expq3 (problem 3.4.2)

Average Error: 60.0 → 3.6

Time: 10.2s

Precision: binary64

• could not determine a ground truth for program body

  1. a = -6.395177241206731e+271
  2. b = -4.040627605452422e-73
  3. eps = -9.968715394893397e-199

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

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 code(double a, double b, double eps) {
	return ((double) (((double) (eps * ((double) (((double) exp(((double) (((double) (a + b)) * eps)))) - 1.0)))) / ((double) (((double) (((double) exp(((double) (a * eps)))) - 1.0)) * ((double) (((double) exp(((double) (b * eps)))) - 1.0))))));
}

↓

double code(double a, double b, double eps) {
	return ((double) (((double) (1.0 / b)) + ((double) (1.0 / a))));
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.0
Target	15.0
Herbie	3.6

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

Derivation

1. Initial program 60.0

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

2. Taylor expanded around 0 57.6

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

3. Simplified 57.5

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

5. Applied associate-*l* 57.4

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

6. Simplified 56.2

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

8. Applied associate-*r* 56.3

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

9. Simplified 56.0

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

10. Taylor expanded around 0 3.6

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

11. Final simplification 3.6

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

Reproduce

herbie shell --seed 2020184 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :precision binary64
  :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))))