exp-w (used to crash)

Percentage Accurate: 99.4% → 99.4%

Time: 17.7s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. w = 1.9814839838300917
  2. l = -7.541030103807927e+165

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(w, l)
use fmin_fmax_functions
    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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(w, l)
use fmin_fmax_functions
    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. 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-exp.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. exp-neg N/A

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

   6. lift-exp.f64 N/A

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

   7. associate-*r/ N/A

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

   8. *-rgt-identity N/A

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

   9. lower-/.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 1.0)
   (*
    (exp (- w))
    (pow
     (pow (pow l -1.0) -1.0)
     (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0)))
   (* (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 <= 1.0) {
		tmp = exp(-w) * pow(pow(pow(l, -1.0), -1.0), fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	} 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 <= 1.0)
		tmp = Float64(exp(Float64(-w)) * (((l ^ -1.0) ^ -1.0) ^ fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0)));
	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, 1.0], N[(N[Exp[(-w)], $MachinePrecision] * N[Power[N[Power[N[Power[l, -1.0], $MachinePrecision], -1.0], $MachinePrecision], N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $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 < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in l around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. exp-prod N/A

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

      4. lower-pow.f64 N/A

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

      5. mul-1-neg N/A

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

      6. log-rec N/A

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

      7. remove-double-neg N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-log.f64 N/A

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

      10. lower-exp.f64 95.0

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 99.4%

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

   8. Taylor expanded in w around 0

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

   10. Applied rewrites 98.9%

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

   if 1 < l

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot w - 1, w, 1\right) \cdot {\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + \frac{1}{2} \cdot w\right)\right)}} \]
   7. 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 98.4

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

   8. Applied rewrites 98.4%

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

4. Final simplification 98.7%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -9.5 \cdot 10^{-6}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\ell}^{\left(e^{w}\right)}}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -9.5e-6)
   (exp (fma (log l) (exp w) (- w)))
   (/
    (pow l (exp w))
    (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0))))

C

double code(double w, double l) {
	double tmp;
	if (w <= -9.5e-6) {
		tmp = exp(fma(log(l), exp(w), -w));
	} else {
		tmp = pow(l, exp(w)) / fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -9.5e-6)
		tmp = exp(fma(log(l), exp(w), Float64(-w)));
	else
		tmp = Float64((l ^ exp(w)) / fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -9.5e-6], N[Exp[N[(N[Log[l], $MachinePrecision] * N[Exp[w], $MachinePrecision] + (-w)), $MachinePrecision]], $MachinePrecision], N[(N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -9.5000000000000005e-6

   1. Initial program 99.5%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 99.5

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

   4. Applied rewrites 99.5%

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

   if -9.5000000000000005e-6 < w

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

Alternative 4: 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(w, l)
use fmin_fmax_functions
    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. Add Preprocessing
3. Add Preprocessing

Alternative 5: 99.0% accurate, 2.2× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 1.0)
   (*
    (fma -1.0 w 1.0)
    (pow l (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0)))
   (* (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 <= 1.0) {
		tmp = fma(-1.0, w, 1.0) * pow(l, fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	} 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 <= 1.0)
		tmp = Float64(fma(-1.0, w, 1.0) * (l ^ fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0)));
	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, 1.0], N[(N[(-1.0 * w + 1.0), $MachinePrecision] * N[Power[l, N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $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 < 1

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 86.8

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

   8. Applied rewrites 86.8%

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

   9. Taylor expanded in w around 0

      \[\leadsto \mathsf{fma}\left(-1, w, 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, w, \frac{1}{2}\right), w, 1\right), w, 1\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.6%

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

   if 1 < l

   1. Initial program 99.5%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot w - 1, w, 1\right) \cdot {\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + \frac{1}{2} \cdot w\right)\right)}} \]
   7. 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 98.4

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

   8. Applied rewrites 98.4%

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

Alternative 6: 98.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 0.7:\\ \;\;\;\;\mathsf{fma}\left(-1, w, 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.7)
   (*
    (fma -1.0 w 1.0)
    (pow l (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0)))
   (* (- 1.0 w) (pow l (fma (fma 0.5 w 1.0) w 1.0)))))

