Average Error: 60.6 → 3.1
Time: 2.2min
Precision: binary64
Cost: 448
\[-1 < \varepsilon \land \varepsilon < 1\]
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\frac{1}{b} + \frac{1}{a}\]
\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}{b} + \frac{1}{a}
(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 b) (/ 1.0 a)))
double code(double a, double b, double eps) {
	return (eps * (exp((a + b) * eps) - 1.0)) / ((exp(a * eps) - 1.0) * (exp(b * eps) - 1.0));
}
↓
double code(double a, double b, double eps) {
	return (1.0 / b) + (1.0 / a);
}

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

Original60.6
Target14.7
Herbie3.1
\[\frac{a + b}{a \cdot b}\]

Alternatives

Alternative 1
Error10.0
Cost1097
\[\begin{array}{l} \mathbf{if}\;a \leq -2.9368846088630596 \cdot 10^{+173}:\\ \;\;\;\;\frac{1}{b}\\ \mathbf{elif}\;a \leq -1.0774485526151028 \cdot 10^{-101} \lor \neg \left(a \leq 2.2873523470155514 \cdot 10^{-128}\right):\\ \;\;\;\;\frac{b + a}{b \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array}\]
Alternative 2
Error18.3
Cost520
\[\begin{array}{l} \mathbf{if}\;a \leq -29973173135.65804 \lor \neg \left(a \leq 2.8471803732139122 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a}\\ \end{array}\]
Alternative 3
Error33.0
Cost192
\[\frac{1}{a}\]
Alternative 4
Error61.0
Cost64
\[0\]
Alternative 5
Error61.9
Cost64
\[1\]

Error

Time

Derivation

  1. Initial program 60.6

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

  2. Taylor expanded around 0 56.8

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

  3. Simplified56.6

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

  4. Taylor expanded around 0 14.7

    \[\leadsto \color{blue}{\frac{a + b}{a \cdot b}}\]

  5. Taylor expanded around 0 3.1

    \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]

  6. Simplified3.1

    \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
  7. Using strategy rm

  8. Applied +-commutative_binary64_10313.1

    \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]

  9. Simplified3.1

    \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]

  10. Final simplification3.1

    \[\leadsto \frac{1}{b} + \frac{1}{a}\]

Reproduce

herbie shell --seed 2021040 
(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))))