Average Error: 58.4 → 0.1
Time: 1.1m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\sqrt[3]{{\left((e^{\varepsilon \cdot a} - 1)^*\right)}^{3}} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\sqrt[3]{{\left((e^{\varepsilon \cdot a} - 1)^*\right)}^{3}} \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 1.4571872936964753 \cdot 10^{+23}:\\ \;\;\;\;\frac{\varepsilon}{\frac{(e^{b \cdot \varepsilon} - 1)^* \cdot (e^{\varepsilon \cdot a} - 1)^*}{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.4
Target14.3
Herbie0.1
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (cbrt (pow (expm1 (* eps a)) 3)) (- (exp (* b eps)) 1))) < -inf.0 or 1.4571872936964753e+23 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (cbrt (pow (expm1 (* eps a)) 3)) (- (exp (* b eps)) 1)))

    1. Initial program 62.2

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

    2. Taylor expanded around 0 0.0

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

    if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (cbrt (pow (expm1 (* eps a)) 3)) (- (exp (* b eps)) 1))) < 1.4571872936964753e+23

    1. Initial program 20.8

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

    3. Applied associate-/l*20.8

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

    4. Applied simplify0.1

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1072330854 3074818769 591214268 3603999196 3863745332 3332387116)' +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))))