C

double code(double w, double l) {
	double tmp;
	if (l <= 0.7) {
		tmp = fma(-1.0, w, 1.0) * pow(l, fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	} else {
		tmp = (1.0 - w) * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (l <= 0.7)
		tmp = Float64(fma(-1.0, w, 1.0) * (l ^ fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0)));
	else
		tmp = Float64(Float64(1.0 - w) * (l ^ fma(fma(0.5, w, 1.0), w, 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[l, 0.7], N[(N[(-1.0 * w + 1.0), $MachinePrecision] * N[Power[l, N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - w), $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 < 0.69999999999999996

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 87.4

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

   8. Applied rewrites 87.4%

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

   9. Taylor expanded in w around 0

      \[\leadsto \mathsf{fma}\left(-1, w, 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{6}, w, \frac{1}{2}\right), w, 1\right), w, 1\right)\right)} \]
   10. Step-by-step derivation

   11. Applied rewrites 98.6%

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

   if 0.69999999999999996 < l

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 60.2

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

   8. Applied rewrites 60.2%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 97.4%

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

   12. Taylor expanded in w around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 98.3

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

   14. Applied rewrites 98.3%

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

Alternative 7: 91.5% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot w, w, 1\right)\\ \mathbf{if}\;w \leq -5.6 \cdot 10^{+145}:\\ \;\;\;\;t\_0 \cdot {\ell}^{1}\\ \mathbf{elif}\;w \leq -3.7 \cdot 10^{+62}:\\ \;\;\;\;t\_0 \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma (* 0.5 w) w 1.0)))
   (if (<= w -5.6e+145)
     (* t_0 (pow l 1.0))
     (if (<= w -3.7e+62)
       (* t_0 (pow l (+ 1.0 w)))
       (* (- 1.0 w) (pow l (fma (fma 0.5 w 1.0) w 1.0)))))))

C

double code(double w, double l) {
	double t_0 = fma((0.5 * w), w, 1.0);
	double tmp;
	if (w <= -5.6e+145) {
		tmp = t_0 * pow(l, 1.0);
	} else if (w <= -3.7e+62) {
		tmp = t_0 * pow(l, (1.0 + w));
	} else {
		tmp = (1.0 - w) * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = fma(Float64(0.5 * w), w, 1.0)
	tmp = 0.0
	if (w <= -5.6e+145)
		tmp = Float64(t_0 * (l ^ 1.0));
	elseif (w <= -3.7e+62)
		tmp = Float64(t_0 * (l ^ Float64(1.0 + w)));
	else
		tmp = Float64(Float64(1.0 - w) * (l ^ fma(fma(0.5, w, 1.0), w, 1.0)));
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(0.5 * w), $MachinePrecision] * w + 1.0), $MachinePrecision]}, If[LessEqual[w, -5.6e+145], N[(t$95$0 * N[Power[l, 1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[w, -3.7e+62], N[(t$95$0 * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if w < -5.5999999999999997e145

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 97.6%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 100.0%

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

   if -5.5999999999999997e145 < w < -3.70000000000000014e62

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 5.7

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 5.7%

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

   9. Taylor expanded in w around 0

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

       1. lower-+.f64 65.5

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

   11. Applied rewrites 65.5%

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

   if -3.70000000000000014e62 < w

   1. Initial program 99.4%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 83.7

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

   8. Applied rewrites 83.7%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 85.4%

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

   12. Taylor expanded in w around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 94.6

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

   14. Applied rewrites 94.6%

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

Alternative 8: 91.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\\ \mathbf{if}\;\ell \leq 0.52:\\ \;\;\;\;\frac{{\ell}^{\left(1 + w\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(1 - w\right) \cdot {\ell}^{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma (fma 0.5 w 1.0) w 1.0)))
   (if (<= l 0.52) (/ (pow l (+ 1.0 w)) t_0) (* (- 1.0 w) (pow l t_0)))))

