NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.2% → 99.9%

Time: 5.9s

Alternatives: 13

Speedup: 2.0×

• 62.0% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -4.9674460008004655e+238
  2. eps = -1.825316277775522e-303

Specification

Math

\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 73.2% accurate, 1.0× speedup

Math

\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (/
  (-
   (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x))))
   (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x)))))
  2.0))

C

double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (((1.0d0 + (1.0d0 / eps)) * exp(-((1.0d0 - eps) * x))) - (((1.0d0 / eps) - 1.0d0) * exp(-((1.0d0 + eps) * x)))) / 2.0d0
end function

Java

public static double code(double x, double eps) {
	return (((1.0 + (1.0 / eps)) * Math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * Math.exp(-((1.0 + eps) * x)))) / 2.0;
}

Python

def code(x, eps):
	return (((1.0 + (1.0 / eps)) * math.exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * math.exp(-((1.0 + eps) * x)))) / 2.0

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * exp(Float64(-Float64(Float64(1.0 - eps) * x)))) - Float64(Float64(Float64(1.0 / eps) - 1.0) * exp(Float64(-Float64(Float64(1.0 + eps) * x))))) / 2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (((1.0 + (1.0 / eps)) * exp(-((1.0 - eps) * x))) - (((1.0 / eps) - 1.0) * exp(-((1.0 + eps) * x)))) / 2.0;
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[(1.0 - eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(1.0 / eps), $MachinePrecision] - 1.0), $MachinePrecision] * N[Exp[(-N[(N[(1.0 + eps), $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 2.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{-\left|\varepsilon\right| \cdot x} + e^{\left(\left|\varepsilon\right| - 1\right) \cdot x}\right) \cdot 0.5\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 2.7e-18)
   (/ (* (fma 2.0 x 2.0) 0.5) (exp x))
   (* (+ (exp (- (* (fabs eps) x))) (exp (* (- (fabs eps) 1.0) x))) 0.5)))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 2.7e-18) {
		tmp = (fma(2.0, x, 2.0) * 0.5) / exp(x);
	} else {
		tmp = (exp(-(fabs(eps) * x)) + exp(((fabs(eps) - 1.0) * x))) * 0.5;
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 2.7e-18)
		tmp = Float64(Float64(fma(2.0, x, 2.0) * 0.5) / exp(x));
	else
		tmp = Float64(Float64(exp(Float64(-Float64(abs(eps) * x))) + exp(Float64(Float64(abs(eps) - 1.0) * x))) * 0.5);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 2.7e-18], N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[(-N[(N[Abs[eps], $MachinePrecision] * x), $MachinePrecision])], $MachinePrecision] + N[Exp[N[(N[(N[Abs[eps], $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 2.69999999999999989e-18

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, e^{-x}, -1 \cdot \left(x \cdot e^{-x}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.7%

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

   6. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 57.7%

         \[\leadsto \frac{\left(\left(x - -1\right) + \left(x - -1\right)\right) \cdot 0.5}{e^{\color{blue}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) + \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      7. count-2 N/A

         \[\leadsto \frac{\left(2 \cdot \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      9. sub-flip N/A

         \[\leadsto \frac{\left(2 \cdot \left(x + \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot \left(x + 1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      11. distribute-lft-in N/A

          \[\leadsto \frac{\left(2 \cdot x + 2 \cdot 1\right) \cdot \frac{1}{2}}{e^{x}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot x + 2\right) \cdot \frac{1}{2}}{e^{x}} \]

      13. lower-fma.f64 57.7%

          \[\leadsto \frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{e^{x}} \]

   8. Applied rewrites 57.7%

      \[\leadsto \frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{\color{blue}{e^{x}}} \]

   if 2.69999999999999989e-18 < eps

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

         \[\leadsto \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \cdot \color{blue}{0.5} \]

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in eps around inf

      \[\leadsto \left(e^{-\varepsilon \cdot x} + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. lower-*.f64 88.9%

         \[\leadsto \left(e^{-\varepsilon \cdot x} + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot 0.5 \]

   9. Applied rewrites 88.9%

      \[\leadsto \left(e^{-\varepsilon \cdot x} + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot 0.5 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.1% accurate, 1.5× speedup

