expq3 (problem 3.4.2)

Average Error: 60.3 → 0.3

Time: 56.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = 2.8364102844858466e+132
  2. b = 1.36934023547096e+82
  3. eps = 1.5035876167361006e-51

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 0.002537838552519357249170894874623627401888:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r4639567 = eps;
        double r4639568 = a;
        double r4639569 = b;
        double r4639570 = r4639568 + r4639569;
        double r4639571 = r4639570 * r4639567;
        double r4639572 = exp(r4639571);
        double r4639573 = 1.0;
        double r4639574 = r4639572 - r4639573;
        double r4639575 = r4639567 * r4639574;
        double r4639576 = r4639568 * r4639567;
        double r4639577 = exp(r4639576);
        double r4639578 = r4639577 - r4639573;
        double r4639579 = r4639569 * r4639567;
        double r4639580 = exp(r4639579);
        double r4639581 = r4639580 - r4639573;
        double r4639582 = r4639578 * r4639581;
        double r4639583 = r4639575 / r4639582;
        return r4639583;
}

↓

double f(double a, double b, double eps) {
        double r4639584 = a;
        double r4639585 = b;
        double r4639586 = r4639584 + r4639585;
        double r4639587 = eps;
        double r4639588 = r4639586 * r4639587;
        double r4639589 = exp(r4639588);
        double r4639590 = 1.0;
        double r4639591 = r4639589 - r4639590;
        double r4639592 = r4639591 * r4639587;
        double r4639593 = r4639587 * r4639585;
        double r4639594 = exp(r4639593);
        double r4639595 = r4639594 - r4639590;
        double r4639596 = r4639587 * r4639584;
        double r4639597 = exp(r4639596);
        double r4639598 = r4639597 - r4639590;
        double r4639599 = r4639595 * r4639598;
        double r4639600 = r4639592 / r4639599;
        double r4639601 = -inf.0;
        bool r4639602 = r4639600 <= r4639601;
        double r4639603 = 1.0;
        double r4639604 = r4639603 / r4639584;
        double r4639605 = r4639603 / r4639585;
        double r4639606 = r4639604 + r4639605;
        double r4639607 = 0.0025378385525193572;
        bool r4639608 = r4639600 <= r4639607;
        double r4639609 = r4639608 ? r4639600 : r4639606;
        double r4639610 = r4639602 ? r4639606 : r4639609;
        return r4639610;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	14.9
Herbie	0.3

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

Derivation

1. Split input into 2 regimes

2. if (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < -inf.0 or 0.0025378385525193572 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program 64.0

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

   2. Taylor expanded around 0 58.1

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

   3. Simplified 58.1

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

   4. Taylor expanded around 0 0.1

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 0.0025378385525193572

   1. Initial program 3.7

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 0.002537838552519357249170894874623627401888:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :pre (and (< -1.0 eps) (< eps 1.0))

  :herbie-target
  (/ (+ a b) (* a b))

  (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))))