expq3 (problem 3.4.2)

Average Error: 60.3 → 0.8

Time: 32.8s

Precision: 64

• could not determine a ground truth for program body

  1. a = -5.278031834180875e+72
  2. b = -5.9461326997729256e+41
  3. eps = -2.3299265771866867e-17

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 1.492763212396486590455423330799142226004 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r83803 = eps;
        double r83804 = a;
        double r83805 = b;
        double r83806 = r83804 + r83805;
        double r83807 = r83806 * r83803;
        double r83808 = exp(r83807);
        double r83809 = 1.0;
        double r83810 = r83808 - r83809;
        double r83811 = r83803 * r83810;
        double r83812 = r83804 * r83803;
        double r83813 = exp(r83812);
        double r83814 = r83813 - r83809;
        double r83815 = r83805 * r83803;
        double r83816 = exp(r83815);
        double r83817 = r83816 - r83809;
        double r83818 = r83814 * r83817;
        double r83819 = r83811 / r83818;
        return r83819;
}

↓

double f(double a, double b, double eps) {
        double r83820 = eps;
        double r83821 = a;
        double r83822 = b;
        double r83823 = r83821 + r83822;
        double r83824 = r83823 * r83820;
        double r83825 = exp(r83824);
        double r83826 = 1.0;
        double r83827 = r83825 - r83826;
        double r83828 = r83820 * r83827;
        double r83829 = r83821 * r83820;
        double r83830 = exp(r83829);
        double r83831 = r83830 - r83826;
        double r83832 = r83822 * r83820;
        double r83833 = exp(r83832);
        double r83834 = r83833 - r83826;
        double r83835 = r83831 * r83834;
        double r83836 = r83828 / r83835;
        double r83837 = -inf.0;
        bool r83838 = r83836 <= r83837;
        double r83839 = 1.4927632123964866e-92;
        bool r83840 = r83836 <= r83839;
        double r83841 = !r83840;
        bool r83842 = r83838 || r83841;
        double r83843 = 1.0;
        double r83844 = r83843 / r83822;
        double r83845 = r83843 / r83821;
        double r83846 = r83844 + r83845;
        double r83847 = r83842 ? r83846 : r83836;
        return r83847;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	15.2
Herbie	0.8

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

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 1.4927632123964866e-92 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program 63.2

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

   2. Taylor expanded around 0 0.7

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 1.4927632123964866e-92

   1. Initial program 3.6

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

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 1.492763212396486590455423330799142226004 \cdot 10^{-92}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(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))))