exp-w (used to crash)

Percentage Accurate: 99.5% → 99.2%

Time: 16.7s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. w = -2.8455683840218908e-86
  2. l = -1.9801551805278478e+54

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: 99.5% 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: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -1.12 \cdot 10^{-14}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(1 + w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -1.12e-14)
   (exp (fma (log l) (exp w) (- w)))
   (* 1.0 (pow l (+ 1.0 w)))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -1.12e-14) {
		tmp = exp(fma(log(l), exp(w), -w));
	} else {
		tmp = 1.0 * pow(l, (1.0 + w));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -1.12e-14)
		tmp = exp(fma(log(l), exp(w), Float64(-w)));
	else
		tmp = Float64(1.0 * (l ^ Float64(1.0 + w)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -1.12e-14], N[Exp[N[(N[Log[l], $MachinePrecision] * N[Exp[w], $MachinePrecision] + (-w)), $MachinePrecision]], $MachinePrecision], N[(1.0 * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -1.12000000000000006e-14

   1. Initial program 99.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-pow.f64 N/A

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

      4. pow-to-exp N/A

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

      5. lift-exp.f64 N/A

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

      6. prod-exp N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-log.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -1.12000000000000006e-14 < w

   1. Initial program 98.7%

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

   3. Taylor expanded in w around 0

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

      1. lower-+.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 99.5%

      \[\leadsto \color{blue}{1} \cdot {\ell}^{\left(1 + w\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. lower-/.f64 N/A

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

   8. exp-neg N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-/.f64 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in w around 0

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

   1. lower-/.f64 97.3

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

7. Applied rewrites 97.3%

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

8. Final simplification 97.3%

   \[\leadsto e^{-w} \cdot {\left({\ell}^{-1}\right)}^{-1} \]
9. Add Preprocessing

Alternative 3: 99.5% 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 99.0%

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

Alternative 4: 97.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \frac{e^{-w}}{{\ell}^{-1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. lower-/.f64 N/A

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

   8. exp-neg N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-/.f64 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. lower-/.f64 98.9

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-neg.f64 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in w around 0

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

   1. lower-/.f64 97.3

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

9. Applied rewrites 97.3%

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

10. Final simplification 97.3%

    \[\leadsto \frac{e^{-w}}{{\ell}^{-1}} \]
11. Add Preprocessing

Alternative 5: 98.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.41:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.41)
   (* (- 1.0 w) (pow l (+ 1.0 w)))
   (* (fma (- (* 0.5 w) 1.0) w 1.0) (pow l (fma (fma 0.5 w 1.0) w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 0.41) {
		tmp = (1.0 - w) * pow(l, (1.0 + w));
	} else {
		tmp = fma(((0.5 * w) - 1.0), w, 1.0) * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 0.41)
		tmp = Float64(Float64(1.0 - w) * (l ^ Float64(1.0 + w)));
	else
		tmp = Float64(fma(Float64(Float64(0.5 * w) - 1.0), w, 1.0) * (l ^ fma(fma(0.5, w, 1.0), w, 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[l, 0.41], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.5 * w), $MachinePrecision] - 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision] * N[Power[l, N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.409999999999999976

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in w around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if 0.409999999999999976 < l

   1. Initial program 97.9%

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

   3. Taylor expanded in w around 0

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

      1. lower-+.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 61.3

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

   8. Applied rewrites 61.3%

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

   9. Taylor expanded in w around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 99.6

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

   11. Applied rewrites 99.6%

       \[\leadsto \mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right) \cdot {\ell}^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 95.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.41:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right) \cdot w - 1, w, 1\right)}{{\ell}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.41)
   (* (- 1.0 w) (pow l (+ 1.0 w)))
   (/
    (fma (- (* (fma -0.16666666666666666 w 0.5) w) 1.0) w 1.0)
    (pow l -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 0.41) {
		tmp = (1.0 - w) * pow(l, (1.0 + w));
	} else {
		tmp = fma(((fma(-0.16666666666666666, w, 0.5) * w) - 1.0), w, 1.0) / pow(l, -1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 0.41)
		tmp = Float64(Float64(1.0 - w) * (l ^ Float64(1.0 + w)));
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, w, 0.5) * w) - 1.0), w, 1.0) / (l ^ -1.0));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[l, 0.41], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.16666666666666666 * w + 0.5), $MachinePrecision] * w), $MachinePrecision] - 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision] / N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.409999999999999976