Math

\[\left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot 0.5 \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (+ (exp (- (fma x eps x))) (exp (* (- eps 1.0) x))) 0.5))

C

double code(double x, double eps) {
	return (exp(-fma(x, eps, x)) + exp(((eps - 1.0) * x))) * 0.5;
}

Julia

function code(x, eps)
	return Float64(Float64(exp(Float64(-fma(x, eps, x))) + exp(Float64(Float64(eps - 1.0) * x))) * 0.5)
end

Wolfram

code[x_, eps_] := N[(N[(N[Exp[(-N[(x * eps + x), $MachinePrecision])], $MachinePrecision] + N[Exp[N[(N[(eps - 1.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 73.2%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

   3. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   5. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   6. lower--.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

   8. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

   9. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   11. lower-+.f64 99.1%

       \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

4. Applied rewrites 99.1%

   \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.1%

      \[\leadsto \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \cdot \color{blue}{0.5} \]

6. Applied rewrites 99.1%

   \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + e^{\left(\varepsilon - 1\right) \cdot x}\right) \cdot \color{blue}{0.5} \]
7. Add Preprocessing

Alternative 3: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-260}:\\ \;\;\;\;\left(e^{-\mathsf{fma}\left(x, \left|\varepsilon\right|, x\right)} + \left(1 + x \cdot -1\right)\right) \cdot 0.5\\ \mathbf{elif}\;x \leq 420:\\ \;\;\;\;0.5 \cdot \left(t\_0 - \left(x \cdot \left(1 + \left|\varepsilon\right|\right) - 1\right)\right)\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(t\_0 - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (- (* x (- 1.0 (fabs eps)))))))
   (if (<= x -2e-260)
     (* (+ (exp (- (fma x (fabs eps) x))) (+ 1.0 (* x -1.0))) 0.5)
     (if (<= x 420.0)
       (* 0.5 (- t_0 (- (* x (+ 1.0 (fabs eps))) 1.0)))
       (if (<= x 6.6e+87)
         (/ (* (fma 2.0 x 2.0) 0.5) (exp x))
         (* 0.5 (- t_0 -1.0)))))))

C

double code(double x, double eps) {
	double t_0 = exp(-(x * (1.0 - fabs(eps))));
	double tmp;
	if (x <= -2e-260) {
		tmp = (exp(-fma(x, fabs(eps), x)) + (1.0 + (x * -1.0))) * 0.5;
	} else if (x <= 420.0) {
		tmp = 0.5 * (t_0 - ((x * (1.0 + fabs(eps))) - 1.0));
	} else if (x <= 6.6e+87) {
		tmp = (fma(2.0, x, 2.0) * 0.5) / exp(x);
	} else {
		tmp = 0.5 * (t_0 - -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	t_0 = exp(Float64(-Float64(x * Float64(1.0 - abs(eps)))))
	tmp = 0.0
	if (x <= -2e-260)
		tmp = Float64(Float64(exp(Float64(-fma(x, abs(eps), x))) + Float64(1.0 + Float64(x * -1.0))) * 0.5);
	elseif (x <= 420.0)
		tmp = Float64(0.5 * Float64(t_0 - Float64(Float64(x * Float64(1.0 + abs(eps))) - 1.0)));
	elseif (x <= 6.6e+87)
		tmp = Float64(Float64(fma(2.0, x, 2.0) * 0.5) / exp(x));
	else
		tmp = Float64(0.5 * Float64(t_0 - -1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[x, -2e-260], N[(N[(N[Exp[(-N[(x * N[Abs[eps], $MachinePrecision] + x), $MachinePrecision])], $MachinePrecision] + N[(1.0 + N[(x * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[x, 420.0], N[(0.5 * N[(t$95$0 - N[(N[(x * N[(1.0 + N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e+87], N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(t$95$0 - -1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.99999999999999992e-260

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

         \[\leadsto \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \cdot \color{blue}{0.5} \]

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot \frac{1}{2} \]

      3. lower--.f64 64.5%

         \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot 0.5 \]

   9. Applied rewrites 64.5%

      \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot 0.5 \]

   10. Taylor expanded in eps around 0

       \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot -1\right)\right) \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 64.1%