C

double code(double w, double l) {
	double t_0 = fma(fma(0.5, w, 1.0), w, 1.0);
	double tmp;
	if (l <= 0.52) {
		tmp = pow(l, (1.0 + w)) / t_0;
	} else {
		tmp = (1.0 - w) * pow(l, t_0);
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = fma(fma(0.5, w, 1.0), w, 1.0)
	tmp = 0.0
	if (l <= 0.52)
		tmp = Float64((l ^ Float64(1.0 + w)) / t_0);
	else
		tmp = Float64(Float64(1.0 - w) * (l ^ t_0));
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]}, If[LessEqual[l, 0.52], N[(N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(1.0 - w), $MachinePrecision] * N[Power[l, t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 0.52000000000000002

   1. Initial program 99.6%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. lift-exp.f64 N/A

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

      7. associate-*r/ N/A

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

      8. *-rgt-identity N/A

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

      9. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 77.5

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

   7. Applied rewrites 77.5%

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

   8. Taylor expanded in w around 0

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

      1. lower-+.f64 89.2

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

   10. Applied rewrites 89.2%

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

   if 0.52000000000000002 < l

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 60.2

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

   8. Applied rewrites 60.2%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 97.4%

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

   12. Taylor expanded in w around 0

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

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

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

       2. metadata-eval N/A

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

       3. *-lft-identity N/A

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

       4. lower--.f64 98.3

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

   14. Applied rewrites 98.3%

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

Alternative 9: 87.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot w, w, 1\right)\\ \mathbf{if}\;\ell \leq 0.7:\\ \;\;\;\;t\_0 \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\ell}^{1}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma (* 0.5 w) w 1.0)))
   (if (<= l 0.7) (* t_0 (pow l (+ 1.0 w))) (* t_0 (pow l 1.0)))))

C

double code(double w, double l) {
	double t_0 = fma((0.5 * w), w, 1.0);
	double tmp;
	if (l <= 0.7) {
		tmp = t_0 * pow(l, (1.0 + w));
	} else {
		tmp = t_0 * pow(l, 1.0);
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = fma(Float64(0.5 * w), w, 1.0)
	tmp = 0.0
	if (l <= 0.7)
		tmp = Float64(t_0 * (l ^ Float64(1.0 + w)));
	else
		tmp = Float64(t_0 * (l ^ 1.0));
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[(N[(0.5 * w), $MachinePrecision] * w + 1.0), $MachinePrecision]}, If[LessEqual[l, 0.7], N[(t$95$0 * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[l, 1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if l < 0.69999999999999996

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 74.0%

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

   9. Taylor expanded in w around 0

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

       1. lower-+.f64 86.3

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

   11. Applied rewrites 86.3%

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

   if 0.69999999999999996 < l

   1. Initial program 99.6%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 79.7

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

   5. Applied rewrites 79.7%

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

   6. Taylor expanded in w around inf

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

   8. Applied rewrites 78.5%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 86.5%

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

Alternative 10: 74.6% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in w around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 76.9

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

5. Applied rewrites 76.9%

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

6. Taylor expanded in w around inf

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

8. Applied rewrites 75.7%

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

9. Taylor expanded in w around 0

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

11. Applied rewrites 71.7%

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

Alternative 11: 57.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in l around inf

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. mul-1-neg N/A

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

   6. log-rec N/A

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

   7. remove-double-neg N/A

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

   8. lower-exp.f64 N/A

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

   9. lower-log.f64 N/A

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

   10. lower-exp.f64 94.9

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

5. Applied rewrites 94.9%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in w around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. mul-1-neg N/A

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

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

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. lower--.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-log.f64 56.2

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

10. Applied rewrites 56.2%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log \ell \cdot \ell - \ell, w, \ell\right)} \]
11. Add Preprocessing

Alternative 12: 3.6% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (w l) :precision binary64 (* (* l (- (log l) 1.0)) w))

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in w around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-log.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 56.2

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

5. Applied rewrites 56.2%

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

6. Taylor expanded in w around inf

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

8. Applied rewrites 3.8%

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

10. Applied rewrites 3.8%

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

Reproduce

herbie shell --seed 2024350 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))