Average Error: 58.4 → 2.4
Time: 2.4m
Precision: 64
Internal Precision: 2368
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{\sqrt[3]{\frac{1}{b} \cdot \frac{1}{b} - \frac{1}{a} \cdot \frac{1}{a}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{1}{b} \cdot \frac{1}{b} - \frac{1}{a} \cdot \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} \cdot \frac{1}{b} - \frac{1}{a} \cdot \frac{1}{a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{b} \cdot \frac{1}{b} - \frac{1}{a} \cdot \frac{1}{a}}}\right)}{\sqrt[3]{\frac{1}{b} - \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} - \frac{1}{a}}} \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le -8.822589555832524 \cdot 10^{-128}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{\sqrt[3]{\frac{1}{b} \cdot \frac{1}{b} - \frac{1}{a} \cdot \frac{1}{a}} \cdot \left(\sqrt[3]{\sqrt[3]{\frac{1}{b} \cdot \frac{1}{b} - \frac{1}{a} \cdot \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} \cdot \frac{1}{b} - \frac{1}{a} \cdot \frac{1}{a}}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{b} \cdot \frac{1}{b} - \frac{1}{a} \cdot \frac{1}{a}}}\right)}{\sqrt[3]{\frac{1}{b} - \frac{1}{a}} \cdot \sqrt[3]{\frac{1}{b} - \frac{1}{a}}} \cdot \sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le 2.068518135812299 \cdot 10^{-135}:\\ \;\;\;\;\frac{(\varepsilon \cdot \left(e^{\varepsilon \cdot \left(a + b\right)}\right) + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.4
Target14.5
Herbie2.4
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (* (/ (* (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))) (* (cbrt (* (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))) (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))))) (cbrt (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a))))))) (* (cbrt (- (/ 1 b) (/ 1 a))) (cbrt (- (/ 1 b) (/ 1 a))))) (cbrt (+ (/ 1 b) (/ 1 a)))) < -8.822589555832524e-128 or 2.068518135812299e-135 < (* (/ (* (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))) (* (cbrt (* (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))) (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))))) (cbrt (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a))))))) (* (cbrt (- (/ 1 b) (/ 1 a))) (cbrt (- (/ 1 b) (/ 1 a))))) (cbrt (+ (/ 1 b) (/ 1 a))))

    1. Initial program 60.6

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

    2. Applied simplify59.9

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

    3. Taylor expanded around 0 1.4

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

    if -8.822589555832524e-128 < (* (/ (* (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))) (* (cbrt (* (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))) (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a)))))) (cbrt (cbrt (- (* (/ 1 b) (/ 1 b)) (* (/ 1 a) (/ 1 a))))))) (* (cbrt (- (/ 1 b) (/ 1 a))) (cbrt (- (/ 1 b) (/ 1 a))))) (cbrt (+ (/ 1 b) (/ 1 a)))) < 2.068518135812299e-135

    1. Initial program 26.7

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

    2. Applied simplify42.9

      \[\leadsto \color{blue}{\frac{(\varepsilon \cdot \left({\left(e^{\varepsilon}\right)}^{\left(a + b\right)}\right) + \left(-\varepsilon\right))_*}{(e^{\varepsilon \cdot b} - 1)^* \cdot (e^{a \cdot \varepsilon} - 1)^*}}\]
    3. Using strategy rm

    4. Applied pow-exp15.8

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

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed '#(1071119240 1686926585 3481876196 78132896 2080707795 3185793749)' +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))))