expq3 (problem 3.4.2)

Average Error: 59.1 → 3.1

Time: 29.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = -5.5526078831285184e+17
  2. b = 3.113802564449468e+295
  3. eps = 2.7616000569276128e-84

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le 9.918390622765604 \cdot 10^{+229}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;a \le 2.194671702223418 \cdot 10^{+284}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r4713529 = eps;
        double r4713530 = a;
        double r4713531 = b;
        double r4713532 = r4713530 + r4713531;
        double r4713533 = r4713532 * r4713529;
        double r4713534 = exp(r4713533);
        double r4713535 = 1.0;
        double r4713536 = r4713534 - r4713535;
        double r4713537 = r4713529 * r4713536;
        double r4713538 = r4713530 * r4713529;
        double r4713539 = exp(r4713538);
        double r4713540 = r4713539 - r4713535;
        double r4713541 = r4713531 * r4713529;
        double r4713542 = exp(r4713541);
        double r4713543 = r4713542 - r4713535;
        double r4713544 = r4713540 * r4713543;
        double r4713545 = r4713537 / r4713544;
        return r4713545;
}

↓

double f(double a, double b, double eps) {
        double r4713546 = a;
        double r4713547 = 9.918390622765604e+229;
        bool r4713548 = r4713546 <= r4713547;
        double r4713549 = 1.0;
        double r4713550 = b;
        double r4713551 = r4713549 / r4713550;
        double r4713552 = r4713549 / r4713546;
        double r4713553 = r4713551 + r4713552;
        double r4713554 = 2.194671702223418e+284;
        bool r4713555 = r4713546 <= r4713554;
        double r4713556 = r4713550 + r4713546;
        double r4713557 = eps;
        double r4713558 = r4713556 * r4713557;
        double r4713559 = expm1(r4713558);
        double r4713560 = r4713557 * r4713546;
        double r4713561 = expm1(r4713560);
        double r4713562 = r4713557 / r4713561;
        double r4713563 = r4713559 * r4713562;
        double r4713564 = r4713550 * r4713557;
        double r4713565 = expm1(r4713564);
        double r4713566 = r4713563 / r4713565;
        double r4713567 = r4713555 ? r4713566 : r4713553;
        double r4713568 = r4713548 ? r4713553 : r4713567;
        return r4713568;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	59.1
Target	13.8
Herbie	3.1

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

Derivation

1. Split input into 2 regimes

2. if a < 9.918390622765604e+229 or 2.194671702223418e+284 < a

   1. Initial program 59.4

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

   2. Simplified 27.9

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

   3. Taylor expanded around 0 2.6

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

   if 9.918390622765604e+229 < a < 2.194671702223418e+284

   1. Initial program 50.7

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

   2. Simplified 18.0

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

   4. Applied associate-*l/ 17.8

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le 9.918390622765604 \cdot 10^{+229}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;a \le 2.194671702223418 \cdot 10^{+284}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(b + a\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019163 +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))))