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in w around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if 0.409999999999999976 < l

   1. Initial program 97.9%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. lower-/.f64 N/A

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

      8. exp-neg N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lower-/.f64 97.8

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-flip N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-neg.f64 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in w around 0

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

      1. lower-/.f64 97.0

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

   9. Applied rewrites 97.0%

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

   10. Taylor expanded in w around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 91.7

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

   12. Applied rewrites 91.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right) \cdot w - 1, w, 1\right)}}{\frac{1}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.41:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right) \cdot w - 1, w, 1\right)}{{\ell}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 95.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.41:\\ \;\;\;\;1 \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right) \cdot w - 1, w, 1\right)}{{\ell}^{-1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.41)
   (* 1.0 (pow l (+ 1.0 w)))
   (/
    (fma (- (* (fma -0.16666666666666666 w 0.5) w) 1.0) w 1.0)
    (pow l -1.0))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 0.41) {
		tmp = 1.0 * pow(l, (1.0 + w));
	} else {
		tmp = fma(((fma(-0.16666666666666666, w, 0.5) * w) - 1.0), w, 1.0) / pow(l, -1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 0.41)
		tmp = Float64(1.0 * (l ^ Float64(1.0 + w)));
	else
		tmp = Float64(fma(Float64(Float64(fma(-0.16666666666666666, w, 0.5) * w) - 1.0), w, 1.0) / (l ^ -1.0));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[l, 0.41], N[(1.0 * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(-0.16666666666666666 * w + 0.5), $MachinePrecision] * w), $MachinePrecision] - 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision] / N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 0.409999999999999976

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 98.3%

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

   if 0.409999999999999976 < l

   1. Initial program 97.9%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

      3. sinh-+-cosh-rev N/A

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

      4. flip-+ N/A

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

      5. sinh-cosh N/A

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

      6. sinh---cosh-rev N/A

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

      7. lower-/.f64 N/A

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

      8. exp-neg N/A

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

      9. pow-to-exp N/A

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

      10. lift-pow.f64 N/A

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

      11. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. *-rgt-identity N/A

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

      5. lower-/.f64 97.8

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

      6. lift-/.f64 N/A

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

      7. lift-pow.f64 N/A

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

      8. pow-flip N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-neg.f64 97.8

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

   6. Applied rewrites 97.8%

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

   7. Taylor expanded in w around 0

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

      1. lower-/.f64 97.0

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

   9. Applied rewrites 97.0%

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

   10. Taylor expanded in w around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower--.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 N/A

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

       7. +-commutative N/A

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

       8. lower-fma.f64 91.7

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

   12. Applied rewrites 91.7%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right) \cdot w - 1, w, 1\right)}}{\frac{1}{\ell}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \leq 0.41:\\ \;\;\;\;1 \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right) \cdot w - 1, w, 1\right)}{{\ell}^{-1}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right) \cdot w - 1, w, 1\right)}{{\ell}^{-1}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/ (fma (- (* (fma -0.16666666666666666 w 0.5) w) 1.0) w 1.0) (pow l -1.0)))

C

double code(double w, double l) {
	return fma(((fma(-0.16666666666666666, w, 0.5) * w) - 1.0), w, 1.0) / pow(l, -1.0);
}

Julia

function code(w, l)
	return Float64(fma(Float64(Float64(fma(-0.16666666666666666, w, 0.5) * w) - 1.0), w, 1.0) / (l ^ -1.0))
end

Wolfram

code[w_, l_] := N[(N[(N[(N[(N[(-0.16666666666666666 * w + 0.5), $MachinePrecision] * w), $MachinePrecision] - 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision] / N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. lower-/.f64 N/A

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

   8. exp-neg N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-/.f64 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. lower-/.f64 98.9

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-neg.f64 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in w around 0

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

   1. lower-/.f64 97.3

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

9. Applied rewrites 97.3%

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

10. Taylor expanded in w around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower--.f64 N/A

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

    5. *-commutative N/A

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

    6. lower-*.f64 N/A

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

    7. +-commutative N/A

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

    8. lower-fma.f64 77.5

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

12. Applied rewrites 77.5%

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

13. Final simplification 77.5%

    \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, w, 0.5\right) \cdot w - 1, w, 1\right)}{{\ell}^{-1}} \]
14. Add Preprocessing

