Average Error: 58.8 → 4.0
Time: 1.2m
Precision: 64
Internal Precision: 128
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;a \le -4.2007210785261985 \cdot 10^{+237} \lor \neg \left(a \le -1.9124168729062723 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^* \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}}{(e^{b \cdot \varepsilon} - 1)^*}\\ \end{array}\]

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

Original58.8
Target14.2
Herbie4.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if a < -4.2007210785261985e+237 or -1.9124168729062723e+102 < a

    1. Initial program 59.4

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

    2. Initial simplification28.4

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

    3. Taylor expanded around 0 2.7

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

    if -4.2007210785261985e+237 < a < -1.9124168729062723e+102

    1. Initial program 53.1

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

    2. Initial simplification17.1

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

    4. Applied associate-*l/16.8

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

  4. Final simplification4.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -4.2007210785261985 \cdot 10^{+237} \lor \neg \left(a \le -1.9124168729062723 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^* \cdot \frac{\varepsilon}{(e^{\varepsilon \cdot a} - 1)^*}}{(e^{b \cdot \varepsilon} - 1)^*}\\ \end{array}\]

Runtime

Time bar (total: 1.2m)Debug logProfile

herbie shell --seed 2018346 +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))))