NMSE Section 6.1 mentioned, A

Average Error: 45.98% → 0.11%

Time: 15.6s

Precision: binary64

Cost: 27972

Math

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

↓

\[\begin{array}{l} t_0 := e^{x \cdot \left(-1 - \varepsilon\right)}\\ t_1 := e^{x \cdot \left(\varepsilon + -1\right)}\\ \mathbf{if}\;\left(1 + \frac{1}{\varepsilon}\right) \cdot t_1 + t_0 \cdot \left(1 - \frac{1}{\varepsilon}\right) \leq 2:\\ \;\;\;\;\frac{\frac{2}{e^{x}} \cdot \left(1 + x\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1 + t_0}{2}\\ \end{array} \]

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))

↓

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (exp (* x (- -1.0 eps)))) (t_1 (exp (* x (+ eps -1.0)))))
   (if (<= (+ (* (+ 1.0 (/ 1.0 eps)) t_1) (* t_0 (- 1.0 (/ 1.0 eps)))) 2.0)
     (/ (* (/ 2.0 (exp x)) (+ 1.0 x)) 2.0)
     (/ (+ t_1 t_0) 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;
}

↓

double code(double x, double eps) {
	double t_0 = exp((x * (-1.0 - eps)));
	double t_1 = exp((x * (eps + -1.0)));
	double tmp;
	if ((((1.0 + (1.0 / eps)) * t_1) + (t_0 * (1.0 - (1.0 / eps)))) <= 2.0) {
		tmp = ((2.0 / exp(x)) * (1.0 + x)) / 2.0;
	} else {
		tmp = (t_1 + t_0) / 2.0;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = exp((x * ((-1.0d0) - eps)))
    t_1 = exp((x * (eps + (-1.0d0))))
    if ((((1.0d0 + (1.0d0 / eps)) * t_1) + (t_0 * (1.0d0 - (1.0d0 / eps)))) <= 2.0d0) then
        tmp = ((2.0d0 / exp(x)) * (1.0d0 + x)) / 2.0d0
    else
        tmp = (t_1 + t_0) / 2.0d0
    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 t_0 = Math.exp((x * (-1.0 - eps)));
	double t_1 = Math.exp((x * (eps + -1.0)));
	double tmp;
	if ((((1.0 + (1.0 / eps)) * t_1) + (t_0 * (1.0 - (1.0 / eps)))) <= 2.0) {
		tmp = ((2.0 / Math.exp(x)) * (1.0 + x)) / 2.0;
	} else {
		tmp = (t_1 + t_0) / 2.0;
	}
	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):
	t_0 = math.exp((x * (-1.0 - eps)))
	t_1 = math.exp((x * (eps + -1.0)))
	tmp = 0
	if (((1.0 + (1.0 / eps)) * t_1) + (t_0 * (1.0 - (1.0 / eps)))) <= 2.0:
		tmp = ((2.0 / math.exp(x)) * (1.0 + x)) / 2.0
	else:
		tmp = (t_1 + t_0) / 2.0
	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)
	t_0 = exp(Float64(x * Float64(-1.0 - eps)))
	t_1 = exp(Float64(x * Float64(eps + -1.0)))
	tmp = 0.0
	if (Float64(Float64(Float64(1.0 + Float64(1.0 / eps)) * t_1) + Float64(t_0 * Float64(1.0 - Float64(1.0 / eps)))) <= 2.0)
		tmp = Float64(Float64(Float64(2.0 / exp(x)) * Float64(1.0 + x)) / 2.0);
	else
		tmp = Float64(Float64(t_1 + t_0) / 2.0);
	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)
	t_0 = exp((x * (-1.0 - eps)));
	t_1 = exp((x * (eps + -1.0)));
	tmp = 0.0;
	if ((((1.0 + (1.0 / eps)) * t_1) + (t_0 * (1.0 - (1.0 / eps)))) <= 2.0)
		tmp = ((2.0 / exp(x)) * (1.0 + x)) / 2.0;
	else
		tmp = (t_1 + t_0) / 2.0;
	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_] := Block[{t$95$0 = N[Exp[N[(x * N[(-1.0 - eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(x * N[(eps + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(N[(N[(1.0 + N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(t$95$0 * N[(1.0 - N[(1.0 / eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[(2.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(t$95$1 + t$95$0), $MachinePrecision] / 2.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x))))) < 2

   1. Initial program 46.72

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

   2. Simplified 67.98

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

      [Start] 46.72	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Taylor expanded in eps around 0 46.72

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

   4. Simplified 0.04

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

      [Start] 46.72	\[ \frac{\left(\frac{e^{-1 \cdot x}}{\varepsilon} + \left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)}{2} \]
      +-commutative [=>] 46.72	\[ \frac{\color{blue}{\left(\left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \frac{e^{-1 \cdot x}}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)}{2} \]
      associate--l+ [=>] 2.06	\[ \frac{\color{blue}{\left(e^{-1 \cdot x} \cdot x + \left(\frac{1}{e^{x}} + e^{-1 \cdot x}\right)\right) + \left(\frac{e^{-1 \cdot x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)}}{2} \]

   5. Taylor expanded in x around inf 0.04

      \[\leadsto \frac{\color{blue}{2 \cdot \frac{x}{e^{x}} + 2 \cdot \frac{1}{e^{x}}}}{2} \]

