Average Error: 58.5 → 2.7
Time: 39.8s
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le -5.3266218346219837 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le 1.0560859396603726 \cdot 10^{-191}:\\ \;\;\;\;\frac{(\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)}\right) + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.5
Target14.7
Herbie2.7
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (/ 1 b) (/ 1 a)) < -5.3266218346219837e-160 or 1.0560859396603726e-191 < (+ (/ 1 b) (/ 1 a))

    1. Initial program 59.7

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

    2. Applied simplify59.8

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

    3. Taylor expanded around 0 2.3

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

    if -5.3266218346219837e-160 < (+ (/ 1 b) (/ 1 a)) < 1.0560859396603726e-191

    1. Initial program 24.3

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

    2. Applied simplify41.8

      \[\leadsto \color{blue}{\frac{(\varepsilon \cdot \left({\left(e^{\varepsilon}\right)}^{\left(a + b\right)}\right) + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}}\]
    3. Using strategy rm

    4. Applied pow-exp14.5

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

Runtime

Time bar (total: 39.8s)Debug logProfile

herbie shell --seed '#(1070578969 3140398606 632207097 462683394 1189254563 964980650)' +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))))