expq3 (problem 3.4.2)

Average Error: 58.5 → 3.7

Time: 59.2s

Precision: 64

• could not determine a ground truth for program body

  1. a = -1.0963667354944655e-215
  2. b = 9.652562031662357e+202
  3. eps = 2.4667630494523382e-135

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;b \le 6.565337873712311 \cdot 10^{+258}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^* \cdot \varepsilon}{\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}} \cdot \frac{\frac{1}{(e^{a \cdot \varepsilon} - 1)^*}}{\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}}\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r13727267 = eps;
        double r13727268 = a;
        double r13727269 = b;
        double r13727270 = r13727268 + r13727269;
        double r13727271 = r13727270 * r13727267;
        double r13727272 = exp(r13727271);
        double r13727273 = 1.0;
        double r13727274 = r13727272 - r13727273;
        double r13727275 = r13727267 * r13727274;
        double r13727276 = r13727268 * r13727267;
        double r13727277 = exp(r13727276);
        double r13727278 = r13727277 - r13727273;
        double r13727279 = r13727269 * r13727267;
        double r13727280 = exp(r13727279);
        double r13727281 = r13727280 - r13727273;
        double r13727282 = r13727278 * r13727281;
        double r13727283 = r13727275 / r13727282;
        return r13727283;
}

↓

double f(double a, double b, double eps) {
        double r13727284 = b;
        double r13727285 = 6.565337873712311e+258;
        bool r13727286 = r13727284 <= r13727285;
        double r13727287 = 1.0;
        double r13727288 = a;
        double r13727289 = r13727287 / r13727288;
        double r13727290 = r13727287 / r13727284;
        double r13727291 = r13727289 + r13727290;
        double r13727292 = eps;
        double r13727293 = r13727284 + r13727288;
        double r13727294 = r13727292 * r13727293;
        double r13727295 = expm1(r13727294);
        double r13727296 = r13727295 * r13727292;
        double r13727297 = r13727292 * r13727284;
        double r13727298 = expm1(r13727297);
        double r13727299 = cbrt(r13727298);
        double r13727300 = r13727299 * r13727299;
        double r13727301 = r13727296 / r13727300;
        double r13727302 = r13727288 * r13727292;
        double r13727303 = expm1(r13727302);
        double r13727304 = r13727287 / r13727303;
        double r13727305 = r13727304 / r13727299;
        double r13727306 = r13727301 * r13727305;
        double r13727307 = r13727286 ? r13727291 : r13727306;
        return r13727307;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.5
Target	14.8
Herbie	3.7

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

Derivation

1. Split input into 2 regimes

2. if b < 6.565337873712311e+258

   1. Initial program 58.8

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

   2. Simplified 35.2

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

   3. Taylor expanded around 0 3.3

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

   if 6.565337873712311e+258 < b

   1. Initial program 47.9

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

   2. Simplified 19.3

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

   4. Applied add-cube-cbrt 19.3

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

   5. Applied div-inv 20.4

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

   6. Applied times-frac 20.5

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \le 6.565337873712311 \cdot 10^{+258}:\\ \;\;\;\;\frac{1}{a} + \frac{1}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{(e^{\varepsilon \cdot \left(b + a\right)} - 1)^* \cdot \varepsilon}{\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*} \cdot \sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}} \cdot \frac{\frac{1}{(e^{a \cdot \varepsilon} - 1)^*}}{\sqrt[3]{(e^{\varepsilon \cdot b} - 1)^*}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 +o rules:numerics
(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))))