expq3 (problem 3.4.2)

Average Error: 60.4 → 0.4

Time: 33.4s

Precision: 64

• could not determine a ground truth for program body

  1. a = 7.706155067127394e+81
  2. b = 6.88703791615021e+280
  3. eps = 8.102269269768392e-109

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 1.570835326132957750197283763877702962205 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r6298414 = eps;
        double r6298415 = a;
        double r6298416 = b;
        double r6298417 = r6298415 + r6298416;
        double r6298418 = r6298417 * r6298414;
        double r6298419 = exp(r6298418);
        double r6298420 = 1.0;
        double r6298421 = r6298419 - r6298420;
        double r6298422 = r6298414 * r6298421;
        double r6298423 = r6298415 * r6298414;
        double r6298424 = exp(r6298423);
        double r6298425 = r6298424 - r6298420;
        double r6298426 = r6298416 * r6298414;
        double r6298427 = exp(r6298426);
        double r6298428 = r6298427 - r6298420;
        double r6298429 = r6298425 * r6298428;
        double r6298430 = r6298422 / r6298429;
        return r6298430;
}

↓

double f(double a, double b, double eps) {
        double r6298431 = a;
        double r6298432 = b;
        double r6298433 = r6298431 + r6298432;
        double r6298434 = eps;
        double r6298435 = r6298433 * r6298434;
        double r6298436 = exp(r6298435);
        double r6298437 = 1.0;
        double r6298438 = r6298436 - r6298437;
        double r6298439 = r6298438 * r6298434;
        double r6298440 = r6298434 * r6298432;
        double r6298441 = exp(r6298440);
        double r6298442 = r6298441 - r6298437;
        double r6298443 = r6298434 * r6298431;
        double r6298444 = exp(r6298443);
        double r6298445 = r6298444 - r6298437;
        double r6298446 = r6298442 * r6298445;
        double r6298447 = r6298439 / r6298446;
        double r6298448 = -inf.0;
        bool r6298449 = r6298447 <= r6298448;
        double r6298450 = 1.0;
        double r6298451 = r6298450 / r6298431;
        double r6298452 = r6298450 / r6298432;
        double r6298453 = r6298451 + r6298452;
        double r6298454 = 1.5708353261329578e-35;
        bool r6298455 = r6298447 <= r6298454;
        double r6298456 = r6298455 ? r6298447 : r6298453;
        double r6298457 = r6298449 ? r6298453 : r6298456;
        return r6298457;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.6
Herbie	0.4

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

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 1.5708353261329578e-35 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program 63.7

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

   2. 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 57.4

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

   4. Taylor expanded around 0 0.3

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 1.5708353261329578e-35

   1. Initial program 3.1

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

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 1.570835326132957750197283763877702962205 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(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))))