expq3 (problem 3.4.2)

Average Error: 58.8 → 3.8

Time: 33.4s

Precision: 64

• could not determine a ground truth for program body

  1. a = -32298.533864343164
  2. b = -5.585987783226601e+189
  3. eps = -3.153647452178881e-153

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le 8.414834018041844 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;a \le 2.937678032680349 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \cdot \mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, b, a \cdot \varepsilon\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r1725733 = eps;
        double r1725734 = a;
        double r1725735 = b;
        double r1725736 = r1725734 + r1725735;
        double r1725737 = r1725736 * r1725733;
        double r1725738 = exp(r1725737);
        double r1725739 = 1.0;
        double r1725740 = r1725738 - r1725739;
        double r1725741 = r1725733 * r1725740;
        double r1725742 = r1725734 * r1725733;
        double r1725743 = exp(r1725742);
        double r1725744 = r1725743 - r1725739;
        double r1725745 = r1725735 * r1725733;
        double r1725746 = exp(r1725745);
        double r1725747 = r1725746 - r1725739;
        double r1725748 = r1725744 * r1725747;
        double r1725749 = r1725741 / r1725748;
        return r1725749;
}

↓

double f(double a, double b, double eps) {
        double r1725750 = a;
        double r1725751 = 8.414834018041844e+137;
        bool r1725752 = r1725750 <= r1725751;
        double r1725753 = 1.0;
        double r1725754 = b;
        double r1725755 = r1725753 / r1725754;
        double r1725756 = r1725753 / r1725750;
        double r1725757 = r1725755 + r1725756;
        double r1725758 = 2.937678032680349e+248;
        bool r1725759 = r1725750 <= r1725758;
        double r1725760 = eps;
        double r1725761 = r1725750 * r1725760;
        double r1725762 = expm1(r1725761);
        double r1725763 = r1725760 / r1725762;
        double r1725764 = sqrt(r1725763);
        double r1725765 = fma(r1725760, r1725754, r1725761);
        double r1725766 = expm1(r1725765);
        double r1725767 = r1725764 * r1725766;
        double r1725768 = r1725754 * r1725760;
        double r1725769 = expm1(r1725768);
        double r1725770 = r1725767 / r1725769;
        double r1725771 = r1725770 * r1725764;
        double r1725772 = r1725759 ? r1725771 : r1725757;
        double r1725773 = r1725752 ? r1725757 : r1725772;
        return r1725773;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.8
Target	14.2
Herbie	3.8

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

Derivation

1. Split input into 2 regimes

2. if a < 8.414834018041844e+137 or 2.937678032680349e+248 < a

   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. Simplified 40.3

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]

   3. Taylor expanded around 0 2.8

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

   if 8.414834018041844e+137 < a < 2.937678032680349e+248

   1. Initial program 53.0

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

   2. Simplified 25.6

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot b\right) \cdot \mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]
   3. Using strategy rm

   4. Applied times-frac 17.4

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 17.5

      \[\leadsto \frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \color{blue}{\left(\sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}} \cdot \sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\right)}\]

   7. Applied associate-*r* 17.5

      \[\leadsto \color{blue}{\left(\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\right) \cdot \sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}}\]

   8. Simplified 17.0

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}} \cdot \mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, b, \varepsilon \cdot a\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}} \cdot \sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 3.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 8.414834018041844 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;a \le 2.937678032680349 \cdot 10^{+248}:\\ \;\;\;\;\frac{\sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}} \cdot \mathsf{expm1}\left(\mathsf{fma}\left(\varepsilon, b, a \cdot \varepsilon\right)\right)}{\mathsf{expm1}\left(b \cdot \varepsilon\right)} \cdot \sqrt{\frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 +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))))