expq3 (problem 3.4.2)

Average Error: 60.4 → 0.7

Time: 35.6s

Precision: 64

• could not determine a ground truth for program body

  1. a = 2.8774590365534544e-144
  2. b = 3.0260304830237114e+184
  3. eps = 1.0722929270979165e-66

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 4.275877483011707453930622974464582369913 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r80233 = eps;
        double r80234 = a;
        double r80235 = b;
        double r80236 = r80234 + r80235;
        double r80237 = r80236 * r80233;
        double r80238 = exp(r80237);
        double r80239 = 1.0;
        double r80240 = r80238 - r80239;
        double r80241 = r80233 * r80240;
        double r80242 = r80234 * r80233;
        double r80243 = exp(r80242);
        double r80244 = r80243 - r80239;
        double r80245 = r80235 * r80233;
        double r80246 = exp(r80245);
        double r80247 = r80246 - r80239;
        double r80248 = r80244 * r80247;
        double r80249 = r80241 / r80248;
        return r80249;
}

↓

double f(double a, double b, double eps) {
        double r80250 = eps;
        double r80251 = a;
        double r80252 = b;
        double r80253 = r80251 + r80252;
        double r80254 = r80253 * r80250;
        double r80255 = exp(r80254);
        double r80256 = 1.0;
        double r80257 = r80255 - r80256;
        double r80258 = r80250 * r80257;
        double r80259 = r80251 * r80250;
        double r80260 = exp(r80259);
        double r80261 = r80260 - r80256;
        double r80262 = r80252 * r80250;
        double r80263 = exp(r80262);
        double r80264 = r80263 - r80256;
        double r80265 = r80261 * r80264;
        double r80266 = r80258 / r80265;
        double r80267 = -inf.0;
        bool r80268 = r80266 <= r80267;
        double r80269 = 4.2758774830117075e-79;
        bool r80270 = r80266 <= r80269;
        double r80271 = !r80270;
        bool r80272 = r80268 || r80271;
        double r80273 = 1.0;
        double r80274 = r80273 / r80252;
        double r80275 = r80273 / r80251;
        double r80276 = r80274 + r80275;
        double r80277 = r80272 ? r80276 : r80266;
        return r80277;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.4
Target	14.8
Herbie	0.7

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

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 4.2758774830117075e-79 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0)))

   1. Initial program 63.4

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

   2. Taylor expanded around 0 0.6

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 4.2758774830117075e-79

   1. Initial program 3.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 4.275877483011707453930622974464582369913 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 +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))))