expq3 (problem 3.4.2)

Average Error: 60.4 → 0.4

Time: 35.8s

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 r5804956 = eps;
        double r5804957 = a;
        double r5804958 = b;
        double r5804959 = r5804957 + r5804958;
        double r5804960 = r5804959 * r5804956;
        double r5804961 = exp(r5804960);
        double r5804962 = 1.0;
        double r5804963 = r5804961 - r5804962;
        double r5804964 = r5804956 * r5804963;
        double r5804965 = r5804957 * r5804956;
        double r5804966 = exp(r5804965);
        double r5804967 = r5804966 - r5804962;
        double r5804968 = r5804958 * r5804956;
        double r5804969 = exp(r5804968);
        double r5804970 = r5804969 - r5804962;
        double r5804971 = r5804967 * r5804970;
        double r5804972 = r5804964 / r5804971;
        return r5804972;
}

↓

double f(double a, double b, double eps) {
        double r5804973 = a;
        double r5804974 = b;
        double r5804975 = r5804973 + r5804974;
        double r5804976 = eps;
        double r5804977 = r5804975 * r5804976;
        double r5804978 = exp(r5804977);
        double r5804979 = 1.0;
        double r5804980 = r5804978 - r5804979;
        double r5804981 = r5804980 * r5804976;
        double r5804982 = r5804976 * r5804974;
        double r5804983 = exp(r5804982);
        double r5804984 = r5804983 - r5804979;
        double r5804985 = r5804976 * r5804973;
        double r5804986 = exp(r5804985);
        double r5804987 = r5804986 - r5804979;
        double r5804988 = r5804984 * r5804987;
        double r5804989 = r5804981 / r5804988;
        double r5804990 = -inf.0;
        bool r5804991 = r5804989 <= r5804990;
        double r5804992 = 1.0;
        double r5804993 = r5804992 / r5804973;
        double r5804994 = r5804992 / r5804974;
        double r5804995 = r5804993 + r5804994;
        double r5804996 = 1.5708353261329578e-35;
        bool r5804997 = r5804989 <= r5804996;
        double r5804998 = r5804997 ? r5804989 : r5804995;
        double r5804999 = r5804991 ? r5804995 : r5804998;
        return r5804999;
}

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

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

   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))))