Kahan's exp quotient

Average Error: 40.0 → 0.3

Time: 3.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0001289306651216294:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}}}{\frac{x}{\frac{{\left(e^{x}\right)}^{2} + -1}{\sqrt[3]{{\left(1 + e^{x}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\sqrt[3]{1 + e^{x}}}}}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.0001289306651216294)
   (/
    (/ 1.0 (* (cbrt (+ 1.0 (exp x))) (cbrt (+ 1.0 (exp x)))))
    (/
     x
     (/
      (+ (pow (exp x) 2.0) -1.0)
      (*
       (cbrt (pow (+ 1.0 (exp x)) 0.6666666666666666))
       (cbrt (cbrt (+ 1.0 (exp x))))))))
   (+ 1.0 (* x (+ 0.5 (* x 0.16666666666666666))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.0001289306651216294) {
		tmp = (1.0 / (cbrt(1.0 + exp(x)) * cbrt(1.0 + exp(x)))) / (x / ((pow(exp(x), 2.0) + -1.0) / (cbrt(pow((1.0 + exp(x)), 0.6666666666666666)) * cbrt(cbrt(1.0 + exp(x))))));
	} else {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	40.0
Target	40.3
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if x < -1.289306651216294e-4

   1. Initial program 0.1

      \[\frac{e^{x} - 1}{x}\]
   2. Using strategy rm

   3. Applied flip--_binary64_1758 0.1

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

   4. Simplified 0.1

      \[\leadsto \frac{\frac{\color{blue}{{\left(e^{x}\right)}^{2} + -1}}{e^{x} + 1}}{x}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt_binary64_1818 0.1

      \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{2} + -1}{\color{blue}{\left(\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}\right) \cdot \sqrt[3]{e^{x} + 1}}}}{x}\]

   7. Applied *-un-lft-identity_binary64_1783 0.1

      \[\leadsto \frac{\frac{\color{blue}{1 \cdot \left({\left(e^{x}\right)}^{2} + -1\right)}}{\left(\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}\right) \cdot \sqrt[3]{e^{x} + 1}}}{x}\]

   8. Applied times-frac_binary64_1789 0.1

      \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}} \cdot \frac{{\left(e^{x}\right)}^{2} + -1}{\sqrt[3]{e^{x} + 1}}}}{x}\]

   9. Applied associate-/l*_binary64_1728 0.1

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}}{\frac{x}{\frac{{\left(e^{x}\right)}^{2} + -1}{\sqrt[3]{e^{x} + 1}}}}}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt_binary64_1818 0.1

       \[\leadsto \frac{\frac{1}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}}{\frac{x}{\frac{{\left(e^{x}\right)}^{2} + -1}{\sqrt[3]{\color{blue}{\left(\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}\right) \cdot \sqrt[3]{e^{x} + 1}}}}}}\]

   12. Applied cbrt-prod_binary64_1814 0.1

       \[\leadsto \frac{\frac{1}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}}{\frac{x}{\frac{{\left(e^{x}\right)}^{2} + -1}{\color{blue}{\sqrt[3]{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}}}}}\]

   13. Simplified 0.1

       \[\leadsto \frac{\frac{1}{\sqrt[3]{e^{x} + 1} \cdot \sqrt[3]{e^{x} + 1}}}{\frac{x}{\frac{{\left(e^{x}\right)}^{2} + -1}{\color{blue}{\sqrt[3]{{\left(e^{x} + 1\right)}^{0.6666666666666666}}} \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}}}}\]

   if -1.289306651216294e-4 < x

   1. Initial program 60.1

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

   2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{0.5 \cdot x + \left(0.16666666666666666 \cdot {x}^{2} + 1\right)}\]

   3. Simplified 0.4

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0001289306651216294:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{1 + e^{x}} \cdot \sqrt[3]{1 + e^{x}}}}{\frac{x}{\frac{{\left(e^{x}\right)}^{2} + -1}{\sqrt[3]{{\left(1 + e^{x}\right)}^{0.6666666666666666}} \cdot \sqrt[3]{\sqrt[3]{1 + e^{x}}}}}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2021011 
(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))