NMSE Section 6.1 mentioned, A

Average Error: 29.2 → 0.7

Time: 15.5s

Precision: binary64

Cost: 6976

Math

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

↓

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

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 (* 0.5 (* 2.0 (/ (+ 1.0 x) (exp x)))))

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) {
	return 0.5 * (2.0 * ((1.0 + x) / exp(x)));
}

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
    code = 0.5d0 * (2.0d0 * ((1.0d0 + x) / exp(x)))
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) {
	return 0.5 * (2.0 * ((1.0 + x) / Math.exp(x)));
}

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):
	return 0.5 * (2.0 * ((1.0 + x) / math.exp(x)))

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)
	return Float64(0.5 * Float64(2.0 * Float64(Float64(1.0 + x) / exp(x))))
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 = code(x, eps)
	tmp = 0.5 * (2.0 * ((1.0 + x) / exp(x)));
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_] := N[(0.5 * N[(2.0 * N[(N[(1.0 + x), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.2

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

2. Simplified 29.2

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

   [Start] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \]
   div-sub [=>] 29.2	\[ \color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}} \]
   associate-/l* [=>] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}}} \]
   *-lft-identity [<=] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{\color{blue}{1 \cdot \left(\frac{1}{\varepsilon} - 1\right)}}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}} \]
   associate-*l/ [<=] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{1}{\frac{2}{e^{-\left(1 + \varepsilon\right) \cdot x}}} \cdot \left(\frac{1}{\varepsilon} - 1\right)} \]
   associate-/r/ [=>] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\left(\frac{1}{2} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)} \cdot \left(\frac{1}{\varepsilon} - 1\right) \]
   associate-*l* [=>] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \color{blue}{\frac{1}{2} \cdot \left(e^{-\left(1 + \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)\right)} \]
   *-commutative [<=] 29.2	\[ \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{2} - \frac{1}{2} \cdot \color{blue}{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)} \]

3. Taylor expanded in eps around 0 29.8

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

4. Simplified 0.7

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

   [Start] 29.8	\[ 0.5 \cdot \left(\left(\frac{1}{e^{x}} + \left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right) \]
   associate--l+ [=>] 25.3	\[ 0.5 \cdot \color{blue}{\left(\frac{1}{e^{x}} + \left(\left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right)} \]
   exp-neg [<=] 25.3	\[ 0.5 \cdot \left(\color{blue}{e^{-x}} + \left(\left(\frac{e^{-x}}{\varepsilon} + \left(e^{-x} + e^{-x} \cdot x\right)\right) - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right) \]
   +-commutative [=>] 25.3	\[ 0.5 \cdot \left(e^{-x} + \left(\color{blue}{\left(\left(e^{-x} + e^{-x} \cdot x\right) + \frac{e^{-x}}{\varepsilon}\right)} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right) \]
   associate--l+ [=>] 2.0	\[ 0.5 \cdot \left(e^{-x} + \color{blue}{\left(\left(e^{-x} + e^{-x} \cdot x\right) + \left(\frac{e^{-x}}{\varepsilon} - \left(\frac{1}{\varepsilon \cdot e^{x}} + -1 \cdot \frac{x}{e^{x}}\right)\right)\right)}\right) \]

5. Taylor expanded in x around inf 0.7

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

6. Simplified 0.7

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

   [Start] 0.7	\[ 0.5 \cdot \left(e^{-x} + \left(e^{-x} + \left(\frac{1}{e^{x}} + e^{-x}\right) \cdot x\right)\right) \]
   *-commutative [=>] 0.7	\[ 0.5 \cdot \left(e^{-x} + \left(e^{-x} + \color{blue}{x \cdot \left(\frac{1}{e^{x}} + e^{-x}\right)}\right)\right) \]
   exp-neg [=>] 0.7	\[ 0.5 \cdot \left(e^{-x} + \left(e^{-x} + x \cdot \left(\frac{1}{e^{x}} + \color{blue}{\frac{1}{e^{x}}}\right)\right)\right) \]
   count-2 [=>] 0.7	\[ 0.5 \cdot \left(e^{-x} + \left(e^{-x} + x \cdot \color{blue}{\left(2 \cdot \frac{1}{e^{x}}\right)}\right)\right) \]
   associate-*r/ [=>] 0.7	\[ 0.5 \cdot \left(e^{-x} + \left(e^{-x} + x \cdot \color{blue}{\frac{2 \cdot 1}{e^{x}}}\right)\right) \]
   metadata-eval [=>] 0.7	\[ 0.5 \cdot \left(e^{-x} + \left(e^{-x} + x \cdot \frac{\color{blue}{2}}{e^{x}}\right)\right) \]

7. Taylor expanded in x around inf 0.7

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

8. Simplified 0.7

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

   [Start] 0.7	\[ 0.5 \cdot \left(2 \cdot \frac{x}{e^{x}} + 2 \cdot e^{-x}\right) \]
   distribute-lft-out [=>] 0.7	\[ 0.5 \cdot \color{blue}{\left(2 \cdot \left(\frac{x}{e^{x}} + e^{-x}\right)\right)} \]

9. Applied egg-rr 16.7

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

10. Simplified 0.7

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

    [Start] 16.7	\[ 0.5 \cdot \left(2 \cdot \frac{\mathsf{fma}\left(x, e^{x}, e^{x}\right)}{e^{x + x}}\right) \]
    fma-udef [=>] 16.7	\[ 0.5 \cdot \left(2 \cdot \frac{\color{blue}{x \cdot e^{x} + e^{x}}}{e^{x + x}}\right) \]
    distribute-lft1-in [=>] 16.7	\[ 0.5 \cdot \left(2 \cdot \frac{\color{blue}{\left(x + 1\right) \cdot e^{x}}}{e^{x + x}}\right) \]
    associate-/l* [=>] 16.7	\[ 0.5 \cdot \left(2 \cdot \color{blue}{\frac{x + 1}{\frac{e^{x + x}}{e^{x}}}}\right) \]
    +-commutative [=>] 16.7	\[ 0.5 \cdot \left(2 \cdot \frac{\color{blue}{1 + x}}{\frac{e^{x + x}}{e^{x}}}\right) \]
    exp-sum [=>] 16.7	\[ 0.5 \cdot \left(2 \cdot \frac{1 + x}{\frac{\color{blue}{e^{x} \cdot e^{x}}}{e^{x}}}\right) \]
    associate-/l* [=>] 16.6	\[ 0.5 \cdot \left(2 \cdot \frac{1 + x}{\color{blue}{\frac{e^{x}}{\frac{e^{x}}{e^{x}}}}}\right) \]
    associate-/r/ [=>] 16.6	\[ 0.5 \cdot \left(2 \cdot \frac{1 + x}{\color{blue}{\frac{e^{x}}{e^{x}} \cdot e^{x}}}\right) \]
    *-inverses [=>] 0.7	\[ 0.5 \cdot \left(2 \cdot \frac{1 + x}{\color{blue}{1} \cdot e^{x}}\right) \]
    *-lft-identity [=>] 0.7	\[ 0.5 \cdot \left(2 \cdot \frac{1 + x}{\color{blue}{e^{x}}}\right) \]

11. Final simplification 0.7

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

Alternatives

Alternative 1
Error	1.1
Cost	580

\[\begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;0.5 \cdot \left(2 - x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]

Alternative 2
Error	1.2
Cost	196

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

Alternative 3
Error	16.7
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023187 
(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))