NMSE Section 6.1 mentioned, A

Percentage Accurate: 73.0% → 99.9%

Time: 7.2s

Alternatives: 3

Speedup: N/A×

• 38.3% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. x = -1.1899360899311763e+119
  2. eps = -1.514862336677036e-20

Specification

Math

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

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: 73.0% accurate, 1.0× speedup

Math

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

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, N/A× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x - -1\right) \cdot e^{-x}\\ \mathbf{if}\;eps\_m \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_0 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{eps\_m}\right) \cdot e^{\left(-1 + eps\_m\right) \cdot x} + \left(\frac{-1}{eps\_m} - -1\right) \cdot e^{e^{\mathsf{log1p}\left(eps\_m\right)} \cdot \left(-x\right)}}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (- x -1.0) (exp (- x)))))
   (if (<= eps_m 4e-17)
     (/ (+ t_0 t_0) 2.0)
     (/
      (+
       (* (+ 1.0 (/ 1.0 eps_m)) (exp (* (+ -1.0 eps_m) x)))
       (* (- (/ -1.0 eps_m) -1.0) (exp (* (exp (log1p eps_m)) (- x)))))
      2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x - -1.0) * exp(-x);
	double tmp;
	if (eps_m <= 4e-17) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * exp(((-1.0 + eps_m) * x))) + (((-1.0 / eps_m) - -1.0) * exp((exp(log1p(eps_m)) * -x)))) / 2.0;
	}
	return tmp;
}

Java

eps_m = Math.abs(eps);
public static double code(double x, double eps_m) {
	double t_0 = (x - -1.0) * Math.exp(-x);
	double tmp;
	if (eps_m <= 4e-17) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = (((1.0 + (1.0 / eps_m)) * Math.exp(((-1.0 + eps_m) * x))) + (((-1.0 / eps_m) - -1.0) * Math.exp((Math.exp(Math.log1p(eps_m)) * -x)))) / 2.0;
	}
	return tmp;
}

Python

eps_m = math.fabs(eps)
def code(x, eps_m):
	t_0 = (x - -1.0) * math.exp(-x)
	tmp = 0
	if eps_m <= 4e-17:
		tmp = (t_0 + t_0) / 2.0
	else:
		tmp = (((1.0 + (1.0 / eps_m)) * math.exp(((-1.0 + eps_m) * x))) + (((-1.0 / eps_m) - -1.0) * math.exp((math.exp(math.log1p(eps_m)) * -x)))) / 2.0
	return tmp

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x - -1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 4e-17)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(Float64(Float64(Float64(1.0 + Float64(1.0 / eps_m)) * exp(Float64(Float64(-1.0 + eps_m) * x))) + Float64(Float64(Float64(-1.0 / eps_m) - -1.0) * exp(Float64(exp(log1p(eps_m)) * Float64(-x))))) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x - -1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 4e-17], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(1.0 + N[(1.0 / eps$95$m), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(-1.0 + eps$95$m), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(-1.0 / eps$95$m), $MachinePrecision] - -1.0), $MachinePrecision] * N[Exp[N[(N[Exp[N[Log[1 + eps$95$m], $MachinePrecision]], $MachinePrecision] * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 4.00000000000000029e-17

   1. Initial program 61.7%

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

   3. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 66.6

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

   5. Applied rewrites 66.6%

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

   if 4.00000000000000029e-17 < eps

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. pow-to-exp N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log1p.f64 100.0

