Average Error: 58.6 → 1.2
Time: 51.0s
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le -8.338296356902215 \cdot 10^{-94}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\frac{1}{b} + \frac{1}{a} \le 3.5698637064956427 \cdot 10^{-130}:\\ \;\;\;\;\frac{\sqrt[3]{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*} \cdot \sqrt[3]{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}{(e^{\varepsilon \cdot a} - 1)^*} \cdot \frac{\sqrt[3]{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}{\frac{(e^{\varepsilon \cdot b} - 1)^*}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.6
Target14.5
Herbie1.2
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (+ (/ 1 b) (/ 1 a)) < -8.338296356902215e-94 or 3.5698637064956427e-130 < (+ (/ 1 b) (/ 1 a))

    1. Initial program 60.8

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

    2. Applied simplify29.7

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

    3. Taylor expanded around 0 1.2

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

    if -8.338296356902215e-94 < (+ (/ 1 b) (/ 1 a)) < 3.5698637064956427e-130

    1. Initial program 32.7

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

    2. Applied simplify1.2

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

    4. Applied div-inv1.3

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

    5. Applied *-un-lft-identity1.3

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

    6. Applied add-cube-cbrt1.6

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

    7. Applied times-frac1.6

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

    8. Applied times-frac1.6

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

    9. Applied simplify1.6

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

    10. Applied simplify1.6

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

Runtime

Time bar (total: 51.0s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +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))))