exp-w (used to crash)

Percentage Accurate: 99.4% → 99.1%

Time: 16.9s

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. w = 2.487911354678651e-191
  2. l = -3.421440353515561e-137

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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-w}\\ \mathbf{if}\;t\_0 \cdot {\ell}^{\left(e^{w}\right)} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t\_0 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \ell, e^{w}, -w\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (exp (- w))))
   (if (<= (* t_0 (pow l (exp w))) 5e+303)
     (* t_0 (pow l (fma (fma 0.5 w 1.0) w 1.0)))
     (exp (fma (log l) (exp w) (- w))))))

C

double code(double w, double l) {
	double t_0 = exp(-w);
	double tmp;
	if ((t_0 * pow(l, exp(w))) <= 5e+303) {
		tmp = t_0 * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	} else {
		tmp = exp(fma(log(l), exp(w), -w));
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = exp(Float64(-w))
	tmp = 0.0
	if (Float64(t_0 * (l ^ exp(w))) <= 5e+303)
		tmp = Float64(t_0 * (l ^ fma(fma(0.5, w, 1.0), w, 1.0)));
	else
		tmp = exp(fma(log(l), exp(w), Float64(-w)));
	end
	return tmp
end

Wolfram

code[w_, l_] := Block[{t$95$0 = N[Exp[(-w)], $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Power[l, N[Exp[w], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 5e+303], N[(t$95$0 * N[Power[l, N[(N[(0.5 * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Log[l], $MachinePrecision] * N[Exp[w], $MachinePrecision] + (-w)), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w))) < 4.9999999999999997e303

   1. Initial program 99.4%

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

   3. Taylor expanded in w around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. +-commutative N/A

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

      4. distribute-rgt1-in N/A

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

      5. distribute-rgt1-in N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 98.5

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

   5. Applied rewrites 98.5%

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

   if 4.9999999999999997e303 < (*.f64 (exp.f64 (neg.f64 w)) (pow.f64 l (exp.f64 w)))

   1. Initial program 97.8%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 100.0

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

   4. Applied rewrites 100.0%

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

Alternative 2: 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 98.8%

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

Alternative 3: 99.0% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 1.0)
   (* (exp (- 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 <= 1.0) {
		tmp = exp(-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 <= 1.0)
		tmp = Float64(exp(Float64(-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, 1.0], N[(N[Exp[(-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 < 1

   1. Initial program 99.4%

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

   3. Taylor expanded in w around 0

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

      1. lower-+.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if 1 < l

   1. Initial program 98.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 80.8

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

   5. Applied rewrites 80.8%

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

   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 99.3

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

   8. Applied rewrites 99.3%

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

Alternative 4: 96.6% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 0.5)
   (exp (fma (log l) (+ 1.0 w) (- 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 <= 0.5) {
		tmp = exp(fma(log(l), (1.0 + w), -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 <= 0.5)
		tmp = exp(fma(log(l), Float64(1.0 + w), Float64(-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, 0.5], N[Exp[N[(N[Log[l], $MachinePrecision] * N[(1.0 + w), $MachinePrecision] + (-w)), $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 < 0.5

   1. Initial program 99.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in w around 0

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

      1. lower-+.f64 94.9

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

   7. Applied rewrites 94.9%

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

   if 0.5 < l

   1. Initial program 98.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 80.8

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

   5. Applied rewrites 80.8%

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

   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 99.3

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

   8. Applied rewrites 99.3%

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

Alternative 5: 91.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right)\\ \mathbf{if}\;\ell \leq 0.38:\\ \;\;\;\;t\_0 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, w, 0.5\right), w, 1\right), w, 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma (- (* 0.5 w) 1.0) w 1.0)))
   (if (<= l 0.38)
     (* t_0 (pow l (fma (fma (fma 0.16666666666666666 w 0.5) w 1.0) w 1.0)))
     (* t_0 (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) - 1.0), w, 1.0);
	double tmp;
	if (l <= 0.38) {
		tmp = t_0 * pow(l, fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0));
	} else {
		tmp = t_0 * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = fma(Float64(Float64(0.5 * w) - 1.0), w, 1.0)
	tmp = 0.0
	if (l <= 0.38)
		tmp = Float64(t_0 * (l ^ fma(fma(fma(0.16666666666666666, w, 0.5), w, 1.0), w, 1.0)));
	else
		tmp = Float64(t_0 * (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[(N[(0.5 * w), $MachinePrecision] - 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]}, If[LessEqual[l, 0.38], N[(t$95$0 * N[Power[l, N[(N[(N[(0.16666666666666666 * w + 0.5), $MachinePrecision] * w + 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$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 < 0.38

