Average Error: 58.6 → 2.5
Time: 37.4s
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;(\left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \left(\sqrt[3]{\frac{1}{b}}\right) + \left(\frac{1}{a}\right))_* \le -8.48095371942162 \cdot 10^{-116}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;(\left(\sqrt[3]{\frac{1}{b}} \cdot \sqrt[3]{\frac{1}{b}}\right) \cdot \left(\sqrt[3]{\frac{1}{b}}\right) + \left(\frac{1}{a}\right))_* \le 9.200663417868296 \cdot 10^{-181}:\\ \;\;\;\;\frac{(\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)}\right) + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target14.6
Herbie2.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (fma (* (cbrt (/ 1 b)) (cbrt (/ 1 b))) (cbrt (/ 1 b)) (/ 1 a)) < -8.48095371942162e-116 or 9.200663417868296e-181 < (fma (* (cbrt (/ 1 b)) (cbrt (/ 1 b))) (cbrt (/ 1 b)) (/ 1 a))

    1. Initial program 60.4

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

    2. Applied simplify59.9

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

    3. Taylor expanded around 0 1.7

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

    if -8.48095371942162e-116 < (fma (* (cbrt (/ 1 b)) (cbrt (/ 1 b))) (cbrt (/ 1 b)) (/ 1 a)) < 9.200663417868296e-181

    1. Initial program 29.4

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

    2. Applied simplify44.8

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

    4. Applied pow-exp17.1

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

Runtime

Time bar (total: 37.4s)Debug logProfile

herbie shell --seed '#(1072107073 2127697367 3936270018 2300570620 2134894798 4023771849)' +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))))