expq3 (problem 3.4.2)

Average Error: 60.2 → 0.3

Time: 9.0s

Precision: 64

• could not determine a ground truth for program body

  1. a = 7.321465291114456e-39
  2. b = -5.705042820112819e+286
  3. eps = -2.4185511553388966e-62

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 0.0087632414871769045\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{e^{a \cdot \varepsilon + \varepsilon \cdot b} + \left(1 - 1 \cdot \left(e^{\varepsilon \cdot b} + e^{a \cdot \varepsilon}\right)\right)}{e^{\left(a + b\right) \cdot \varepsilon} - 1}}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r86219 = eps;
        double r86220 = a;
        double r86221 = b;
        double r86222 = r86220 + r86221;
        double r86223 = r86222 * r86219;
        double r86224 = exp(r86223);
        double r86225 = 1.0;
        double r86226 = r86224 - r86225;
        double r86227 = r86219 * r86226;
        double r86228 = r86220 * r86219;
        double r86229 = exp(r86228);
        double r86230 = r86229 - r86225;
        double r86231 = r86221 * r86219;
        double r86232 = exp(r86231);
        double r86233 = r86232 - r86225;
        double r86234 = r86230 * r86233;
        double r86235 = r86227 / r86234;
        return r86235;
}

↓

double f(double a, double b, double eps) {
        double r86236 = eps;
        double r86237 = a;
        double r86238 = b;
        double r86239 = r86237 + r86238;
        double r86240 = r86239 * r86236;
        double r86241 = exp(r86240);
        double r86242 = 1.0;
        double r86243 = r86241 - r86242;
        double r86244 = r86236 * r86243;
        double r86245 = r86237 * r86236;
        double r86246 = exp(r86245);
        double r86247 = r86246 - r86242;
        double r86248 = r86238 * r86236;
        double r86249 = exp(r86248);
        double r86250 = r86249 - r86242;
        double r86251 = r86247 * r86250;
        double r86252 = r86244 / r86251;
        double r86253 = -inf.0;
        bool r86254 = r86252 <= r86253;
        double r86255 = 0.008763241487176904;
        bool r86256 = r86252 <= r86255;
        double r86257 = !r86256;
        bool r86258 = r86254 || r86257;
        double r86259 = 1.0;
        double r86260 = r86259 / r86238;
        double r86261 = r86259 / r86237;
        double r86262 = r86260 + r86261;
        double r86263 = r86236 * r86238;
        double r86264 = r86245 + r86263;
        double r86265 = exp(r86264);
        double r86266 = exp(r86263);
        double r86267 = r86266 + r86246;
        double r86268 = r86242 * r86267;
        double r86269 = r86242 - r86268;
        double r86270 = r86265 + r86269;
        double r86271 = r86270 / r86243;
        double r86272 = r86236 / r86271;
        double r86273 = r86258 ? r86262 : r86272;
        return r86273;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	60.2
Target	15.2
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 0.008763241487176904 < (/ (* 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 0.0

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

   if -inf.0 < (/ (* eps (- (exp (* (+ a b) eps)) 1.0)) (* (- (exp (* a eps)) 1.0) (- (exp (* b eps)) 1.0))) < 0.008763241487176904

   1. Initial program 4.1

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

   2. Taylor expanded around inf 29.9

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

   3. Simplified 4.4

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

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} = -\infty \lor \neg \left(\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)} \le 0.0087632414871769045\right):\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\varepsilon}{\frac{e^{a \cdot \varepsilon + \varepsilon \cdot b} + \left(1 - 1 \cdot \left(e^{\varepsilon \cdot b} + e^{a \cdot \varepsilon}\right)\right)}{e^{\left(a + b\right) \cdot \varepsilon} - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(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))))