Average Error: 58.6 → 0.6
Time: 57.1s
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \left(\sqrt[3]{\frac{1}{b}}\right) + \left(\frac{1}{a}\right))_* \le -2.1394519520469999 \cdot 10^{-97}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;(\left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \left(\sqrt[3]{\frac{1}{b}}\right) + \left(\frac{1}{a}\right))_* \le 1.1557750579452375 \cdot 10^{-09}:\\ \;\;\;\;\frac{\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{(e^{\varepsilon \cdot b} - 1)^*}}{(e^{a \cdot \varepsilon} - 1)^* \cdot \frac{1}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target14.0
Herbie0.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (fma (* (cbrt (/ 1 b)) (cbrt (/ 1 b))) (cbrt (/ 1 b)) (/ 1 a)) < -2.1394519520469999e-97 or 1.1557750579452375e-09 < (fma (* (cbrt (/ 1 b)) (cbrt (/ 1 b))) (cbrt (/ 1 b)) (/ 1 a))

    1. Initial program 61.6

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

    2. Applied simplify32.2

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

    3. Taylor expanded around 0 0.5

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

    if -2.1394519520469999e-97 < (fma (* (cbrt (/ 1 b)) (cbrt (/ 1 b))) (cbrt (/ 1 b)) (/ 1 a)) < 1.1557750579452375e-09

    1. Initial program 41.6

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

    2. Applied simplify0.9

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

    4. Applied div-inv1.0

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

Runtime

Time bar (total: 57.1s)Debug logProfile

herbie shell --seed '#(1064173506 2580572819 2847706409 4129882574 1125180799 1845288547)' +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))))