Average Error: 58.5 → 4.0
Time: 37.6s
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;b \le 1.5030710836592605 \cdot 10^{+80} \lor \neg \left(b \le 1.750898411564071 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{b \cdot \varepsilon} - 1)^*} \cdot \frac{1}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{\varepsilon}}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original58.5
Target14.1
Herbie4.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if b < 1.5030710836592605e+80 or 1.750898411564071e+160 < b

    1. Initial program 58.7

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

    2. Taylor expanded around 0 3.3

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

    if 1.5030710836592605e+80 < b < 1.750898411564071e+160

    1. Initial program 54.6

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

    3. Applied times-frac54.7

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

    4. Simplified44.4

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

    5. Simplified16.2

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

    7. Applied clear-num16.1

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 1.5030710836592605 \cdot 10^{+80} \lor \neg \left(b \le 1.750898411564071 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^*}{(e^{b \cdot \varepsilon} - 1)^*} \cdot \frac{1}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{\varepsilon}}\\ \end{array}\]

Runtime

Time bar (total: 37.6s)Debug logProfile

herbie shell --seed 2018221 +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))))