expq3 (problem 3.4.2)

Average Error: 60.4 → 3.3

Time: 10.3s

Precision: 64

• could not determine a ground truth for program body

  1. a = -2.0882778719170116e-105
  2. b = -1.1140740905072972e+272
  3. eps = -8.795263419266061e-104

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

Math

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

↓

\[\frac{1}{b} + \frac{1}{a}\]

C

double f(double a, double b, double eps) {
        double r94360 = eps;
        double r94361 = a;
        double r94362 = b;
        double r94363 = r94361 + r94362;
        double r94364 = r94363 * r94360;
        double r94365 = exp(r94364);
        double r94366 = 1.0;
        double r94367 = r94365 - r94366;
        double r94368 = r94360 * r94367;
        double r94369 = r94361 * r94360;
        double r94370 = exp(r94369);
        double r94371 = r94370 - r94366;
        double r94372 = r94362 * r94360;
        double r94373 = exp(r94372);
        double r94374 = r94373 - r94366;
        double r94375 = r94371 * r94374;
        double r94376 = r94368 / r94375;
        return r94376;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r94377 = 1.0;
        double r94378 = b;
        double r94379 = r94377 / r94378;
        double r94380 = a;
        double r94381 = r94377 / r94380;
        double r94382 = r94379 + r94381;
        return r94382;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	15.2
Herbie	3.3

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

Derivation

1. Initial program 60.4

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

2. Taylor expanded around 0 3.3

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

3. Final simplification 3.3

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

Reproduce

herbie shell --seed 2019362 
(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))))