Average Error: 58.5 → 0.2
Time: 54.7s
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le -5.879714522759736 \cdot 10^{-21}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 6.302490121618446 \cdot 10^{+111}:\\ \;\;\;\;\frac{(\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)}\right) + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.5
Target13.6
Herbie0.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -5.879714522759736e-21 or 6.302490121618446e+111 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

    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. Applied simplify60.7

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

    3. Taylor expanded around 0 0.2

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

    if -5.879714522759736e-21 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 6.302490121618446e+111

    1. Initial program 4.2

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

    2. Applied simplify32.1

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

    4. Applied pow-exp0.3

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

Runtime

Time bar (total: 54.7s)Debug logProfile

herbie shell --seed 2018199 +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))))