Kahan's exp quotient

Average Error: 39.8 → 0.2

Time: 14.6s

Precision: binary64

Cost: 7684

Math

\[\frac{e^{x} - 1}{x} \]

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0195:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{1}{x} + \left(x \cdot 0.08333333333333333 + \left({x}^{3} \cdot -0.001388888888888889 + -0.5\right)\right)}\\ \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) 1.0) x))

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.0195)
   (/ (- (exp x) 1.0) x)
   (/
    (/ 1.0 x)
    (+
     (/ 1.0 x)
     (+
      (* x 0.08333333333333333)
      (+ (* (pow x 3.0) -0.001388888888888889) -0.5))))))

C

double code(double x) {
	return (exp(x) - 1.0) / x;
}

↓

double code(double x) {
	double tmp;
	if (x <= -0.0195) {
		tmp = (exp(x) - 1.0) / x;
	} else {
		tmp = (1.0 / x) / ((1.0 / x) + ((x * 0.08333333333333333) + ((pow(x, 3.0) * -0.001388888888888889) + -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 1.0d0) / x
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.0195d0)) then
        tmp = (exp(x) - 1.0d0) / x
    else
        tmp = (1.0d0 / x) / ((1.0d0 / x) + ((x * 0.08333333333333333d0) + (((x ** 3.0d0) * (-0.001388888888888889d0)) + (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 1.0) / x;
}

↓

public static double code(double x) {
	double tmp;
	if (x <= -0.0195) {
		tmp = (Math.exp(x) - 1.0) / x;
	} else {
		tmp = (1.0 / x) / ((1.0 / x) + ((x * 0.08333333333333333) + ((Math.pow(x, 3.0) * -0.001388888888888889) + -0.5)));
	}
	return tmp;
}

Python

def code(x):
	return (math.exp(x) - 1.0) / x

↓

def code(x):
	tmp = 0
	if x <= -0.0195:
		tmp = (math.exp(x) - 1.0) / x
	else:
		tmp = (1.0 / x) / ((1.0 / x) + ((x * 0.08333333333333333) + ((math.pow(x, 3.0) * -0.001388888888888889) + -0.5)))
	return tmp

Julia

function code(x)
	return Float64(Float64(exp(x) - 1.0) / x)
end

↓

function code(x)
	tmp = 0.0
	if (x <= -0.0195)
		tmp = Float64(Float64(exp(x) - 1.0) / x);
	else
		tmp = Float64(Float64(1.0 / x) / Float64(Float64(1.0 / x) + Float64(Float64(x * 0.08333333333333333) + Float64(Float64((x ^ 3.0) * -0.001388888888888889) + -0.5))));
	end
	return tmp
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 1.0) / x;
end

↓

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.0195)
		tmp = (exp(x) - 1.0) / x;
	else
		tmp = (1.0 / x) / ((1.0 / x) + ((x * 0.08333333333333333) + (((x ^ 3.0) * -0.001388888888888889) + -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision]

↓

code[x_] := If[LessEqual[x, -0.0195], N[(N[(N[Exp[x], $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] / N[(N[(1.0 / x), $MachinePrecision] + N[(N[(x * 0.08333333333333333), $MachinePrecision] + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.001388888888888889), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	39.8
Target	40.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;x < 1 \land x > -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -0.0195

   1. Initial program 0.0

      \[\frac{e^{x} - 1}{x} \]

   if -0.0195 < x

   1. Initial program 59.9

      \[\frac{e^{x} - 1}{x} \]

   2. Applied egg-rr 59.9

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

   3. Taylor expanded in x around 0 0.5

      \[\leadsto \frac{1}{\color{blue}{\left(-0.001388888888888889 \cdot {x}^{3} + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)\right) - 0.5}} \cdot \frac{1}{x} \]

   4. Simplified 0.5

      \[\leadsto \frac{1}{\color{blue}{\left(\left(\frac{1}{x} + x \cdot 0.08333333333333333\right) + {x}^{3} \cdot -0.001388888888888889\right) - 0.5}} \cdot \frac{1}{x} \]
      Proof

      [Start] 0.5	\[ \frac{1}{\left(-0.001388888888888889 \cdot {x}^{3} + \left(0.08333333333333333 \cdot x + \frac{1}{x}\right)\right) - 0.5} \cdot \frac{1}{x} \]
      rational.json-simplify-1 [=>] 0.5	\[ \frac{1}{\color{blue}{\left(\left(0.08333333333333333 \cdot x + \frac{1}{x}\right) + -0.001388888888888889 \cdot {x}^{3}\right)} - 0.5} \cdot \frac{1}{x} \]
      rational.json-simplify-1 [=>] 0.5	\[ \frac{1}{\left(\color{blue}{\left(\frac{1}{x} + 0.08333333333333333 \cdot x\right)} + -0.001388888888888889 \cdot {x}^{3}\right) - 0.5} \cdot \frac{1}{x} \]
      rational.json-simplify-2 [=>] 0.5	\[ \frac{1}{\left(\left(\frac{1}{x} + \color{blue}{x \cdot 0.08333333333333333}\right) + -0.001388888888888889 \cdot {x}^{3}\right) - 0.5} \cdot \frac{1}{x} \]
      rational.json-simplify-2 [=>] 0.5	\[ \frac{1}{\left(\left(\frac{1}{x} + x \cdot 0.08333333333333333\right) + \color{blue}{{x}^{3} \cdot -0.001388888888888889}\right) - 0.5} \cdot \frac{1}{x} \]

   5. Applied egg-rr 0.4

      \[\leadsto \color{blue}{\frac{\frac{1}{x}}{\frac{1}{x} + \left(x \cdot 0.08333333333333333 + \left({x}^{3} \cdot -0.001388888888888889 + -0.5\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0195:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{x}}{\frac{1}{x} + \left(x \cdot 0.08333333333333333 + \left({x}^{3} \cdot -0.001388888888888889 + -0.5\right)\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.3
Cost	6852

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00185:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{x}}{\frac{2}{x} + \left(x \cdot 0.16666666666666666 + -1\right)}\\ \end{array} \]

Alternative 2
Error	17.9
Cost	452

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot x + 1\\ \end{array} \]

Alternative 3
Error	18.0
Cost	384

\[\frac{2}{2 + \left(-x\right)} \]

Alternative 4
Error	18.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;x \leq -2:\\ \;\;\;\;\frac{-2}{x}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	21.3
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023068 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))