expq3 (problem 3.4.2)

Average Error: 58.6 → 4.9

Time: 34.3s

Precision: 64

• could not determine a ground truth for program body

  1. a = -5.3203362579500095e-06
  2. b = 8.982188186591226e+192
  3. eps = 7.519418569700105e-169

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le -1.6805430101467526 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\\ \mathbf{elif}\;b \le 7.650457866843507 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;b \le 1.4324980532913986 \cdot 10^{+284}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r4204819 = eps;
        double r4204820 = a;
        double r4204821 = b;
        double r4204822 = r4204820 + r4204821;
        double r4204823 = r4204822 * r4204819;
        double r4204824 = exp(r4204823);
        double r4204825 = 1.0;
        double r4204826 = r4204824 - r4204825;
        double r4204827 = r4204819 * r4204826;
        double r4204828 = r4204820 * r4204819;
        double r4204829 = exp(r4204828);
        double r4204830 = r4204829 - r4204825;
        double r4204831 = r4204821 * r4204819;
        double r4204832 = exp(r4204831);
        double r4204833 = r4204832 - r4204825;
        double r4204834 = r4204830 * r4204833;
        double r4204835 = r4204827 / r4204834;
        return r4204835;
}

↓

double f(double a, double b, double eps) {
        double r4204836 = b;
        double r4204837 = -1.6805430101467526e+103;
        bool r4204838 = r4204836 <= r4204837;
        double r4204839 = eps;
        double r4204840 = a;
        double r4204841 = r4204836 + r4204840;
        double r4204842 = r4204839 * r4204841;
        double r4204843 = expm1(r4204842);
        double r4204844 = r4204839 * r4204836;
        double r4204845 = expm1(r4204844);
        double r4204846 = r4204843 / r4204845;
        double r4204847 = r4204839 * r4204840;
        double r4204848 = expm1(r4204847);
        double r4204849 = r4204839 / r4204848;
        double r4204850 = r4204846 * r4204849;
        double r4204851 = 7.650457866843507e+127;
        bool r4204852 = r4204836 <= r4204851;
        double r4204853 = 1.0;
        double r4204854 = r4204853 / r4204836;
        double r4204855 = r4204853 / r4204840;
        double r4204856 = r4204854 + r4204855;
        double r4204857 = 1.4324980532913986e+284;
        bool r4204858 = r4204836 <= r4204857;
        double r4204859 = r4204858 ? r4204850 : r4204856;
        double r4204860 = r4204852 ? r4204856 : r4204859;
        double r4204861 = r4204838 ? r4204850 : r4204860;
        return r4204861;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.6
Target	14.0
Herbie	4.9

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

Derivation

1. Split input into 2 regimes

2. if b < -1.6805430101467526e+103 or 7.650457866843507e+127 < b < 1.4324980532913986e+284

   1. Initial program 51.4

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

   2. Simplified 16.6

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]

   3. Taylor expanded around -inf 30.4

      \[\leadsto \frac{\color{blue}{e^{\left(a + b\right) \cdot \varepsilon} - 1}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\]

   4. Simplified 16.6

      \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\varepsilon \cdot \left(a + b\right)\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\]

   if -1.6805430101467526e+103 < b < 7.650457866843507e+127 or 1.4324980532913986e+284 < b

   1. Initial program 60.9

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

   2. Simplified 30.2

      \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]

   3. Taylor expanded around 0 1.1

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le -1.6805430101467526 \cdot 10^{+103}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\\ \mathbf{elif}\;b \le 7.650457866843507 \cdot 10^{+127}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{elif}\;b \le 1.4324980532913986 \cdot 10^{+284}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\varepsilon \cdot \left(b + a\right)\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 +o rules:numerics
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))