NMSE Section 6.1 mentioned, A

Average Accuracy: 54.6% → 99.1%

Time: 11.7s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

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 360.0) (/ (* 2.0 (cosh (* x eps))) 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 <= 360.0) {
		tmp = (2.0 * cosh((x * eps))) / 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 <= 360.0d0) then
        tmp = (2.0d0 * cosh((x * eps))) / 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 <= 360.0) {
		tmp = (2.0 * Math.cosh((x * eps))) / 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 <= 360.0:
		tmp = (2.0 * math.cosh((x * eps))) / 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 <= 360.0)
		tmp = Float64(Float64(2.0 * cosh(Float64(x * eps))) / 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 <= 360.0)
		tmp = (2.0 * cosh((x * eps))) / 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, 360.0], N[(N[(2.0 * N[Cosh[N[(x * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 360

   1. Initial program 38.9%

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

   2. Simplified 38.9%

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

      [Start] 38.9	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [=>] 38.9	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
      +-rgt-identity [<=] 38.9	\[ \frac{\color{blue}{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
      div-sub [<=] 38.9	\[ \color{blue}{\frac{\left(\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} + 0\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]

   3. Taylor expanded in eps around inf 98.1%

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

   4. Taylor expanded in eps around -inf 98.1%

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

   5. Simplified 98.1%

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

      [Start] 98.1	\[ \frac{e^{-1 \cdot \left(\left(1 + -1 \cdot \varepsilon\right) \cdot x\right)} - -1 \cdot e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}{2} \]
      associate-*r* [=>] 98.1	\[ \frac{e^{\color{blue}{\left(-1 \cdot \left(1 + -1 \cdot \varepsilon\right)\right) \cdot x}} - -1 \cdot e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}{2} \]
      mul-1-neg [=>] 98.1	\[ \frac{e^{\left(-1 \cdot \left(1 + \color{blue}{\left(-\varepsilon\right)}\right)\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}{2} \]
      sub-neg [<=] 98.1	\[ \frac{e^{\left(-1 \cdot \color{blue}{\left(1 - \varepsilon\right)}\right) \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}{2} \]
      mul-1-neg [=>] 98.1	\[ \frac{e^{\color{blue}{\left(-\left(1 - \varepsilon\right)\right)} \cdot x} - -1 \cdot e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}}{2} \]
      mul-1-neg [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \color{blue}{\left(-e^{-1 \cdot \left(\left(1 - -1 \cdot \varepsilon\right) \cdot x\right)}\right)}}{2} \]
      mul-1-neg [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{\color{blue}{-\left(1 - -1 \cdot \varepsilon\right) \cdot x}}\right)}{2} \]
      distribute-rgt-neg-in [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{\color{blue}{\left(1 - -1 \cdot \varepsilon\right) \cdot \left(-x\right)}}\right)}{2} \]
      cancel-sign-sub-inv [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{\color{blue}{\left(1 + \left(--1\right) \cdot \varepsilon\right)} \cdot \left(-x\right)}\right)}{2} \]
      metadata-eval [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{\left(1 + \color{blue}{1} \cdot \varepsilon\right) \cdot \left(-x\right)}\right)}{2} \]
      *-lft-identity [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{\left(1 + \color{blue}{\varepsilon}\right) \cdot \left(-x\right)}\right)}{2} \]
      +-commutative [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{\color{blue}{\left(\varepsilon + 1\right)} \cdot \left(-x\right)}\right)}{2} \]
      distribute-rgt-neg-in [<=] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{\color{blue}{-\left(\varepsilon + 1\right) \cdot x}}\right)}{2} \]
      *-commutative [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{-\color{blue}{x \cdot \left(\varepsilon + 1\right)}}\right)}{2} \]
      +-commutative [<=] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{-x \cdot \color{blue}{\left(1 + \varepsilon\right)}}\right)}{2} \]
      *-lft-identity [<=] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{-x \cdot \left(1 + \color{blue}{1 \cdot \varepsilon}\right)}\right)}{2} \]
      metadata-eval [<=] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{-x \cdot \left(1 + \color{blue}{\left(--1\right)} \cdot \varepsilon\right)}\right)}{2} \]
      cancel-sign-sub-inv [<=] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{-x \cdot \color{blue}{\left(1 - -1 \cdot \varepsilon\right)}}\right)}{2} \]
      mul-1-neg [=>] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{-x \cdot \left(1 - \color{blue}{\left(-\varepsilon\right)}\right)}\right)}{2} \]

   6. Applied egg-rr 98.8%

      \[\leadsto \frac{\color{blue}{2 \cdot \cosh \left(x + \varepsilon \cdot x\right)}}{2} \]
      Proof

