\ell + \left(-1 \cdot \left(\ell \cdot w\right) + 0.5 \cdot \left(\ell \cdot {w}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. neg-mul-1 73.1%

      \[\leadsto \ell + \left(\color{blue}{\left(-\ell \cdot w\right)} + 0.5 \cdot \left(\ell \cdot {w}^{2}\right)\right) \]

   2. associate-+r+ 73.1%

      \[\leadsto \color{blue}{\left(\ell + \left(-\ell \cdot w\right)\right) + 0.5 \cdot \left(\ell \cdot {w}^{2}\right)} \]

   3. sub-neg 73.1%

      \[\leadsto \color{blue}{\left(\ell - \ell \cdot w\right)} + 0.5 \cdot \left(\ell \cdot {w}^{2}\right) \]

   4. *-commutative 73.1%

      \[\leadsto \left(\ell - \color{blue}{w \cdot \ell}\right) + 0.5 \cdot \left(\ell \cdot {w}^{2}\right) \]

   5. unpow2 73.1%

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

5. Simplified 73.1%

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

6. Final simplification 73.1%

   \[\leadsto \left(\ell - w \cdot \ell\right) + 0.5 \cdot \left(\ell \cdot \left(w \cdot w\right)\right) \]

Alternative 7: 73.8% accurate, 23.5× speedup

Math

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

FPCore

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

C

double code(double w, double l) {
	return l - ((w * 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 * l) + ((w * w) * (l * (-0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in w around 0 98.1%

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

5. Taylor expanded in w around 0 73.1%

   \[\leadsto \color{blue}{\ell + \left(-1 \cdot \left(\ell \cdot w\right) + -1 \cdot \left({w}^{2} \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-out 73.1%

      \[\leadsto \ell + \color{blue}{-1 \cdot \left(\ell \cdot w + {w}^{2} \cdot \left(-1 \cdot \ell + 0.5 \cdot \ell\right)\right)} \]

   2. unpow2 73.1%

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

   3. distribute-rgt-out 73.1%

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

   4. metadata-eval 73.1%

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

7. Simplified 73.1%

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

8. Final simplification 73.1%

   \[\leadsto \ell - \left(w \cdot \ell + \left(w \cdot w\right) \cdot \left(\ell \cdot -0.5\right)\right) \]

Alternative 8: 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 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in w around 0 98.1%

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

5. Taylor expanded in w around 0 64.2%

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

   1. mul-1-neg 64.2%

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

   2. unsub-neg 64.2%

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

7. Simplified 64.2%

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

8. Taylor expanded in l around 0 64.2%

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

9. Final simplification 64.2%

   \[\leadsto \ell \cdot \left(1 - w\right) \]

Alternative 9: 11.1% accurate, 76.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in w around 0 98.1%

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

5. Taylor expanded in w around 0 64.2%

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

   1. mul-1-neg 64.2%

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

   2. unsub-neg 64.2%

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

7. Simplified 64.2%

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

8. Taylor expanded in w around inf 10.1%

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

   1. neg-mul-1 10.1%

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

   2. distribute-rgt-neg-in 10.1%

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

10. Simplified 10.1%

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

11. Final simplification 10.1%

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

Alternative 10: 3.1% accurate, 101.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. exp-neg 99.6%

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

   2. associate-*l/ 99.6%

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

   3. *-lft-identity 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in w around 0 98.1%

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

5. Taylor expanded in w around 0 64.2%

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

   1. mul-1-neg 64.2%

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

   2. unsub-neg 64.2%

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

7. Simplified 64.2%

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

   1. *-un-lft-identity 64.2%

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

   2. prod-diff 57.9%

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

   3. *-commutative 57.9%

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

   4. fma-neg 57.9%

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

   5. *-un-lft-identity 57.9%

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

   6. add-sqr-sqrt 29.1%

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

   7. sqrt-unprod 65.3%

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

   8. sqr-neg 65.3%

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

   9. sqrt-unprod 28.9%

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

   10. add-sqr-sqrt 57.3%

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

   11. *-commutative 57.3%

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

9. Applied egg-rr 57.3%

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

    1. *-commutative 57.3%

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

    2. fma-udef 57.3%

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

    3. *-commutative 57.3%

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

    4. distribute-lft-out 57.3%

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

11. Simplified 57.3%

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

12. Taylor expanded in w around inf 2.9%

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

    1. *-commutative 2.9%

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

    2. sub-neg 2.9%

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

    3. neg-mul-1 2.9%

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

    4. distribute-rgt-out 2.9%

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

    5. metadata-eval 2.9%

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

    6. *-rgt-identity 2.9%

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

14. Simplified 2.9%

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

15. Final simplification 2.9%

    \[\leadsto w \cdot \ell \]

Reproduce

herbie shell --seed 2023292 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))