Average Error: 58.7 → 0.5
Time: 1.1m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 2.0358482214599566 \cdot 10^{-35}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

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

Original58.7
Target13.8
Herbie0.5
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -inf.0 or 2.0358482214599566e-35 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

    1. Initial program 61.9

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

    2. Taylor expanded around 0 0.2

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

    if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 2.0358482214599566e-35

    1. Initial program 4.4

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1072936661 1621281212 3440817831 3219514234 460296804 1258167384)' 
(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))))