      [Start] 98.1	\[ \frac{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} - \left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)}{2} \]
      sub-neg [=>] 98.1	\[ \frac{\color{blue}{e^{\left(-\left(1 - \varepsilon\right)\right) \cdot x} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}}{2} \]
      *-commutative [=>] 98.1	\[ \frac{e^{\color{blue}{x \cdot \left(-\left(1 - \varepsilon\right)\right)}} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      exp-prod [=>] 96.3	\[ \frac{\color{blue}{{\left(e^{x}\right)}^{\left(-\left(1 - \varepsilon\right)\right)}} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      add-sqr-sqrt [=>] 95.2	\[ \frac{{\left(e^{x}\right)}^{\left(-\left(1 - \color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}\right)\right)} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      sqrt-unprod [=>] 96.3	\[ \frac{{\left(e^{x}\right)}^{\left(-\left(1 - \color{blue}{\sqrt{\varepsilon \cdot \varepsilon}}\right)\right)} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      sqr-neg [<=] 96.3	\[ \frac{{\left(e^{x}\right)}^{\left(-\left(1 - \sqrt{\color{blue}{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}}\right)\right)} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      sqrt-unprod [<=] 95.3	\[ \frac{{\left(e^{x}\right)}^{\left(-\left(1 - \color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}\right)\right)} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      add-sqr-sqrt [<=] 96.3	\[ \frac{{\left(e^{x}\right)}^{\left(-\left(1 - \color{blue}{\left(-\varepsilon\right)}\right)\right)} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      exp-prod [<=] 96.1	\[ \frac{\color{blue}{e^{x \cdot \left(-\left(1 - \left(-\varepsilon\right)\right)\right)}} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      distribute-rgt-neg-in [<=] 96.1	\[ \frac{e^{\color{blue}{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}} + \left(-\left(-e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}\right)\right)}{2} \]
      remove-double-neg [=>] 96.1	\[ \frac{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)} + \color{blue}{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)}}}{2} \]
      distribute-rgt-neg-in [=>] 96.1	\[ \frac{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)} + e^{\color{blue}{x \cdot \left(-\left(1 - \left(-\varepsilon\right)\right)\right)}}}{2} \]
      exp-prod [=>] 96.3	\[ \frac{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)} + \color{blue}{{\left(e^{x}\right)}^{\left(-\left(1 - \left(-\varepsilon\right)\right)\right)}}}{2} \]
      add-sqr-sqrt [=>] 95.3	\[ \frac{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)} + {\left(e^{x}\right)}^{\left(-\left(1 - \color{blue}{\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}}\right)\right)}}{2} \]
      sqrt-unprod [=>] 96.3	\[ \frac{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)} + {\left(e^{x}\right)}^{\left(-\left(1 - \color{blue}{\sqrt{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}}\right)\right)}}{2} \]
      sqr-neg [=>] 96.3	\[ \frac{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)} + {\left(e^{x}\right)}^{\left(-\left(1 - \sqrt{\color{blue}{\varepsilon \cdot \varepsilon}}\right)\right)}}{2} \]
      sqrt-unprod [<=] 95.2	\[ \frac{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)} + {\left(e^{x}\right)}^{\left(-\left(1 - \color{blue}{\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}}\right)\right)}}{2} \]
      add-sqr-sqrt [<=] 96.3	\[ \frac{e^{-x \cdot \left(1 - \left(-\varepsilon\right)\right)} + {\left(e^{x}\right)}^{\left(-\left(1 - \color{blue}{\varepsilon}\right)\right)}}{2} \]

   7. Taylor expanded in eps around inf 98.8%

      \[\leadsto \frac{2 \cdot \cosh \color{blue}{\left(\varepsilon \cdot x\right)}}{2} \]

   if 360 < x

   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. Simplified 100.0%

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

      [Start] 100.0	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]

   3. Taylor expanded in eps around 0 100.0%

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

   4. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{0}}{2} \]
      Proof

      [Start] 100.0	\[ \frac{\frac{e^{-1 \cdot x} - \frac{1}{e^{x}}}{\varepsilon}}{2} \]
      div-sub [=>] 100.0	\[ \frac{\color{blue}{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\frac{1}{e^{x}}}{\varepsilon}}}{2} \]
      rec-exp [=>] 100.0	\[ \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{\color{blue}{e^{-x}}}{\varepsilon}}{2} \]
      mul-1-neg [<=] 100.0	\[ \frac{\frac{e^{-1 \cdot x}}{\varepsilon} - \frac{e^{\color{blue}{-1 \cdot x}}}{\varepsilon}}{2} \]
      +-inverses [=>] 100.0	\[ \frac{\color{blue}{0}}{2} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 360:\\ \;\;\;\;\frac{2 \cdot \cosh \left(x \cdot \varepsilon\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.9%
Cost	13632

\[\begin{array}{l} t_0 := \frac{x + 1}{e^{x}}\\ \frac{t_0 + t_0}{2} \end{array} \]

Alternative 2
Accuracy	98.2%
Cost	1220

\[\begin{array}{l} t_0 := 1 + -0.5 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;\frac{t_0 + t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 3
Accuracy	98.1%
Cost	836

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

Alternative 4
Accuracy	98.0%
Cost	196

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

Alternative 5
Accuracy	28.0%
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2023142 
(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))