NMSE Section 6.1 mentioned, A

Average Error: 29.2 → 1.0

Time: 20.6s

Precision: binary64

Cost: 7556

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(1 + -0.5 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

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))

↓

(FPCore (x eps)
 :precision binary64
 (if (<= x 1.75)
   (/ (+ (* (+ x 1.0) (exp (- x))) (+ 1.0 (* -0.5 (* x x)))) 2.0)
   0.0))

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;
}

↓

double code(double x, double eps) {
	double tmp;
	if (x <= 1.75) {
		tmp = (((x + 1.0) * exp(-x)) + (1.0 + (-0.5 * (x * x)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    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

↓

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (x <= 1.75d0) then
        tmp = (((x + 1.0d0) * exp(-x)) + (1.0d0 + ((-0.5d0) * (x * x)))) / 2.0d0
    else
        tmp = 0.0d0
    end if
    code = tmp
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;
}

↓

public static double code(double x, double eps) {
	double tmp;
	if (x <= 1.75) {
		tmp = (((x + 1.0) * Math.exp(-x)) + (1.0 + (-0.5 * (x * x)))) / 2.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

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

↓

def code(x, eps):
	tmp = 0
	if x <= 1.75:
		tmp = (((x + 1.0) * math.exp(-x)) + (1.0 + (-0.5 * (x * x)))) / 2.0
	else:
		tmp = 0.0
	return tmp

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

↓

function code(x, eps)
	tmp = 0.0
	if (x <= 1.75)
		tmp = Float64(Float64(Float64(Float64(x + 1.0) * exp(Float64(-x))) + Float64(1.0 + Float64(-0.5 * Float64(x * x)))) / 2.0);
	else
		tmp = 0.0;
	end
	return tmp
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

↓

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = (((x + 1.0) * exp(-x)) + (1.0 + (-0.5 * (x * x)))) / 2.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
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]

↓

code[x_, eps_] := If[LessEqual[x, 1.75], N[(N[(N[(N[(x + 1.0), $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] + N[(1.0 + N[(-0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 38.6

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

   2. Simplified 38.6

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

      [Start] 38.6	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      distribute-rgt-neg-in [=>] 38.6	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      sub-neg [=>] 38.6	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      metadata-eval [=>] 38.6	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      distribute-rgt-neg-in [=>] 38.6	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Taylor expanded in eps around 0 0.8

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

   4. Simplified 0.8

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

      [Start] 0.8	\[ \frac{\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      *-commutative [=>] 0.8	\[ \frac{\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right) - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      distribute-lft1-in [=>] 0.8	\[ \frac{\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      mul-1-neg [=>] 0.8	\[ \frac{\left(x + 1\right) \cdot e^{\color{blue}{-x}} - \left(-1 \cdot \left(e^{-1 \cdot x} \cdot x\right) + -1 \cdot e^{-1 \cdot x}\right)}{2} \]
      distribute-lft-out [=>] 0.8	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{-1 \cdot \left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)}}{2} \]
      mul-1-neg [=>] 0.8	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \color{blue}{\left(-\left(e^{-1 \cdot x} \cdot x + e^{-1 \cdot x}\right)\right)}}{2} \]
      *-commutative [=>] 0.8	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(\color{blue}{x \cdot e^{-1 \cdot x}} + e^{-1 \cdot x}\right)\right)}{2} \]
      distribute-lft1-in [=>] 0.8	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\color{blue}{\left(x + 1\right) \cdot e^{-1 \cdot x}}\right)}{2} \]
      mul-1-neg [=>] 0.8	\[ \frac{\left(x + 1\right) \cdot e^{-x} - \left(-\left(x + 1\right) \cdot e^{\color{blue}{-x}}\right)}{2} \]

   5. Taylor expanded in x around 0 1.1

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   6. Simplified 1.1

      \[\leadsto \frac{\color{blue}{\left(1 + -0.5 \cdot \left(x \cdot x\right)\right)} - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]
      Proof

      [Start] 1.1	\[ \frac{\left(1 + -0.5 \cdot {x}^{2}\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]
      unpow2 [=>] 1.1	\[ \frac{\left(1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) - \left(-\left(x + 1\right) \cdot e^{-x}\right)}{2} \]

   if 1.75 < x

   1. Initial program 0.7

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

   2. Simplified 0.7

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

      [Start] 0.7	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 0.7	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      associate-/l* [=>] 0.7	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}}} \]
      *-lft-identity [<=] 0.7	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\color{blue}{1 \cdot \left(\frac{1}{\varepsilon} - 1\right)}}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}} \]
      associate-*l/ [<=] 0.7	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{1}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \]
      associate-/r/ [=>] 0.7	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\left(\frac{1}{2} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right) \]
      associate-*l* [=>] 0.7	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{1}{2} \cdot \left(e^{-\left(1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]
      *-commutative [<=] 0.7	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{1}{2} \cdot \color{blue}{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)} \]

   3. Taylor expanded in eps around 0 0.7

      \[\leadsto 0.5 \cdot \color{blue}{\frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon}} \]

   4. Simplified 0.7

      \[\leadsto 0.5 \cdot \color{blue}{0} \]
      Proof

      [Start] 0.7	\[ 0.5 \cdot \frac{e^{-x} - \frac{1}{e^{x}}}{\varepsilon} \]
      div-sub [=>] 0.7	\[ 0.5 \cdot \color{blue}{\left(\frac{e^{-x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right)} \]
      exp-neg [=>] 0.7	\[ 0.5 \cdot \left(\frac{\color{blue}{\frac{1}{e^{x}}}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]
      associate-/l/ [=>] 0.7	\[ 0.5 \cdot \left(\color{blue}{\frac{1}{\varepsilon \cdot e^{x}}} - \frac{\frac{1}{e^{x}}}{\varepsilon}\right) \]
      associate-/l/ [=>] 0.7	\[ 0.5 \cdot \left(\frac{1}{\varepsilon \cdot e^{x}} - \color{blue}{\frac{1}{\varepsilon \cdot e^{x}}}\right) \]
      +-inverses [=>] 0.7	\[ 0.5 \cdot \color{blue}{0} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\frac{\left(x + 1\right) \cdot e^{-x} + \left(1 + -0.5 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternatives

Alternative 1
Error	0.6
Cost	13760

\[\begin{array}{l} t_0 := \left(x + 1\right) \cdot e^{-x}\\ \frac{t_0 + t_0}{2} \end{array} \]

Alternative 2
Error	0.9
Cost	13632

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

Alternative 3
Error	1.2
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;\frac{1 + \left(1 + -0.5 \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 4
Error	1.2
Cost	196

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	46.8
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023038 
(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))