expq3 (problem 3.4.2)

Average Error: 58.6 → 3.4

Time: 32.2s

Precision: 64

• could not determine a ground truth for program body

  1. a = -6.47945612629242e+112
  2. b = -6.334028728260259e-236
  3. eps = -3.442085074769599e-36

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

Math

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

↓

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

C

double f(double a, double b, double eps) {
        double r3195102 = eps;
        double r3195103 = a;
        double r3195104 = b;
        double r3195105 = r3195103 + r3195104;
        double r3195106 = r3195105 * r3195102;
        double r3195107 = exp(r3195106);
        double r3195108 = 1.0;
        double r3195109 = r3195107 - r3195108;
        double r3195110 = r3195102 * r3195109;
        double r3195111 = r3195103 * r3195102;
        double r3195112 = exp(r3195111);
        double r3195113 = r3195112 - r3195108;
        double r3195114 = r3195104 * r3195102;
        double r3195115 = exp(r3195114);
        double r3195116 = r3195115 - r3195108;
        double r3195117 = r3195113 * r3195116;
        double r3195118 = r3195110 / r3195117;
        return r3195118;
}

↓

double f(double a, double b, double __attribute__((unused)) eps) {
        double r3195119 = 1.0;
        double r3195120 = a;
        double r3195121 = r3195119 / r3195120;
        double r3195122 = b;
        double r3195123 = r3195119 / r3195122;
        double r3195124 = r3195121 + r3195123;
        return r3195124;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.6
Target	14.3
Herbie	3.4

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

Derivation

1. Initial program 58.6

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

2. Taylor expanded around 0 3.4

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

3. Final simplification 3.4

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

Reproduce

herbie shell --seed 2019137 
(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))))