       \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot -1\right)\right) \cdot 0.5 \]

   if -1.99999999999999992e-260 < x < 420

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \]

      3. lower-+.f64 63.9%

         \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - \left(x \cdot \left(1 + \varepsilon\right) - 1\right)\right) \]

   7. Applied rewrites 63.9%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - \left(x \cdot \left(1 + \varepsilon\right) - \color{blue}{1}\right)\right) \]

   if 420 < x < 6.6000000000000003e87

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, e^{-x}, -1 \cdot \left(x \cdot e^{-x}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.7%

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

   6. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 57.7%

         \[\leadsto \frac{\left(\left(x - -1\right) + \left(x - -1\right)\right) \cdot 0.5}{e^{\color{blue}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) + \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      7. count-2 N/A

         \[\leadsto \frac{\left(2 \cdot \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      9. sub-flip N/A

         \[\leadsto \frac{\left(2 \cdot \left(x + \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot \left(x + 1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      11. distribute-lft-in N/A

          \[\leadsto \frac{\left(2 \cdot x + 2 \cdot 1\right) \cdot \frac{1}{2}}{e^{x}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot x + 2\right) \cdot \frac{1}{2}}{e^{x}} \]

      13. lower-fma.f64 57.7%

          \[\leadsto \frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{e^{x}} \]

   8. Applied rewrites 57.7%

      \[\leadsto \frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{\color{blue}{e^{x}}} \]

   if 6.6000000000000003e87 < x

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.7%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-260}:\\ \;\;\;\;\left(e^{-\mathsf{fma}\left(x, \left|\varepsilon\right|, x\right)} + \left(1 + x \cdot -1\right)\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)} - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -2e-260)
   (* (+ (exp (- (fma x (fabs eps) x))) (+ 1.0 (* x -1.0))) 0.5)
   (* 0.5 (- (exp (- (* x (- 1.0 (fabs eps))))) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -2e-260) {
		tmp = (exp(-fma(x, fabs(eps), x)) + (1.0 + (x * -1.0))) * 0.5;
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - fabs(eps)))) - -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -2e-260)
		tmp = Float64(Float64(exp(Float64(-fma(x, abs(eps), x))) + Float64(1.0 + Float64(x * -1.0))) * 0.5);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - abs(eps))))) - -1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -2e-260], N[(N[(N[Exp[(-N[(x * N[Abs[eps], $MachinePrecision] + x), $MachinePrecision])], $MachinePrecision] + N[(1.0 + N[(x * -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.99999999999999992e-260

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.1%

         \[\leadsto \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \cdot \color{blue}{0.5} \]

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in x around 0

      \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot 0.5 \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot \frac{1}{2} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot \frac{1}{2} \]

      3. lower--.f64 64.5%

         \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot 0.5 \]

   9. Applied rewrites 64.5%

      \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot \left(\varepsilon - 1\right)\right)\right) \cdot 0.5 \]

   10. Taylor expanded in eps around 0

       \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot -1\right)\right) \cdot 0.5 \]
   11. Step-by-step derivation

   12. Applied rewrites 64.1%

       \[\leadsto \left(e^{-\mathsf{fma}\left(x, \varepsilon, x\right)} + \left(1 + x \cdot -1\right)\right) \cdot 0.5 \]

