Kahan's exp quotient

Average Error: 40.1 → 0.2

Time: 3.0s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0008178455324951667:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + e^{x}}} \cdot \frac{{\left(e^{2}\right)}^{x} - 1}{\sqrt{1 + e^{x}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.0008178455324951667)
   (/
    (*
     (/ 1.0 (sqrt (+ 1.0 (exp x))))
     (/ (- (pow (exp 2.0) x) 1.0) (sqrt (+ 1.0 (exp x)))))
    x)
   (+
    1.0
    (* x (+ 0.5 (* x (+ 0.16666666666666666 (* x 0.041666666666666664))))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -0.0008178455324951667) {
		tmp = ((1.0 / sqrt(1.0 + exp(x))) * ((pow(exp(2.0), x) - 1.0) / sqrt(1.0 + exp(x)))) / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * (0.16666666666666666 + (x * 0.041666666666666664)))));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	40.1
Target	40.5
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 < -8.1784553249516665e-4

   1. Initial program 0.0

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

   3. Applied add-cbrt-cube_binary64 0.0

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

   4. Simplified 0.0

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

   6. Applied flip--_binary64 0.0

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

   7. Simplified 0.0

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

   9. Applied add-sqr-sqrt_binary64 0.0

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

   10. Applied *-un-lft-identity_binary64 0.0

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

   11. Applied times-frac_binary64 0.0

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

   12. Applied unpow-prod-down_binary64 0.0

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

   13. Applied cbrt-prod_binary64 0.0

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

   14. Simplified 0.0

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

   15. Simplified 0.0

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

   if -8.1784553249516665e-4 < x

   1. Initial program 60.0

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

   2. Taylor expanded around 0 0.3

      \[\leadsto \color{blue}{0.5 \cdot x + \left(0.16666666666666666 \cdot {x}^{2} + \left(0.041666666666666664 \cdot {x}^{3} + 1\right)\right)}\]

   3. Simplified 0.3

      \[\leadsto \color{blue}{1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0008178455324951667:\\ \;\;\;\;\frac{\frac{1}{\sqrt{1 + e^{x}}} \cdot \frac{{\left(e^{2}\right)}^{x} - 1}{\sqrt{1 + e^{x}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot \left(0.16666666666666666 + x \cdot 0.041666666666666664\right)\right)\\ \end{array}\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(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))