Average Error: 58.8 → 0.8
Time: 2.2m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le -2.7852926185370486 \cdot 10^{-33}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 1.782816446386849 \cdot 10^{+308}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{\left(\varepsilon \cdot \varepsilon\right) \cdot (e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}{\frac{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}}}{\frac{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}{\varepsilon} \cdot \frac{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}{(e^{\left(a + b\right) \cdot \varepsilon} - 1)^*}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

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

Original58.8
Target14.4
Herbie0.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < -2.7852926185370486e-33 or 1.782816446386849e+308 < (/ (* 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 0.3

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

    if -2.7852926185370486e-33 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (- (exp (* a eps)) 1) (- (exp (* b eps)) 1))) < 1.782816446386849e+308

    1. Initial program 4.3

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

    3. Applied add-cbrt-cube11.9

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

    4. Applied simplify9.4

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

Runtime

Time bar (total: 2.2m)Debug logProfile

herbie shell --seed 2018166 +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))))