Average Error: 58.7 → 0.4
Time: 1.1m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{a} + \frac{1}{b} \le -3.616429613952119 \cdot 10^{-18} \lor \neg \left(\frac{1}{a} + \frac{1}{b} \le 3.0085043408699374 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\varepsilon \cdot \left(a + b\right)} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.7
Target14.3
Herbie0.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (/ 1 b) (/ 1 a)) < -3.616429613952119e-18 or 3.0085043408699374e-32 < (+ (/ 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 -3.616429613952119e-18 < (+ (/ 1 b) (/ 1 a)) < 3.0085043408699374e-32

    1. Initial program 43.5

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

    3. Applied times-frac43.5

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

    4. Applied simplify33.8

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

    5. Applied simplify1.5

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

  4. Applied simplify0.4

    \[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{1}{a} + \frac{1}{b} \le -3.616429613952119 \cdot 10^{-18} \lor \neg \left(\frac{1}{a} + \frac{1}{b} \le 3.0085043408699374 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\varepsilon \cdot \left(a + b\right)} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*} \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}\\ \end{array}}\]

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1071501266 3581234924 1086666455 2685055582 1243441566 1802958749)' +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))))