Alternative 9: 73.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right)}{{\ell}^{-1}} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (/ (fma (- (* 0.5 w) 1.0) w 1.0) (pow l -1.0)))

C

double code(double w, double l) {
	return fma(((0.5 * w) - 1.0), w, 1.0) / pow(l, -1.0);
}

Julia

function code(w, l)
	return Float64(fma(Float64(Float64(0.5 * w) - 1.0), w, 1.0) / (l ^ -1.0))
end

Wolfram

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

Derivation

1. Initial program 99.0%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. lower-/.f64 N/A

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

   8. exp-neg N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-/.f64 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. lower-/.f64 98.9

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-neg.f64 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in w around 0

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

   1. lower-/.f64 97.3

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

9. Applied rewrites 97.3%

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

10. Taylor expanded in w around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower--.f64 N/A

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

    5. lower-*.f64 73.9

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

12. Applied rewrites 73.9%

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

13. Final simplification 73.9%

    \[\leadsto \frac{\mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right)}{{\ell}^{-1}} \]
14. Add Preprocessing

Alternative 10: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 8.5e-10)
   (* (- 1.0 w) (pow l (+ 1.0 w)))
   (* 1.0 (pow l (fma (fma 0.5 w 1.0) w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 8.5e-10) {
		tmp = (1.0 - w) * pow(l, (1.0 + w));
	} else {
		tmp = 1.0 * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 8.5e-10)
		tmp = Float64(Float64(1.0 - w) * (l ^ Float64(1.0 + w)));
	else
		tmp = Float64(1.0 * (l ^ fma(fma(0.5, w, 1.0), w, 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[l, 8.5e-10], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[l, N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if l < 8.4999999999999996e-10

   1. Initial program 99.8%

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

   3. Taylor expanded in w around 0

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

      1. lower-+.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in w around 0

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. lower--.f64 98.5

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

   8. Applied rewrites 98.5%

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

   if 8.4999999999999996e-10 < l

   1. Initial program 98.0%

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

   3. Taylor expanded in w around 0

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

      1. lower-+.f64 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in w around 0

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

   8. Applied rewrites 62.2%

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

   9. Taylor expanded in w around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 99.2

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

   11. Applied rewrites 99.2%

       \[\leadsto 1 \cdot {\ell}^{\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1 - w}{{\ell}^{-1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := N[(N[(1.0 - w), $MachinePrecision] / N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. lower-/.f64 N/A

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

   8. exp-neg N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-/.f64 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. lower-/.f64 98.9

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-neg.f64 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in w around 0

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

   1. lower-/.f64 97.3

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

9. Applied rewrites 97.3%

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

10. Taylor expanded in w around 0

    \[\leadsto \frac{\color{blue}{1 + -1 \cdot w}}{\frac{1}{\ell}} \]
11. Step-by-step derivation

    1. fp-cancel-sign-sub-inv N/A

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

    2. metadata-eval N/A

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

    3. *-lft-identity N/A

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

    4. lower--.f64 62.9

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

12. Applied rewrites 62.9%

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

13. Final simplification 62.9%

    \[\leadsto \frac{1 - w}{{\ell}^{-1}} \]
14. Add Preprocessing

Alternative 12: 57.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\ell}^{-1}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[w_, l_] := N[(1.0 / N[Power[l, -1.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.0%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. sinh-+-cosh-rev N/A

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

   6. sinh---cosh-rev N/A

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

   7. lower-/.f64 N/A

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

   8. exp-neg N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. lower-/.f64 98.9

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

4. Applied rewrites 98.9%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. *-rgt-identity N/A

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

   5. lower-/.f64 98.9

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

   6. lift-/.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-flip N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-neg.f64 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in w around 0

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

   1. lower-/.f64 97.3

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

9. Applied rewrites 97.3%

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

10. Taylor expanded in w around 0

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

12. Applied rewrites 55.9%

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

13. Final simplification 55.9%

    \[\leadsto \frac{1}{{\ell}^{-1}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))