   if -1.99999999999999992e-260 < x

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.7%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 78.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -600:\\ \;\;\;\;\frac{\frac{e^{-x}}{\left|\varepsilon\right|} - \left(\frac{1}{\left|\varepsilon\right|} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)} - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x -600.0)
   (/ (- (/ (exp (- x)) (fabs eps)) (- (/ 1.0 (fabs eps)) 1.0)) 2.0)
   (* 0.5 (- (exp (- (* x (- 1.0 (fabs eps))))) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= -600.0) {
		tmp = ((exp(-x) / fabs(eps)) - ((1.0 / fabs(eps)) - 1.0)) / 2.0;
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - fabs(eps)))) - -1.0);
	}
	return tmp;
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= (-600.0d0)) then
        tmp = ((exp(-x) / abs(eps)) - ((1.0d0 / abs(eps)) - 1.0d0)) / 2.0d0
    else
        tmp = 0.5d0 * (exp(-(x * (1.0d0 - abs(eps)))) - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (x <= -600.0) {
		tmp = ((Math.exp(-x) / Math.abs(eps)) - ((1.0 / Math.abs(eps)) - 1.0)) / 2.0;
	} else {
		tmp = 0.5 * (Math.exp(-(x * (1.0 - Math.abs(eps)))) - -1.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if x <= -600.0:
		tmp = ((math.exp(-x) / math.fabs(eps)) - ((1.0 / math.fabs(eps)) - 1.0)) / 2.0
	else:
		tmp = 0.5 * (math.exp(-(x * (1.0 - math.fabs(eps)))) - -1.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= -600.0)
		tmp = Float64(Float64(Float64(exp(Float64(-x)) / abs(eps)) - Float64(Float64(1.0 / abs(eps)) - 1.0)) / 2.0);
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - abs(eps))))) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= -600.0)
		tmp = ((exp(-x) / abs(eps)) - ((1.0 / abs(eps)) - 1.0)) / 2.0;
	else
		tmp = 0.5 * (exp(-(x * (1.0 - abs(eps)))) - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[x, -600.0], N[(N[(N[(N[Exp[(-x)], $MachinePrecision] / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[Abs[eps], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -600

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in x around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 37.6%

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   4. Applied rewrites 37.6%

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

   5. Taylor expanded in eps around 0

      \[\leadsto \frac{\color{blue}{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(x\right)}}{\color{blue}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

      2. lower-exp.f64 N/A

         \[\leadsto \frac{\frac{e^{\mathsf{neg}\left(x\right)}}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

      3. lower-neg.f64 12.1%

         \[\leadsto \frac{\frac{e^{-x}}{\varepsilon} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   7. Applied rewrites 12.1%

      \[\leadsto \frac{\color{blue}{\frac{e^{-x}}{\varepsilon}} - \left(\frac{1}{\varepsilon} - 1\right)}{2} \]

   if -600 < x

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.7%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 0.0005:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(e^{-x \cdot \left(1 - \left|\varepsilon\right|\right)} - -1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 0.0005)
   (/ (* (fma 2.0 x 2.0) 0.5) (exp x))
   (* 0.5 (- (exp (- (* x (- 1.0 (fabs eps))))) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 0.0005) {
		tmp = (fma(2.0, x, 2.0) * 0.5) / exp(x);
	} else {
		tmp = 0.5 * (exp(-(x * (1.0 - fabs(eps)))) - -1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 0.0005)
		tmp = Float64(Float64(fma(2.0, x, 2.0) * 0.5) / exp(x));
	else
		tmp = Float64(0.5 * Float64(exp(Float64(-Float64(x * Float64(1.0 - abs(eps))))) - -1.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 0.0005], N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(N[Exp[(-N[(x * N[(1.0 - N[Abs[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 5.0000000000000001e-4

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, e^{-x}, -1 \cdot \left(x \cdot e^{-x}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.7%

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

   6. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 57.7%

         \[\leadsto \frac{\left(\left(x - -1\right) + \left(x - -1\right)\right) \cdot 0.5}{e^{\color{blue}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) + \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      7. count-2 N/A

         \[\leadsto \frac{\left(2 \cdot \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      9. sub-flip N/A

         \[\leadsto \frac{\left(2 \cdot \left(x + \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot \left(x + 1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      11. distribute-lft-in N/A

          \[\leadsto \frac{\left(2 \cdot x + 2 \cdot 1\right) \cdot \frac{1}{2}}{e^{x}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot x + 2\right) \cdot \frac{1}{2}}{e^{x}} \]

      13. lower-fma.f64 57.7%

          \[\leadsto \frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{e^{x}} \]

   8. Applied rewrites 57.7%

      \[\leadsto \frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{\color{blue}{e^{x}}} \]

   if 5.0000000000000001e-4 < eps

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 63.7%

      \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 68.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{1 \cdot 1 - x \cdot x}{x - -1}\\ \mathbf{elif}\;x \leq 6.6 \cdot 10^{+87}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.66)
   (/ (- (* 1.0 1.0) (* x x)) (- x -1.0))
   (if (<= x 6.6e+87)
     (/ x (exp x))
     (fma (fma 0.3333333333333333 x -0.5) (sqrt (* (* x x) (* x x))) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.66) {
		tmp = ((1.0 * 1.0) - (x * x)) / (x - -1.0);
	} else if (x <= 6.6e+87) {
		tmp = x / exp(x);
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), sqrt(((x * x) * (x * x))), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(x * x)) / Float64(x - -1.0));
	elseif (x <= 6.6e+87)
		tmp = Float64(x / exp(x));
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), sqrt(Float64(Float64(x * x) * Float64(x * x))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.66], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.6e+87], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.660000000000000031

