Average Error: 58.8 → 2.8
Time: 1.5m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}}\right) \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le -2.5485283913268967 \cdot 10^{-220}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\left(\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}}\right) \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le 1.1771941667951555 \cdot 10^{-182}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.8
Target14.1
Herbie2.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (+ (/ 1 b) (/ 1 a)))) < -2.5485283913268967e-220 or 1.1771941667951555e-182 < (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (+ (/ 1 b) (/ 1 a))))

    1. Initial program 59.6

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

    2. Taylor expanded around 0 2.4

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

    if -2.5485283913268967e-220 < (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (+ (/ 1 b) (/ 1 a)))) < 1.1771941667951555e-182

    1. Initial program 20.3

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

Runtime

Time bar (total: 1.5m)Debug logProfile

herbie shell --seed '#(1070833653 108281690 3330367898 3632331308 3494323072 43156186)' 
(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))))