Average Error: 60.2 → 11.0
Time: 10.8s
Precision: binary64
\[-1 < \varepsilon \land \varepsilon < 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{a} + \left(0.5 \cdot \frac{a \cdot \varepsilon}{b} + \frac{1}{b}\right)\]
\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}
↓
\frac{1}{a} + \left(0.5 \cdot \frac{a \cdot \varepsilon}{b} + \frac{1}{b}\right)
(FPCore (a b eps)
 :precision binary64
 (/
  (* eps (- (exp (* (+ a b) eps)) 1.0))
  (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))
↓
(FPCore (a b eps)
 :precision binary64
 (+ (/ 1.0 a) (+ (* 0.5 (/ (* a eps) b)) (/ 1.0 b))))
double code(double a, double b, double eps) {
	return (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) exp(((double) (b * eps)))) - 1.0)))));
}
↓
double code(double a, double b, double eps) {
	return ((double) ((1.0 / a) + ((double) (((double) (0.5 * (((double) (a * eps)) / b))) + (1.0 / b)))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original60.2
Target15.5
Herbie11.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Initial program 60.2

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

  2. Taylor expanded around 0 58.1

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(0.16666666666666666 \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(0.5 \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  3. Simplified58.1

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(0.16666666666666666 \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \varepsilon \cdot \left(a + \varepsilon \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  4. Using strategy rm

  5. Applied pow-prod-down_binary6457.4

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(0.16666666666666666 \cdot \color{blue}{{\left(a \cdot \varepsilon\right)}^{3}} + \varepsilon \cdot \left(a + \varepsilon \cdot \left(0.5 \cdot \left(a \cdot a\right)\right)\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  6. Taylor expanded around 0 56.7

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(0.16666666666666666 \cdot {\left(a \cdot \varepsilon\right)}^{3} + \color{blue}{a \cdot \varepsilon}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  7. Simplified56.7

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(0.16666666666666666 \cdot {\left(a \cdot \varepsilon\right)}^{3} + \color{blue}{\varepsilon \cdot a}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

  8. Taylor expanded around 0 11.0

    \[\leadsto \color{blue}{\frac{1}{a} + \left(0.5 \cdot \frac{a \cdot \varepsilon}{b} + \frac{1}{b}\right)}\]

  9. Final simplification11.0

    \[\leadsto \frac{1}{a} + \left(0.5 \cdot \frac{a \cdot \varepsilon}{b} + \frac{1}{b}\right)\]

Reproduce

herbie shell --seed 2020210 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))