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.4%

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.4%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

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

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

      4. flip-- N/A

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

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + x} \]

      10. lower-unsound-+.f32 N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + x} \]

      11. lower-+.f32 N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{x + 1} \]

      13. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{x + \left(\mathsf{neg}\left(-1\right)\right)} \]

      14. sub-flip N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{x - -1} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{x - -1} \]

      16. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 50.1%

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

   if 0.660000000000000031 < x < 6.6000000000000003e87

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, e^{-x}, -1 \cdot \left(x \cdot e^{-x}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.7%

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

   6. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot 0.5} \]

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{x}} \]

      2. lower-exp.f64 16.5%

         \[\leadsto \frac{x}{e^{x}} \]

   9. Applied rewrites 16.5%

      \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]

   if 6.6000000000000003e87 < x

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \color{blue}{\frac{1}{2}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

      5. lower-*.f64 52.7%

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.7%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

      3. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      5. lower-fma.f64 52.7%

         \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, {x}^{\color{blue}{2}}, 1\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \frac{-1}{2}, {x}^{2}, 1\right) \]

      10. lower-fma.f64 52.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), {x}^{2}, 1\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), x \cdot x, 1\right) \]

      13. lower-*.f64 52.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right) \]

   9. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), \sqrt{x \cdot x} \cdot \sqrt{x \cdot x}, 1\right) \]

       2. sqrt-unprod N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right) \]

       4. lower-*.f64 53.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right) \]

   11. Applied rewrites 53.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 64.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|\varepsilon\right| \leq 0.108:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (fabs eps) 0.108)
   (/ (* (fma 2.0 x 2.0) 0.5) (exp x))
   (fma (fma 0.3333333333333333 x -0.5) (sqrt (* (* x x) (* x x))) 1.0)))

C

double code(double x, double eps) {
	double tmp;
	if (fabs(eps) <= 0.108) {
		tmp = (fma(2.0, x, 2.0) * 0.5) / exp(x);
	} else {
		tmp = fma(fma(0.3333333333333333, x, -0.5), sqrt(((x * x) * (x * x))), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (abs(eps) <= 0.108)
		tmp = Float64(Float64(fma(2.0, x, 2.0) * 0.5) / exp(x));
	else
		tmp = fma(fma(0.3333333333333333, x, -0.5), sqrt(Float64(Float64(x * x) * Float64(x * x))), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[Abs[eps], $MachinePrecision], 0.108], N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] * 0.5), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * N[Sqrt[N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < 0.107999999999999999

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, e^{-x}, -1 \cdot \left(x \cdot e^{-x}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.7%

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

   6. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot 0.5} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

         \[\leadsto \frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 57.7%

         \[\leadsto \frac{\left(\left(x - -1\right) + \left(x - -1\right)\right) \cdot 0.5}{e^{\color{blue}{x}}} \]

      6. lift-+.f64 N/A

         \[\leadsto \frac{\left(\left(x - -1\right) + \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      7. count-2 N/A

         \[\leadsto \frac{\left(2 \cdot \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      8. lift--.f64 N/A

         \[\leadsto \frac{\left(2 \cdot \left(x - -1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      9. sub-flip N/A

         \[\leadsto \frac{\left(2 \cdot \left(x + \left(\mathsf{neg}\left(-1\right)\right)\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot \left(x + 1\right)\right) \cdot \frac{1}{2}}{e^{x}} \]

      11. distribute-lft-in N/A

          \[\leadsto \frac{\left(2 \cdot x + 2 \cdot 1\right) \cdot \frac{1}{2}}{e^{x}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(2 \cdot x + 2\right) \cdot \frac{1}{2}}{e^{x}} \]

      13. lower-fma.f64 57.7%

          \[\leadsto \frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{e^{x}} \]

