Average Error: 58.3 → 2.8
Time: 2.5m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right) + \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right) + \varepsilon \cdot a\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le -2.699458588543734 \cdot 10^{-243}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {a}^{2}\right) + \left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {a}^{3}\right) + \varepsilon \cdot a\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 6.474925335532127 \cdot 10^{-204}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{\varepsilon}{e^{\varepsilon \cdot b} - 1} \cdot \frac{e^{\left(a + b\right) \cdot \varepsilon} - 1}{e^{a \cdot \varepsilon} - 1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.3
Target14.4
Herbie2.8
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* 1/2 (* (pow eps 2) (pow a 2))) (+ (* 1/6 (* (pow eps 3) (pow a 3))) (* eps a))) (- (exp (* b eps)) 1))) < -2.699458588543734e-243 or 6.474925335532127e-204 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* 1/2 (* (pow eps 2) (pow a 2))) (+ (* 1/6 (* (pow eps 3) (pow a 3))) (* eps a))) (- (exp (* b eps)) 1)))

    1. Initial program 59.2

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

    2. Taylor expanded around 0 2.8

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

    if -2.699458588543734e-243 < (/ (* eps (- (exp (* (+ a b) eps)) 1)) (* (+ (* 1/2 (* (pow eps 2) (pow a 2))) (+ (* 1/6 (* (pow eps 3) (pow a 3))) (* eps a))) (- (exp (* b eps)) 1))) < 6.474925335532127e-204

    1. Initial program 1.6

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

    3. Applied add-cbrt-cube2.1

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

    4. Applied add-cbrt-cube3.5

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

    5. Applied cbrt-undiv3.5

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

    6. Applied simplify3.5

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

Runtime

Time bar (total: 2.5m)Debug logProfile

herbie shell --seed '#(1070609872 3456127585 2380521889 2328837196 1765472538 734540918)' 
(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))))