Average Error: 58.6 → 0.3
Time: 2.0m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{a} + \frac{1}{b} \le -6.406869154277889 \cdot 10^{+19} \lor \neg \left(\frac{1}{a} + \frac{1}{b} \le 1.8624249245372534 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\varepsilon \cdot \left(a + b\right)} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target14.5
Herbie0.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (/ 1 b) (/ 1 a)) < -6.406869154277889e+19 or 1.8624249245372534e-29 < (+ (/ 1 b) (/ 1 a))

    1. Initial program 62.0

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

    2. Taylor expanded around 0 0.1

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

    if -6.406869154277889e+19 < (+ (/ 1 b) (/ 1 a)) < 1.8624249245372534e-29

    1. Initial program 45.7

      \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
    2. Using strategy rm

    3. Applied times-frac45.7

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

    4. Applied simplify36.9

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

    5. Applied simplify1.1

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

  4. Applied simplify0.3

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{1}{a} + \frac{1}{b} \le -6.406869154277889 \cdot 10^{+19} \lor \neg \left(\frac{1}{a} + \frac{1}{b} \le 1.8624249245372534 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\varepsilon \cdot \left(a + b\right)} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\\ \end{array}}\]

Runtime

Time bar (total: 2.0m)Debug logProfile

herbie shell --seed '#(1072361757 3390613284 2339397988 1175251238 145061547 3101881848)' +o rules:numerics
(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))))