   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 70.2

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

   5. Applied rewrites 70.2%

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

   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 84.5

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

   8. Applied rewrites 84.5%

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

   if 0.38 < l

   1. Initial program 98.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot w - 1, w, 1\right) \cdot {\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + \frac{1}{2} \cdot w\right)\right)}} \]
   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 98.5

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

   8. Applied rewrites 98.5%

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

Alternative 6: 91.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right)\\ \mathbf{if}\;\ell \leq 0.38:\\ \;\;\;\;t\_0 \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (let* ((t_0 (fma (- (* 0.5 w) 1.0) w 1.0)))
   (if (<= l 0.38)
     (* t_0 (pow l (+ 1.0 w)))
     (* t_0 (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) - 1.0), w, 1.0);
	double tmp;
	if (l <= 0.38) {
		tmp = t_0 * pow(l, (1.0 + w));
	} else {
		tmp = t_0 * pow(l, fma(fma(0.5, w, 1.0), w, 1.0));
	}
	return tmp;
}

Julia

function code(w, l)
	t_0 = fma(Float64(Float64(0.5 * w) - 1.0), w, 1.0)
	tmp = 0.0
	if (l <= 0.38)
		tmp = Float64(t_0 * (l ^ Float64(1.0 + w)));
	else
		tmp = Float64(t_0 * (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[(N[(0.5 * w), $MachinePrecision] - 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision]}, If[LessEqual[l, 0.38], N[(t$95$0 * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$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 < 0.38

   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 70.2

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

   5. Applied rewrites 70.2%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 84.1

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

   8. Applied rewrites 84.1%

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

   if 0.38 < l

   1. Initial program 98.1%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 80.1

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

   5. Applied rewrites 80.1%

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

   6. Taylor expanded in w around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2} \cdot w - 1, w, 1\right) \cdot {\ell}^{\color{blue}{\left(1 + w \cdot \left(1 + \frac{1}{2} \cdot w\right)\right)}} \]
   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 98.5

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

   8. Applied rewrites 98.5%

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

Alternative 7: 75.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -7 \cdot 10^{+230} \lor \neg \left(w \leq -3 \cdot 10^{+162}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right) \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \ell \cdot \ell, w, \left(-\ell\right) \cdot w\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (or (<= w -7e+230) (not (<= w -3e+162)))
   (* (fma (- (* 0.5 w) 1.0) w 1.0) (pow l (+ 1.0 w)))
   (fma (* (log l) l) w (* (- l) w))))

C

double code(double w, double l) {
	double tmp;
	if ((w <= -7e+230) || !(w <= -3e+162)) {
		tmp = fma(((0.5 * w) - 1.0), w, 1.0) * pow(l, (1.0 + w));
	} else {
		tmp = fma((log(l) * l), w, (-l * w));
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if ((w <= -7e+230) || !(w <= -3e+162))
		tmp = Float64(fma(Float64(Float64(0.5 * w) - 1.0), w, 1.0) * (l ^ Float64(1.0 + w)));
	else
		tmp = fma(Float64(log(l) * l), w, Float64(Float64(-l) * w));
	end
	return tmp
end

Wolfram

code[w_, l_] := If[Or[LessEqual[w, -7e+230], N[Not[LessEqual[w, -3e+162]], $MachinePrecision]], N[(N[(N[(N[(0.5 * w), $MachinePrecision] - 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision] * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[l], $MachinePrecision] * l), $MachinePrecision] * w + N[((-l) * w), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -7.0000000000000001e230 or -2.9999999999999998e162 < w

   1. Initial program 98.7%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\left(1 + w \cdot \left(\frac{1}{2} \cdot w - 1\right)\right)} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 72.8

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 75.4

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

   8. Applied rewrites 75.4%

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

   if -7.0000000000000001e230 < w < -2.9999999999999998e162

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \ell + \ell \cdot \log \ell\right)} \]
   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 0.6

