Average Error: 58.4 → 0.6
Time: 1.1m
Precision: 64
Internal Precision: 2432
\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le -1.3039721609812887 \cdot 10^{-22}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{if}\;\sqrt[3]{\frac{1}{b} + \frac{1}{a}} \le 4.651414381084025 \cdot 10^{-18}:\\ \;\;\;\;\frac{\sqrt[3]{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*} \cdot \sqrt[3]{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}{(e^{\varepsilon \cdot a} - 1)^*} \cdot \frac{\sqrt[3]{(e^{\left(b + a\right) \cdot \varepsilon} - 1)^*}}{\frac{(e^{\varepsilon \cdot b} - 1)^*}{\varepsilon}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original58.4
Target14.1
Herbie0.6
\[\frac{a + b}{a \cdot b}\]

Derivation

  1. Split input into 2 regimes

  2. if (cbrt (+ (/ 1 b) (/ 1 a))) < -1.3039721609812887e-22 or 4.651414381084025e-18 < (cbrt (+ (/ 1 b) (/ 1 a)))

    1. Initial program 61.6

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

    2. Applied simplify31.5

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

    3. Taylor expanded around 0 0.5

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

    if -1.3039721609812887e-22 < (cbrt (+ (/ 1 b) (/ 1 a))) < 4.651414381084025e-18

    1. Initial program 37.0

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

    2. Applied simplify0.9

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

    4. Applied div-inv1.0

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

    5. Applied *-un-lft-identity1.0

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

    6. Applied add-cube-cbrt1.4

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

    7. Applied times-frac1.4

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

    8. Applied times-frac1.3

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

    9. Applied simplify1.3

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

    10. Applied simplify1.3

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

Runtime

Time bar (total: 1.1m)Debug logProfile

herbie shell --seed '#(1064397287 3527694221 3797617954 1138343853 2854031332 1153838279)' +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))))