expq3 (problem 3.4.2)

Average Error: 60.2 → 53.5

Time: 12.9s

Precision: 64

• could not determine a ground truth for program body

  1. a = 1.3277980151911176e-254
  2. b = 2.418429179592415e+245
  3. eps = 5.185358054714819e-68

\[-1 \lt \varepsilon \land \varepsilon \lt 1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.969337513991444315376627752626075518669 \cdot 10^{72} \lor \neg \left(a \le 9.71614083792645651101853201912203907943 \cdot 10^{126}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \sqrt[3]{{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}^{3}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left(\left({\varepsilon}^{3} \cdot b\right) \cdot b\right) \cdot {\left(\sqrt[3]{b}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r104705 = eps;
        double r104706 = a;
        double r104707 = b;
        double r104708 = r104706 + r104707;
        double r104709 = r104708 * r104705;
        double r104710 = exp(r104709);
        double r104711 = 1.0;
        double r104712 = r104710 - r104711;
        double r104713 = r104705 * r104712;
        double r104714 = r104706 * r104705;
        double r104715 = exp(r104714);
        double r104716 = r104715 - r104711;
        double r104717 = r104707 * r104705;
        double r104718 = exp(r104717);
        double r104719 = r104718 - r104711;
        double r104720 = r104716 * r104719;
        double r104721 = r104713 / r104720;
        return r104721;
}

↓

double f(double a, double b, double eps) {
        double r104722 = a;
        double r104723 = -1.9693375139914443e+72;
        bool r104724 = r104722 <= r104723;
        double r104725 = 9.716140837926457e+126;
        bool r104726 = r104722 <= r104725;
        double r104727 = !r104726;
        bool r104728 = r104724 || r104727;
        double r104729 = eps;
        double r104730 = b;
        double r104731 = r104722 + r104730;
        double r104732 = r104731 * r104729;
        double r104733 = exp(r104732);
        double r104734 = 1.0;
        double r104735 = r104733 - r104734;
        double r104736 = 3.0;
        double r104737 = pow(r104735, r104736);
        double r104738 = cbrt(r104737);
        double r104739 = r104729 * r104738;
        double r104740 = r104722 * r104729;
        double r104741 = exp(r104740);
        double r104742 = r104741 - r104734;
        double r104743 = 0.16666666666666666;
        double r104744 = pow(r104729, r104736);
        double r104745 = r104744 * r104730;
        double r104746 = r104745 * r104730;
        double r104747 = cbrt(r104730);
        double r104748 = pow(r104747, r104736);
        double r104749 = r104746 * r104748;
        double r104750 = 0.5;
        double r104751 = 2.0;
        double r104752 = pow(r104729, r104751);
        double r104753 = pow(r104730, r104751);
        double r104754 = r104752 * r104753;
        double r104755 = r104729 * r104730;
        double r104756 = fma(r104750, r104754, r104755);
        double r104757 = fma(r104743, r104749, r104756);
        double r104758 = r104742 * r104757;
        double r104759 = r104739 / r104758;
        double r104760 = r104729 * r104735;
        double r104761 = pow(r104722, r104736);
        double r104762 = r104761 * r104744;
        double r104763 = pow(r104722, r104751);
        double r104764 = r104763 * r104752;
        double r104765 = fma(r104750, r104764, r104740);
        double r104766 = fma(r104743, r104762, r104765);
        double r104767 = r104730 * r104729;
        double r104768 = exp(r104767);
        double r104769 = r104768 - r104734;
        double r104770 = r104766 * r104769;
        double r104771 = r104760 / r104770;
        double r104772 = r104728 ? r104759 : r104771;
        return r104772;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	14.9
Herbie	53.5

\[\frac{a + b}{a \cdot b}\]

Derivation

1. Split input into 2 regimes

2. if a < -1.9693375139914443e+72 or 9.716140837926457e+126 < a

   1. Initial program 52.9

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

   2. Taylor expanded around 0 46.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\left(\frac{1}{6} \cdot \left({\varepsilon}^{3} \cdot {b}^{3}\right) + \left(\frac{1}{2} \cdot \left({\varepsilon}^{2} \cdot {b}^{2}\right) + \varepsilon \cdot b\right)\right)}}\]

   3. Simplified 46.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {b}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}}\]
   4. Using strategy rm

   5. Applied add-cube-cbrt 46.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot {\color{blue}{\left(\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right) \cdot \sqrt[3]{b}\right)}}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\]

   6. Applied unpow-prod-down 46.9

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, {\varepsilon}^{3} \cdot \color{blue}{\left({\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{3} \cdot {\left(\sqrt[3]{b}\right)}^{3}\right)}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\]

   7. Applied associate-*r* 46.0

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left({\varepsilon}^{3} \cdot {\left(\sqrt[3]{b} \cdot \sqrt[3]{b}\right)}^{3}\right) \cdot {\left(\sqrt[3]{b}\right)}^{3}}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\]

   8. Simplified 45.3

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \color{blue}{\left(\left({\varepsilon}^{3} \cdot b\right) \cdot b\right)} \cdot {\left(\sqrt[3]{b}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\]
   9. Using strategy rm

   10. Applied add-cbrt-cube 45.4

       \[\leadsto \frac{\varepsilon \cdot \color{blue}{\sqrt[3]{\left(\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)\right) \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left(\left({\varepsilon}^{3} \cdot b\right) \cdot b\right) \cdot {\left(\sqrt[3]{b}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\]

   11. Simplified 45.4

       \[\leadsto \frac{\varepsilon \cdot \sqrt[3]{\color{blue}{{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}^{3}}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left(\left({\varepsilon}^{3} \cdot b\right) \cdot b\right) \cdot {\left(\sqrt[3]{b}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\]

   if -1.9693375139914443e+72 < a < 9.716140837926457e+126

   1. Initial program 63.0

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

   2. Taylor expanded around 0 56.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

   3. Simplified 56.5

      \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 53.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.969337513991444315376627752626075518669 \cdot 10^{72} \lor \neg \left(a \le 9.71614083792645651101853201912203907943 \cdot 10^{126}\right):\\ \;\;\;\;\frac{\varepsilon \cdot \sqrt[3]{{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}^{3}}}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \mathsf{fma}\left(\frac{1}{6}, \left(\left({\varepsilon}^{3} \cdot b\right) \cdot b\right) \cdot {\left(\sqrt[3]{b}\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {\varepsilon}^{2} \cdot {b}^{2}, \varepsilon \cdot b\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :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))))