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

   5. Applied rewrites 0.6%

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

   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 0.6%

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

   10. Applied rewrites 61.7%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;w \leq -7 \cdot 10^{+230} \lor \neg \left(w \leq -3 \cdot 10^{+162}\right):\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right) \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\log \ell \cdot \ell, w, \left(-\ell\right) \cdot w\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\ell \leq 1.5 \cdot 10^{-51}:\\ \;\;\;\;\mathsf{fma}\left(0.5 \cdot w - 1, w, 1\right) \cdot {\ell}^{\left(1 + w\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, w, 1\right) \cdot {\ell}^{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, w, 1\right), w, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= l 1.5e-51)
   (* (fma (- (* 0.5 w) 1.0) w 1.0) (pow l (+ 1.0 w)))
   (* (fma -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 <= 1.5e-51) {
		tmp = fma(((0.5 * w) - 1.0), w, 1.0) * pow(l, (1.0 + w));
	} else {
		tmp = fma(-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 <= 1.5e-51)
		tmp = Float64(fma(Float64(Float64(0.5 * w) - 1.0), w, 1.0) * (l ^ Float64(1.0 + w)));
	else
		tmp = Float64(fma(-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, 1.5e-51], N[(N[(N[(N[(0.5 * w), $MachinePrecision] - 1.0), $MachinePrecision] * w + 1.0), $MachinePrecision] * N[Power[l, N[(1.0 + w), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * 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 < 1.50000000000000001e-51

   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 69.2

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

   5. Applied rewrites 69.2%

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

   6. Taylor expanded in w around 0

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

      1. lower-+.f64 84.9

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

   8. Applied rewrites 84.9%

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

   if 1.50000000000000001e-51 < l

   1. Initial program 98.4%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 79.0

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

   5. Applied rewrites 79.0%

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

   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 92.5

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

   8. Applied rewrites 92.5%

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

   9. Taylor expanded in w around 0

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

   11. Applied rewrites 94.6%

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

Alternative 9: 62.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;w \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\log \ell \cdot \ell, w, \left(-\ell\right) \cdot w\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\log \ell, \ell, -\ell\right), w, \ell\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (w l)
 :precision binary64
 (if (<= w -2e+64)
   (fma (* (log l) l) w (* (- l) w))
   (fma (fma (log l) l (- l)) w l)))

C

double code(double w, double l) {
	double tmp;
	if (w <= -2e+64) {
		tmp = fma((log(l) * l), w, (-l * w));
	} else {
		tmp = fma(fma(log(l), l, -l), w, l);
	}
	return tmp;
}

Julia

function code(w, l)
	tmp = 0.0
	if (w <= -2e+64)
		tmp = fma(Float64(log(l) * l), w, Float64(Float64(-l) * w));
	else
		tmp = fma(fma(log(l), l, Float64(-l)), w, l);
	end
	return tmp
end

Wolfram

code[w_, l_] := If[LessEqual[w, -2e+64], N[(N[(N[Log[l], $MachinePrecision] * l), $MachinePrecision] * w + N[((-l) * w), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[l], $MachinePrecision] * l + (-l)), $MachinePrecision] * w + l), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if w < -2.00000000000000004e64

   1. Initial program 100.0%

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

   3. Taylor expanded in w around 0

      \[\leadsto \color{blue}{\ell + w \cdot \left(-1 \cdot \ell + \ell \cdot \log \ell\right)} \]
   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 1.6

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

   5. Applied rewrites 1.6%

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

   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 1.6%

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

   10. Applied rewrites 31.7%

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

   if -2.00000000000000004e64 < w

   1. Initial program 98.3%

      \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
   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 70.7

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

   5. Applied rewrites 70.7%

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

Alternative 10: 56.1% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 51.0

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

5. Applied rewrites 51.0%

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

Alternative 11: 56.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 51.0

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

5. Applied rewrites 51.0%

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

6. Taylor expanded in l around 0

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

8. Applied rewrites 51.0%

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

Alternative 12: 3.6% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.8%

   \[e^{-w} \cdot {\ell}^{\left(e^{w}\right)} \]
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 51.0

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

5. Applied rewrites 51.0%

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

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 3.1%

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

Reproduce

herbie shell --seed 2024364 
(FPCore (w l)
  :name "exp-w (used to crash)"
  :precision binary64
  (* (exp (- w)) (pow l (exp w))))