   8. Applied rewrites 57.7%

      \[\leadsto \frac{\mathsf{fma}\left(2, x, 2\right) \cdot 0.5}{\color{blue}{e^{x}}} \]

   if 0.107999999999999999 < eps

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \color{blue}{\frac{1}{2}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

      5. lower-*.f64 52.7%

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.7%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

      3. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      5. lower-fma.f64 52.7%

         \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, {x}^{\color{blue}{2}}, 1\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \frac{-1}{2}, {x}^{2}, 1\right) \]

      10. lower-fma.f64 52.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), {x}^{2}, 1\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), x \cdot x, 1\right) \]

      13. lower-*.f64 52.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right) \]

   9. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)} \]
   10. Step-by-step derivation

       1. rem-square-sqrt N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), \sqrt{x \cdot x} \cdot \sqrt{x \cdot x}, 1\right) \]

       2. sqrt-unprod N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right) \]

       4. lower-*.f64 53.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right) \]

   11. Applied rewrites 53.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), \sqrt{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 63.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;\frac{1 \cdot 1 - x \cdot x}{x - -1}\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+95}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.66)
   (/ (- (* 1.0 1.0) (* x x)) (- x -1.0))
   (if (<= x 1.22e+95)
     (/ x (exp x))
     (fma (* 0.3333333333333333 x) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.66) {
		tmp = ((1.0 * 1.0) - (x * x)) / (x - -1.0);
	} else if (x <= 1.22e+95) {
		tmp = x / exp(x);
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64(Float64(Float64(1.0 * 1.0) - Float64(x * x)) / Float64(x - -1.0));
	elseif (x <= 1.22e+95)
		tmp = Float64(x / exp(x));
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.66], N[(N[(N[(1.0 * 1.0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.22e+95], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.660000000000000031

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.4%

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.4%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

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

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

      4. flip-- N/A

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

      5. distribute-lft-neg-out N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + \left(\mathsf{neg}\left(-1 \cdot x\right)\right)} \]

      8. mul-1-neg N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      9. remove-double-neg N/A

         \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + x} \]

      10. lower-unsound-+.f32 N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + x} \]

      11. lower-+.f32 N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{1 + x} \]

      12. +-commutative N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{x + 1} \]

      13. metadata-eval N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{x + \left(\mathsf{neg}\left(-1\right)\right)} \]

      14. sub-flip N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{x - -1} \]

      15. lift--.f64 N/A

          \[\leadsto \frac{1 \cdot 1 - \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right) \cdot \left(\left(\mathsf{neg}\left(-1\right)\right) \cdot x\right)}{x - -1} \]

      16. lower-unsound-/.f64 N/A

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

   9. Applied rewrites 50.1%

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

   if 0.660000000000000031 < x < 1.22000000000000007e95

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, e^{-x}, -1 \cdot \left(x \cdot e^{-x}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.7%

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

   6. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot 0.5} \]

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{x}} \]

      2. lower-exp.f64 16.5%

         \[\leadsto \frac{x}{e^{x}} \]

   9. Applied rewrites 16.5%

      \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]

   if 1.22000000000000007e95 < x

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \color{blue}{\frac{1}{2}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

      5. lower-*.f64 52.7%

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.7%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

      3. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      5. lower-fma.f64 52.7%

         \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, {x}^{\color{blue}{2}}, 1\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \frac{-1}{2}, {x}^{2}, 1\right) \]

      10. lower-fma.f64 52.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), {x}^{2}, 1\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), x \cdot x, 1\right) \]

      13. lower-*.f64 52.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right) \]

   9. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 52.5%

          \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

   12. Applied rewrites 52.5%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 57.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 0.66:\\ \;\;\;\;1 - x\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{+95}:\\ \;\;\;\;\frac{x}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right)\\ \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= x 0.66)
   (- 1.0 x)
   (if (<= x 1.22e+95)
     (/ x (exp x))
     (fma (* 0.3333333333333333 x) (* x x) 1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (x <= 0.66) {
		tmp = 1.0 - x;
	} else if (x <= 1.22e+95) {
		tmp = x / exp(x);
	} else {
		tmp = fma((0.3333333333333333 * x), (x * x), 1.0);
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (x <= 0.66)
		tmp = Float64(1.0 - x);
	elseif (x <= 1.22e+95)
		tmp = Float64(x / exp(x));
	else
		tmp = fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0);
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[x, 0.66], N[(1.0 - x), $MachinePrecision], If[LessEqual[x, 1.22e+95], N[(x / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.660000000000000031

