Average Error: 58.5 → 1.0
Time: 2.1m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le -5.175057551918931 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 3.8013137572657295 \cdot 10^{+88}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\frac{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{\frac{(e^{a \cdot \varepsilon} - 1)^*}{\varepsilon}}}{(e^{\varepsilon \cdot b} - 1)^*}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

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

Original58.5
Target14.0
Herbie1.0
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -5.175057551918931e-28 or 3.8013137572657295e+88 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1)))

    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. Taylor expanded around 0 0.3

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

    if -5.175057551918931e-28 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 3.8013137572657295e+88

    1. Initial program 2.7

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

    3. Applied add-cbrt-cube2.7

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\sqrt[3]{\left(\left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right) \cdot \left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \cdot \left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)}}}\]

    4. Applied add-cbrt-cube13.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)\right) \cdot \left(\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)\right)\right) \cdot \left(\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)\right)}}}{\sqrt[3]{\left(\left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right) \cdot \left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \cdot \left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)}}\]

    5. Applied cbrt-undiv13.5

      \[\leadsto \color{blue}{\sqrt[3]{\frac{\left(\left(\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)\right) \cdot \left(\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)\right)\right) \cdot \left(\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)\right)}{\left(\left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right) \cdot \left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)\right) \cdot \left(\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)\right)}}}\]

    6. Applied simplify12.1

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

Runtime

Time bar (total: 2.1m)Debug logProfile

herbie shell --seed 2018207 +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))))