expq3 (problem 3.4.2)

Average Error: 60.3 → 0.3

Time: 56.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = 2.8364102844858466e+132
  2. b = 1.36934023547096e+82
  3. eps = 1.5035876167361006e-51

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 0.002537838552519357249170894874623627401888:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r6429178 = eps;
        double r6429179 = a;
        double r6429180 = b;
        double r6429181 = r6429179 + r6429180;
        double r6429182 = r6429181 * r6429178;
        double r6429183 = exp(r6429182);
        double r6429184 = 1.0;
        double r6429185 = r6429183 - r6429184;
        double r6429186 = r6429178 * r6429185;
        double r6429187 = r6429179 * r6429178;
        double r6429188 = exp(r6429187);
        double r6429189 = r6429188 - r6429184;
        double r6429190 = r6429180 * r6429178;
        double r6429191 = exp(r6429190);
        double r6429192 = r6429191 - r6429184;
        double r6429193 = r6429189 * r6429192;
        double r6429194 = r6429186 / r6429193;
        return r6429194;
}

↓

double f(double a, double b, double eps) {
        double r6429195 = a;
        double r6429196 = b;
        double r6429197 = r6429195 + r6429196;
        double r6429198 = eps;
        double r6429199 = r6429197 * r6429198;
        double r6429200 = exp(r6429199);
        double r6429201 = 1.0;
        double r6429202 = r6429200 - r6429201;
        double r6429203 = r6429202 * r6429198;
        double r6429204 = r6429198 * r6429196;
        double r6429205 = exp(r6429204);
        double r6429206 = r6429205 - r6429201;
        double r6429207 = r6429198 * r6429195;
        double r6429208 = exp(r6429207);
        double r6429209 = r6429208 - r6429201;
        double r6429210 = r6429206 * r6429209;
        double r6429211 = r6429203 / r6429210;
        double r6429212 = -inf.0;
        bool r6429213 = r6429211 <= r6429212;
        double r6429214 = 1.0;
        double r6429215 = r6429214 / r6429195;
        double r6429216 = r6429214 / r6429196;
        double r6429217 = r6429215 + r6429216;
        double r6429218 = 0.0025378385525193572;
        bool r6429219 = r6429211 <= r6429218;
        double r6429220 = r6429219 ? r6429211 : r6429217;
        double r6429221 = r6429213 ? r6429217 : r6429220;
        return r6429221;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.3
Target	14.9
Herbie	0.3

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

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 0.0025378385525193572 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program 64.0

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

   2. Taylor expanded around 0 58.1

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

   3. Simplified 58.1

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

   4. Taylor expanded around 0 0.1

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 0.0025378385525193572

   1. Initial program 3.7

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} = -\infty:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{elif}\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)} \le 0.002537838552519357249170894874623627401888:\\ \;\;\;\;\frac{\left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right) \cdot \varepsilon}{\left(e^{\varepsilon \cdot b} - 1\right) \cdot \left(e^{\varepsilon \cdot a} - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \end{array}\]

Reproduce

herbie shell --seed 2019168 
(FPCore (a b eps)
  :name "expq3 (problem 3.4.2)"
  :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))))