Average Error: 58.5 → 2.8
Time: 1.2m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le -7.60456563691005 \cdot 10^{-167}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le 1.0252251552436592 \cdot 10^{-191}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\left(\sqrt[3]{e^{a \cdot \varepsilon} - 1} \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \sqrt[3]{e^{a \cdot \varepsilon} - 1}\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.5
Target14.8
Herbie2.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (/ 1 b) (/ 1 a)) < -7.60456563691005e-167 or 1.0252251552436592e-191 < (+ (/ 1 b) (/ 1 a))

    1. Initial program 59.8

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

    2. Taylor expanded around 0 2.2

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

    if -7.60456563691005e-167 < (+ (/ 1 b) (/ 1 a)) < 1.0252251552436592e-191

    1. Initial program 21.0

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

    3. Applied add-cube-cbrt21.0

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070258749 1877548225 2229079127 1588002776 3179087814 1886870650)' 
(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))))