Average Error: 58.4 → 2.9
Time: 1.7m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\left(\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}}\right) \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le -2.664967209614443 \cdot 10^{-192}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\left(\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}}\right) \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le 5.64983165118109 \cdot 10^{-181}:\\ \;\;\;\;\frac{\varepsilon \cdot \frac{e^{\left(\varepsilon + \varepsilon\right) \cdot \left(b + a\right)} - 1}{e^{\left(a + b\right) \cdot \varepsilon} + 1}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.4
Target13.9
Herbie2.9
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (+ (/ 1 b) (/ 1 a)))) < -2.664967209614443e-192 or 5.64983165118109e-181 < (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (+ (/ 1 b) (/ 1 a))))

    1. Initial program 59.5

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

    2. Taylor expanded around 0 2.5

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

    if -2.664967209614443e-192 < (* (* (cbrt (+ (/ 1 b) (/ 1 a))) (cbrt (+ (/ 1 b) (/ 1 a)))) (cbrt (+ (/ 1 b) (/ 1 a)))) < 5.64983165118109e-181

    1. Initial program 18.6

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

    3. Applied flip--18.6

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

    4. Applied simplify18.5

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

Runtime

Time bar (total: 1.7m)Debug logProfile

herbie shell --seed '#(1070960995 739739648 2531964651 3069671617 351857262 3877178482)' 
(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))))