   6. Applied egg-rr 2.15

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

   7. Simplified 0.03

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

      [Start] 2.15	\[ \frac{e^{\mathsf{log1p}\left(\frac{2}{e^{x}} \cdot \left(1 + x\right)\right)} - 1}{2} \]
      expm1-def [=>] 1.99	\[ \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{e^{x}} \cdot \left(1 + x\right)\right)\right)}}{2} \]
      expm1-log1p [=>] 0.03	\[ \frac{\color{blue}{\frac{2}{e^{x}} \cdot \left(1 + x\right)}}{2} \]

   if 2 < (-.f64 (*.f64 (+.f64 1 (/.f64 1 eps)) (exp.f64 (neg.f64 (*.f64 (-.f64 1 eps) x)))) (*.f64 (-.f64 (/.f64 1 eps) 1) (exp.f64 (neg.f64 (*.f64 (+.f64 1 eps) x)))))

   1. Initial program 5.19

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

   2. Simplified 5.19

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

      [Start] 5.19	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      distribute-rgt-neg-in [=>] 5.19	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\color{blue}{\left(1 - \varepsilon\right) \cdot \left(-x\right)}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      sub-neg [=>] 5.19	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \color{blue}{\left(\frac{1}{\varepsilon} + \left(-1\right)\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      metadata-eval [=>] 5.19	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + \color{blue}{-1}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      distribute-rgt-neg-in [=>] 5.19	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)} - \left(\frac{1}{\varepsilon} + -1\right) \cdot e^{\color{blue}{\left(1 + \varepsilon\right) \cdot \left(-x\right)}}}{2} \]

   3. Taylor expanded in eps around inf 4.57

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

   4. Simplified 4.57

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

      [Start] 4.57	\[ \frac{e^{-1 \cdot \left(\left(1 - \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      associate-*r* [=>] 4.57	\[ \frac{e^{\color{blue}{\left(-1 \cdot \left(1 - \varepsilon\right)\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      sub-neg [=>] 4.57	\[ \frac{e^{\left(-1 \cdot \color{blue}{\left(1 + \left(-\varepsilon\right)\right)}\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      mul-1-neg [<=] 4.57	\[ \frac{e^{\left(-1 \cdot \left(1 + \color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      associate-*r* [<=] 4.57	\[ \frac{e^{\color{blue}{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      *-commutative [=>] 4.57	\[ \frac{e^{-1 \cdot \color{blue}{\left(x \cdot \left(1 + -1 \cdot \varepsilon\right)\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      mul-1-neg [=>] 4.57	\[ \frac{e^{-1 \cdot \left(x \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      sub-neg [<=] 4.57	\[ \frac{e^{-1 \cdot \left(x \cdot \color{blue}{\left(1 - \varepsilon\right)}\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      associate-*r* [=>] 4.57	\[ \frac{e^{\color{blue}{\left(-1 \cdot x\right) \cdot \left(1 - \varepsilon\right)}} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      mul-1-neg [=>] 4.57	\[ \frac{e^{\color{blue}{\left(-x\right)} \cdot \left(1 - \varepsilon\right)} - -1 \cdot e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}}{2} \]
      mul-1-neg [=>] 4.57	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \color{blue}{\left(-e^{-1 \cdot \left(\left(\varepsilon + 1\right) \cdot x\right)}\right)}}{2} \]
      exp-prod [=>] 4.56	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{{\left(e^{-1}\right)}^{\left(\left(\varepsilon + 1\right) \cdot x\right)}}\right)}{2} \]
      +-commutative [<=] 4.56	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-{\left(e^{-1}\right)}^{\left(\color{blue}{\left(1 + \varepsilon\right)} \cdot x\right)}\right)}{2} \]
      exp-prod [<=] 4.57	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-\color{blue}{e^{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}{2} \]
      associate-*r* [=>] 4.57	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{\left(-1 \cdot \left(1 + \varepsilon\right)\right) \cdot x}}\right)}{2} \]
      *-commutative [=>] 4.57	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{\color{blue}{x \cdot \left(-1 \cdot \left(1 + \varepsilon\right)\right)}}\right)}{2} \]
      distribute-lft-in [=>] 4.57	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{x \cdot \color{blue}{\left(-1 \cdot 1 + -1 \cdot \varepsilon\right)}}\right)}{2} \]
      metadata-eval [=>] 4.57	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{x \cdot \left(\color{blue}{-1} + -1 \cdot \varepsilon\right)}\right)}{2} \]
      mul-1-neg [=>] 4.57	\[ \frac{e^{\left(-x\right) \cdot \left(1 - \varepsilon\right)} - \left(-e^{x \cdot \left(-1 + \color{blue}{\left(-\varepsilon\right)}\right)}\right)}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.11

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

Alternatives

Alternative 1
Error	1.53%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 1.46:\\ \;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(-1 + x \cdot 0.6666666666666666\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{e^{x}}}{2}\\ \end{array} \]

Alternative 2
Error	2.2%
Cost	6980

\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{1 + e^{\varepsilon \cdot x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x \cdot 2}{e^{x}}}{2}\\ \end{array} \]

Alternative 3
Error	0.95%
Cost	6976

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

Alternative 4
Error	1.56%
Cost	964

\[\begin{array}{l} \mathbf{if}\;x \leq 350:\\ \;\;\;\;\frac{2 + \left(x \cdot x\right) \cdot \left(-1 + x \cdot 0.6666666666666666\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 5
Error	1.68%
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;\frac{2 - x \cdot x}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 6
Error	1.87%
Cost	196

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

Alternative 7
Error	73.01%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023090 
(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))