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      3. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(x \cdot \left(1 - \varepsilon\right)\right)} - \color{blue}{-1} \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      6. lower--.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot \color{blue}{e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}}\right) \]

      8. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{\mathsf{neg}\left(x \cdot \left(1 + \varepsilon\right)\right)}\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{1}{2} \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

      11. lower-+.f64 99.1%

          \[\leadsto 0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{-x \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-x \cdot \left(1 + \varepsilon\right)}\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lower-*.f64 43.4%

         \[\leadsto 1 + -1 \cdot x \]

   7. Applied rewrites 43.4%

      \[\leadsto 1 + \color{blue}{-1 \cdot x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + -1 \cdot \color{blue}{x} \]

      2. lift-*.f64 N/A

         \[\leadsto 1 + -1 \cdot x \]

      3. mul-1-neg N/A

         \[\leadsto 1 + \left(\mathsf{neg}\left(x\right)\right) \]

      4. sub-flip-reverse N/A

         \[\leadsto 1 - x \]

      5. lower--.f64 43.4%

         \[\leadsto 1 - x \]

   9. Applied rewrites 43.4%

      \[\leadsto \color{blue}{1 - x} \]

   if 0.660000000000000031 < x < 1.22000000000000007e95

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{0.5 \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \mathsf{fma}\left(-1, e^{-x}, -1 \cdot \left(x \cdot e^{-x}\right)\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 57.7%

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

   6. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{\left(x - -1\right) + \left(x - -1\right)}{e^{x}} \cdot 0.5} \]

   7. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{x}{e^{x}} \]

      2. lower-exp.f64 16.5%

         \[\leadsto \frac{x}{e^{x}} \]

   9. Applied rewrites 16.5%

      \[\leadsto \frac{x}{\color{blue}{e^{x}}} \]

   if 1.22000000000000007e95 < x

   1. Initial program 73.2%

      \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   2. Taylor expanded in eps around 0

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      6. lower-*.f64 N/A

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

      7. lower-exp.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in x around 0

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \color{blue}{\frac{1}{2}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

      4. lower--.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

      5. lower-*.f64 52.7%

         \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

   7. Applied rewrites 52.7%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

      2. +-commutative N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

      3. lift-*.f64 N/A

         \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

      4. *-commutative N/A

         \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

      5. lower-fma.f64 52.7%

         \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, {x}^{\color{blue}{2}}, 1\right) \]

      6. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

      7. sub-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \frac{-1}{2}, {x}^{2}, 1\right) \]

      10. lower-fma.f64 52.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), {x}^{2}, 1\right) \]

      11. lift-pow.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), x \cdot x, 1\right) \]

      13. lower-*.f64 52.7%

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right) \]

   9. Applied rewrites 52.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)} \]

   10. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 52.5%

          \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

   12. Applied rewrites 52.5%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 52.7% accurate, 4.2× speedup

Math

\[\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (* (fma 0.3333333333333333 x -0.5) x) x 1.0))

C

double code(double x, double eps) {
	return fma((fma(0.3333333333333333, x, -0.5) * x), x, 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(fma(0.3333333333333333, x, -0.5) * x), x, 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(0.3333333333333333 * x + -0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]

Derivation

1. Initial program 73.2%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   6. lower-*.f64 N/A

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

   7. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   8. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   9. lower-fma.f64 N/A

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

4. Applied rewrites 57.7%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \color{blue}{\frac{1}{2}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

   4. lower--.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

   5. lower-*.f64 52.7%

      \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

7. Applied rewrites 52.7%

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

   3. lift-*.f64 N/A

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

   5. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot \left(x \cdot x\right) + 1 \]

   7. associate-*r* N/A

      \[\leadsto \left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x\right) \cdot x + 1 \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x, x, 1\right) \]

   9. lower-*.f64 52.7%

      \[\leadsto \mathsf{fma}\left(\left(0.3333333333333333 \cdot x - 0.5\right) \cdot x, x, 1\right) \]

