Average Error: 58.5 → 3.3
Time: 44.1s
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le -2.2156921429409975 \cdot 10^{-83}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le 1.1434035551063597 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{e^{\left(a + b\right) \cdot \varepsilon}}{(e^{\varepsilon \cdot b} - 1)^*}}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{\varepsilon}} - \frac{\frac{\varepsilon}{(e^{b \cdot \varepsilon} - 1)^*}}{(e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.5
Target14.8
Herbie3.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (/ 1 b) (/ 1 a)) < -2.2156921429409975e-83 or 1.1434035551063597e-50 < (+ (/ 1 b) (/ 1 a))

    1. Initial program 61.5

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

    2. Applied simplify30.9

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

    3. Taylor expanded around 0 0.6

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

    if -2.2156921429409975e-83 < (+ (/ 1 b) (/ 1 a)) < 1.1434035551063597e-50

    1. Initial program 37.7

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

    2. Applied simplify1.2

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

    4. Applied expm1-udef22.3

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

    5. Applied div-sub21.9

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

    6. Applied div-sub22.4

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

    7. Applied simplify22.3

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

Runtime

Time bar (total: 44.1s)Debug logProfile

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