expq3 (problem 3.4.2)

Average Error: 60.4 → 0.4

Time: 31.9s

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 r4395395 = eps;
        double r4395396 = a;
        double r4395397 = b;
        double r4395398 = r4395396 + r4395397;
        double r4395399 = r4395398 * r4395395;
        double r4395400 = exp(r4395399);
        double r4395401 = 1.0;
        double r4395402 = r4395400 - r4395401;
        double r4395403 = r4395395 * r4395402;
        double r4395404 = r4395396 * r4395395;
        double r4395405 = exp(r4395404);
        double r4395406 = r4395405 - r4395401;
        double r4395407 = r4395397 * r4395395;
        double r4395408 = exp(r4395407);
        double r4395409 = r4395408 - r4395401;
        double r4395410 = r4395406 * r4395409;
        double r4395411 = r4395403 / r4395410;
        return r4395411;
}

↓

double f(double a, double b, double eps) {
        double r4395412 = a;
        double r4395413 = b;
        double r4395414 = r4395412 + r4395413;
        double r4395415 = eps;
        double r4395416 = r4395414 * r4395415;
        double r4395417 = exp(r4395416);
        double r4395418 = 1.0;
        double r4395419 = r4395417 - r4395418;
        double r4395420 = r4395419 * r4395415;
        double r4395421 = r4395415 * r4395413;
        double r4395422 = exp(r4395421);
        double r4395423 = r4395422 - r4395418;
        double r4395424 = r4395415 * r4395412;
        double r4395425 = exp(r4395424);
        double r4395426 = r4395425 - r4395418;
        double r4395427 = r4395423 * r4395426;
        double r4395428 = r4395420 / r4395427;
        double r4395429 = -inf.0;
        bool r4395430 = r4395428 <= r4395429;
        double r4395431 = 1.0;
        double r4395432 = r4395431 / r4395412;
        double r4395433 = r4395431 / r4395413;
        double r4395434 = r4395432 + r4395433;
        double r4395435 = 1.5708353261329578e-35;
        bool r4395436 = r4395428 <= r4395435;
        double r4395437 = r4395436 ? r4395428 : r4395434;
        double r4395438 = r4395430 ? r4395434 : r4395437;
        return r4395438;
}

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

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

   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 +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))))