Average Error: 58.5 → 0.3
Time: 1.4m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le -1.7836187891635008 \cdot 10^{+308}:\\ \;\;\;\;\frac{\frac{1}{b}}{1 - \frac{b}{a}} - \frac{\frac{1}{a}}{\frac{a}{b} - 1}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 1.781894697276848 \cdot 10^{+308}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{b}}{1 - \frac{b}{a}} - \frac{\frac{1}{a}}{\frac{a}{b} - 1}\\ \end{array}\]

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

Original58.5
Target14.7
Herbie0.3
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -1.7836187891635008e+308 or 1.781894697276848e+308 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

    1. Initial program 62.2

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

    2. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{1}{b} + \frac{1}{a}}\]
    3. Using strategy rm

    4. Applied flip-+31.5

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

    5. Applied simplify31.6

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{b}}{b} - \frac{\frac{1}{a}}{a}}}{\frac{1}{b} - \frac{1}{a}}\]
    6. Using strategy rm

    7. Applied div-sub31.6

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

    8. Applied simplify18.8

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

    9. Applied simplify0.0

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

    if -1.7836187891635008e+308 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 1.781894697276848e+308

    1. Initial program 3.8

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018178 
(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))))