   10. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot x, x, 1\right) \]

   11. sub-flip N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot x, x, 1\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right) \cdot x, x, 1\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(\left(\frac{1}{3} \cdot x + \frac{-1}{2}\right) \cdot x, x, 1\right) \]

   14. lower-fma.f64 52.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]

9. Applied rewrites 52.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right) \cdot x, x, 1\right) \]
10. Add Preprocessing

Alternative 12: 52.5% accurate, 4.9× speedup

Math

\[\mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

FPCore

(FPCore (x eps) :precision binary64 (fma (* 0.3333333333333333 x) (* x x) 1.0))

C

double code(double x, double eps) {
	return fma((0.3333333333333333 * x), (x * x), 1.0);
}

Julia

function code(x, eps)
	return fma(Float64(0.3333333333333333 * x), Float64(x * x), 1.0)
end

Wolfram

code[x_, eps_] := N[(N[(0.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 73.2%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(e^{\mathsf{neg}\left(x\right)} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower--.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-exp.f64 N/A

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

   5. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   6. lower-*.f64 N/A

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

   7. lower-exp.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{\mathsf{neg}\left(x\right)}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   8. lower-neg.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \left(\left(e^{-x} + x \cdot e^{-x}\right) - \left(-1 \cdot e^{\mathsf{neg}\left(x\right)} + -1 \cdot \left(x \cdot e^{\mathsf{neg}\left(x\right)}\right)\right)\right) \]

   9. lower-fma.f64 N/A

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

4. Applied rewrites 57.7%

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

5. Taylor expanded in x around 0

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]
6. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \color{blue}{\frac{1}{2}}\right) \]

   3. lower-pow.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

   4. lower--.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \]

   5. lower-*.f64 52.7%

      \[\leadsto 1 + {x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right) \]

7. Applied rewrites 52.7%

   \[\leadsto 1 + \color{blue}{{x}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.5\right)} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto 1 + {x}^{2} \cdot \color{blue}{\left(\frac{1}{3} \cdot x - \frac{1}{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

   3. lift-*.f64 N/A

      \[\leadsto {x}^{2} \cdot \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) + 1 \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{3} \cdot x - \frac{1}{2}\right) \cdot {x}^{2} + 1 \]

   5. lower-fma.f64 52.7%

      \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x - 0.5, {x}^{\color{blue}{2}}, 1\right) \]

   6. lift--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x - \frac{1}{2}, {x}^{2}, 1\right) \]

   7. sub-flip N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), {x}^{2}, 1\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x + \frac{-1}{2}, {x}^{2}, 1\right) \]

   10. lower-fma.f64 52.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), {x}^{2}, 1\right) \]

   11. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), {x}^{2}, 1\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3}, x, \frac{-1}{2}\right), x \cdot x, 1\right) \]

   13. lower-*.f64 52.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right) \]

9. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, -0.5\right), x \cdot x, 1\right)} \]

10. Taylor expanded in x around inf

    \[\leadsto \mathsf{fma}\left(\frac{1}{3} \cdot x, x \cdot x, 1\right) \]
11. Step-by-step derivation

    1. lower-*.f64 52.5%

       \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]

12. Applied rewrites 52.5%

    \[\leadsto \mathsf{fma}\left(0.3333333333333333 \cdot x, x \cdot x, 1\right) \]
13. Add Preprocessing

Alternative 13: 43.9% accurate, 58.4× speedup

Math

\[1 \]

FPCore

(FPCore (x eps) :precision binary64 1.0)

C

double code(double x, double eps) {
	return 1.0;
}

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(x, eps)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0;
}

Python

def code(x, eps):
	return 1.0

Julia

function code(x, eps)
	return 1.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0;
end

Wolfram

code[x_, eps_] := 1.0

Derivation

1. Initial program 73.2%

   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 43.9%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025179 
(FPCore (x eps)
  :name "NMSE Section 6.1 mentioned, A"
  :precision binary64
  (/ (- (* (+ 1.0 (/ 1.0 eps)) (exp (- (* (- 1.0 eps) x)))) (* (- (/ 1.0 eps) 1.0) (exp (- (* (+ 1.0 eps) x))))) 2.0))