Average Error: 58.6 → 3.6
Time: 29.3s
Precision: 64
Internal Precision: 128
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le -3.254099443235512 \cdot 10^{+72}:\\ \;\;\;\;\frac{b + a}{b} \cdot \frac{1}{a}\\ \mathbf{elif}\;b \le 1.2151906213357434 \cdot 10^{-45}:\\ \;\;\;\;\frac{1 + \frac{b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a}{b} \cdot \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target14.4
Herbie3.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if b < -3.254099443235512e+72 or 1.2151906213357434e-45 < b

    1. Initial program 54.2

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

    2. Taylor expanded around 0 7.6

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

    4. Applied frac-add13.4

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

    5. Simplified13.4

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

    7. Applied *-un-lft-identity13.4

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

    8. Applied times-frac7.8

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

    if -3.254099443235512e+72 < b < 1.2151906213357434e-45

    1. Initial program 61.7

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

    2. Taylor expanded around 0 0.3

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

    4. Applied frac-add15.1

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

    5. Simplified15.1

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

    7. Applied associate-/r*0.7

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

    8. Taylor expanded around inf 0.7

      \[\leadsto \frac{\color{blue}{1 + \frac{b}{a}}}{b}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification3.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -3.254099443235512 \cdot 10^{+72}:\\ \;\;\;\;\frac{b + a}{b} \cdot \frac{1}{a}\\ \mathbf{elif}\;b \le 1.2151906213357434 \cdot 10^{-45}:\\ \;\;\;\;\frac{1 + \frac{b}{a}}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{b + a}{b} \cdot \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019053 
(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))))