Average Error: 58.6 → 0.3
Time: 1.2m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{a} + \frac{1}{b} \le -2586.5134666990707:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{if}\;\frac{1}{a} + \frac{1}{b} \le 1.6980593969186866 \cdot 10^{-48}:\\ \;\;\;\;\frac{\varepsilon}{\frac{(e^{\varepsilon \cdot b} - 1)^*}{1} \cdot \frac{(e^{a \cdot \varepsilon} - 1)^*}{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target13.8
Herbie0.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (/ 1 a) (/ 1 b)) < -2586.5134666990707 or 1.6980593969186866e-48 < (+ (/ 1 a) (/ 1 b))

    1. Initial program 62.0

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

    2. Applied simplify44.1

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

    3. Taylor expanded around 0 0.1

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

    if -2586.5134666990707 < (+ (/ 1 a) (/ 1 b)) < 1.6980593969186866e-48

    1. Initial program 43.6

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

    2. Applied simplify6.6

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

    4. Applied *-un-lft-identity6.6

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

    5. Applied times-frac1.3

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

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed '#(1070355188 2193211668 3977393919 3454156579 3755371326 1656365382)' +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))))