         \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-e^{\color{blue}{\mathsf{log1p}\left(\varepsilon\right)} \cdot 1} \cdot x}}{2} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\color{blue}{e^{\mathsf{log1p}\left(\varepsilon\right) \cdot 1}} \cdot x}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(x - -1\right) \cdot e^{-x} + \left(x - -1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(-1 + \varepsilon\right) \cdot x} + \left(\frac{-1}{\varepsilon} - -1\right) \cdot e^{e^{\mathsf{log1p}\left(\varepsilon\right)} \cdot \left(-x\right)}}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 67.1% accurate, N/A× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \begin{array}{l} t_0 := \left(x - -1\right) \cdot e^{-x}\\ \mathbf{if}\;eps\_m \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_0 + t\_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right)}{2}\\ \end{array} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (let* ((t_0 (* (- x -1.0) (exp (- x)))))
   (if (<= eps_m 1.15e-6)
     (/ (+ t_0 t_0) 2.0)
     (/ (fma (- (* 0.6666666666666666 x) 1.0) (* x x) 2.0) 2.0))))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	double t_0 = (x - -1.0) * exp(-x);
	double tmp;
	if (eps_m <= 1.15e-6) {
		tmp = (t_0 + t_0) / 2.0;
	} else {
		tmp = fma(((0.6666666666666666 * x) - 1.0), (x * x), 2.0) / 2.0;
	}
	return tmp;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	t_0 = Float64(Float64(x - -1.0) * exp(Float64(-x)))
	tmp = 0.0
	if (eps_m <= 1.15e-6)
		tmp = Float64(Float64(t_0 + t_0) / 2.0);
	else
		tmp = Float64(fma(Float64(Float64(0.6666666666666666 * x) - 1.0), Float64(x * x), 2.0) / 2.0);
	end
	return tmp
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := Block[{t$95$0 = N[(N[(x - -1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps$95$m, 1.15e-6], N[(N[(t$95$0 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] - 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < 1.15e-6

   1. Initial program 61.9%

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

   3. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 1.15e-6 < eps

   1. Initial program 100.0%

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

   3. Taylor expanded in eps around 0

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

      1. lower--.f64 N/A

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

      2. distribute-rgt1-in N/A

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

      3. lower-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. distribute-lft-out N/A

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

      8. lower-*.f64 N/A

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

      9. distribute-rgt1-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lower-exp.f64 N/A

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

      13. lower-neg.f64 24.5

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

   5. Applied rewrites 24.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 42.4

         \[\leadsto \frac{\mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right)}{2} \]

   8. Applied rewrites 42.4%

      \[\leadsto \frac{\mathsf{fma}\left(0.6666666666666666 \cdot x - 1, \color{blue}{x \cdot x}, 2\right)}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq 1.15 \cdot 10^{-6}:\\ \;\;\;\;\frac{\left(x - -1\right) \cdot e^{-x} + \left(x - -1\right) \cdot e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right)}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.5% accurate, N/A× speedup

Math

\[\begin{array}{l} eps_m = \left|\varepsilon\right| \\ \frac{\mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right)}{2} \end{array} \]

FPCore

eps_m = (fabs.f64 eps)
(FPCore (x eps_m)
 :precision binary64
 (/ (fma (- (* 0.6666666666666666 x) 1.0) (* x x) 2.0) 2.0))

C

eps_m = fabs(eps);
double code(double x, double eps_m) {
	return fma(((0.6666666666666666 * x) - 1.0), (x * x), 2.0) / 2.0;
}

Julia

eps_m = abs(eps)
function code(x, eps_m)
	return Float64(fma(Float64(Float64(0.6666666666666666 * x) - 1.0), Float64(x * x), 2.0) / 2.0)
end

Wolfram

eps_m = N[Abs[eps], $MachinePrecision]
code[x_, eps$95$m_] := N[(N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] - 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 72.8%

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

3. Taylor expanded in eps around 0

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

   1. lower--.f64 N/A

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

   2. distribute-rgt1-in N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-exp.f64 N/A

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

   6. lower-neg.f64 N/A

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

   7. distribute-lft-out N/A

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

   8. lower-*.f64 N/A

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

   9. distribute-rgt1-in N/A

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

   10. lower-*.f64 N/A

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

   11. lower-+.f64 N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-neg.f64 54.7

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

5. Applied rewrites 54.7%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 52.3

      \[\leadsto \frac{\mathsf{fma}\left(0.6666666666666666 \cdot x - 1, x \cdot x, 2\right)}{2} \]

8. Applied rewrites 52.3%

   \[\leadsto \frac{\mathsf{fma}\left(0.6666666666666666 \cdot x - 1, \color{blue}{x \cdot x}, 2\right)}{2} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025059 
(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))