Average Error: 58.7 → 0.5
Time: 41.3s
Precision: 64
Internal Precision: 128
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 1.3598821982296037 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

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

Original58.7
Target14.4
Herbie0.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -inf.0 or 1.3598821982296037e-31 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

    1. Initial program 61.9

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

    2. Taylor expanded around 0 0.3

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

    if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 1.3598821982296037e-31

    1. Initial program 4.0

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

    2. Taylor expanded around -inf 4.0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(e^{a \cdot \varepsilon} - 1\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 1.3598821982296037 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019026 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1 eps) (< eps 1))

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

  (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))))