Kahan's exp quotient

Average Error: 39.8 → 0.3

Time: 2.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -0.00014159864022581637:\\ \;\;\;\;\frac{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{3} + -1}{\left({\left(\sqrt[3]{e^{x}}\right)}^{8} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{4} + \left({\left(e^{x}\right)}^{2} + 1\right)\right) \cdot \left(e^{x} + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array}\]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -0.00014159864022581637)
   (/
    (/
     (+ (pow (pow (exp x) 2.0) 3.0) -1.0)
     (*
      (+
       (* (pow (cbrt (exp x)) 8.0) (pow (cbrt (exp x)) 4.0))
       (+ (pow (exp x) 2.0) 1.0))
      (+ (exp x) 1.0)))
    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 <= -0.00014159864022581637) {
		tmp = ((pow(pow(exp(x), 2.0), 3.0) + -1.0) / (((pow(cbrt(exp(x)), 8.0) * pow(cbrt(exp(x)), 4.0)) + (pow(exp(x), 2.0) + 1.0)) * (exp(x) + 1.0))) / x;
	} else {
		tmp = 1.0 + (x * (0.5 + (x * 0.16666666666666666)));
	}
	return tmp;
}

Error

Bits error versus x

Target

Original	39.8
Target	40.2
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 < -1.4159864022581637e-4

   1. Initial program 0.1

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

   3. Applied flip--_binary64 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 flip3-+_binary64 0.1

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

   7. Applied associate-/l/_binary64 0.1

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

   8. Simplified 0.1

      \[\leadsto \frac{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{3} + {-1}^{3}}{\color{blue}{\left({\left(e^{x}\right)}^{4} + \left({\left(e^{x}\right)}^{2} + 1\right)\right) \cdot \left(e^{x} + 1\right)}}}{x}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt_binary64 0.1

       \[\leadsto \frac{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{3} + {-1}^{3}}{\left({\color{blue}{\left(\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}\right)}}^{4} + \left({\left(e^{x}\right)}^{2} + 1\right)\right) \cdot \left(e^{x} + 1\right)}}{x}\]

   11. Applied unpow-prod-down_binary64 0.1

       \[\leadsto \frac{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{3} + {-1}^{3}}{\left(\color{blue}{{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)}^{4} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{4}} + \left({\left(e^{x}\right)}^{2} + 1\right)\right) \cdot \left(e^{x} + 1\right)}}{x}\]

   12. Simplified 0.1

       \[\leadsto \frac{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{3} + {-1}^{3}}{\left(\color{blue}{{\left(\sqrt[3]{e^{x}}\right)}^{8}} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{4} + \left({\left(e^{x}\right)}^{2} + 1\right)\right) \cdot \left(e^{x} + 1\right)}}{x}\]

   if -1.4159864022581637e-4 < x

   1. Initial program 60.2

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

   2. Taylor expanded around 0 0.4

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

   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

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00014159864022581637:\\ \;\;\;\;\frac{\frac{{\left({\left(e^{x}\right)}^{2}\right)}^{3} + -1}{\left({\left(\sqrt[3]{e^{x}}\right)}^{8} \cdot {\left(\sqrt[3]{e^{x}}\right)}^{4} + \left({\left(e^{x}\right)}^{2} + 1\right)\right) \cdot \left(e^{x} + 1\right)}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(0.5 + x \cdot 0.16666666666666666\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020232 
(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))