expq3 (problem 3.4.2)

Average Error: 60.4 → 3.3

Time: 37.2s

Precision: 64

• could not determine a ground truth for program body

  1. a = 1.771691118882549e+275
  2. b = 2.9315320292795686e-164
  3. eps = 1.414912756308087e-16

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

Math

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

↓

\[\frac{1}{b} + \frac{1}{a}\]

C

double f(double a, double b, double eps) {
        double r62068 = eps;
        double r62069 = a;
        double r62070 = b;
        double r62071 = r62069 + r62070;
        double r62072 = r62071 * r62068;
        double r62073 = exp(r62072);
        double r62074 = 1.0;
        double r62075 = r62073 - r62074;
        double r62076 = r62068 * r62075;
        double r62077 = r62069 * r62068;
        double r62078 = exp(r62077);
        double r62079 = r62078 - r62074;
        double r62080 = r62070 * r62068;
        double r62081 = exp(r62080);
        double r62082 = r62081 - r62074;
        double r62083 = r62079 * r62082;
        double r62084 = r62076 / r62083;
        return r62084;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r62085 = 1.0;
        double r62086 = b;
        double r62087 = r62085 / r62086;
        double r62088 = a;
        double r62089 = r62085 / r62088;
        double r62090 = r62087 + r62089;
        return r62090;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.7
Herbie	3.3

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

Derivation

1. Initial program 60.4

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

2. Taylor expanded around 0 58.2

   \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\left(\frac{1}{6} \cdot \left({a}^{3} \cdot {\varepsilon}^{3}\right) + \left(\frac{1}{2} \cdot \left({a}^{2} \cdot {\varepsilon}^{2}\right) + a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

3. Simplified 58.2

   \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {a}^{3} \cdot {\varepsilon}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right)} \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
4. Using strategy rm

5. Applied pow-prod-down 57.8

   \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{{\left(a \cdot \varepsilon\right)}^{3}}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
6. Using strategy rm

7. Applied add-cube-cbrt 57.8

   \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {\left(a \cdot \varepsilon\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(e^{b \cdot \varepsilon} - \color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}\right)}\]

8. Applied add-sqr-sqrt 57.9

   \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {\left(a \cdot \varepsilon\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(\color{blue}{\sqrt{e^{b \cdot \varepsilon}} \cdot \sqrt{e^{b \cdot \varepsilon}}} - \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}\right)}\]

9. Applied prod-diff 58.0

   \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {\left(a \cdot \varepsilon\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\sqrt{e^{b \cdot \varepsilon}}, \sqrt{e^{b \cdot \varepsilon}}, -\sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right) + \mathsf{fma}\left(-\sqrt[3]{1}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)\right)}}\]

10. Simplified 58.2

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {\left(a \cdot \varepsilon\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(\color{blue}{\left({\left(e^{b}\right)}^{\varepsilon} - 1\right)} + \mathsf{fma}\left(-\sqrt[3]{1}, \sqrt[3]{1} \cdot \sqrt[3]{1}, \sqrt[3]{1} \cdot \left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right)\right)\right)}\]

11. Simplified 58.2

    \[\leadsto \frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\mathsf{fma}\left(\frac{1}{6}, {\left(a \cdot \varepsilon\right)}^{3}, \mathsf{fma}\left(\frac{1}{2}, {a}^{2} \cdot {\varepsilon}^{2}, a \cdot \varepsilon\right)\right) \cdot \left(\left({\left(e^{b}\right)}^{\varepsilon} - 1\right) + \color{blue}{\mathsf{fma}\left(1, -1, 1\right)}\right)}\]

12. Taylor expanded around 0 3.3

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

13. Final simplification 3.3

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

Reproduce

herbie shell --seed 2019235 +o rules:numerics
(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))))