expq3 (problem 3.4.2)

Average Error: 58.7 → 2.9

Time: 56.5s

Precision: 64

• could not determine a ground truth for program body

  1. a = -7.124526950587564e+301
  2. b = -6.9271890162179504e-155
  3. eps = -2.632034696100061e-274

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5467614379857191 \cdot 10^{-65}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}} \cdot \left(\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}}\right)\\ \mathbf{elif}\;\varepsilon \le 7.3489842320315626 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}} \cdot \left(\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}}\right)\\ \end{array}\]

C

double f(double a, double b, double eps) {
        double r8461275 = eps;
        double r8461276 = a;
        double r8461277 = b;
        double r8461278 = r8461276 + r8461277;
        double r8461279 = r8461278 * r8461275;
        double r8461280 = exp(r8461279);
        double r8461281 = 1.0;
        double r8461282 = r8461280 - r8461281;
        double r8461283 = r8461275 * r8461282;
        double r8461284 = r8461276 * r8461275;
        double r8461285 = exp(r8461284);
        double r8461286 = r8461285 - r8461281;
        double r8461287 = r8461277 * r8461275;
        double r8461288 = exp(r8461287);
        double r8461289 = r8461288 - r8461281;
        double r8461290 = r8461286 * r8461289;
        double r8461291 = r8461283 / r8461290;
        return r8461291;
}

↓

double f(double a, double b, double eps) {
        double r8461292 = eps;
        double r8461293 = -1.5467614379857191e-65;
        bool r8461294 = r8461292 <= r8461293;
        double r8461295 = a;
        double r8461296 = b;
        double r8461297 = r8461295 + r8461296;
        double r8461298 = r8461297 * r8461292;
        double r8461299 = expm1(r8461298);
        double r8461300 = r8461299 * r8461292;
        double r8461301 = r8461295 * r8461292;
        double r8461302 = expm1(r8461301);
        double r8461303 = r8461300 / r8461302;
        double r8461304 = r8461296 * r8461292;
        double r8461305 = expm1(r8461304);
        double r8461306 = r8461303 / r8461305;
        double r8461307 = cbrt(r8461306);
        double r8461308 = r8461307 * r8461307;
        double r8461309 = r8461307 * r8461308;
        double r8461310 = 7.3489842320315626e-65;
        bool r8461311 = r8461292 <= r8461310;
        double r8461312 = 1.0;
        double r8461313 = r8461312 / r8461296;
        double r8461314 = r8461312 / r8461295;
        double r8461315 = r8461313 + r8461314;
        double r8461316 = r8461311 ? r8461315 : r8461309;
        double r8461317 = r8461294 ? r8461309 : r8461316;
        return r8461317;
}

Error

Bits error versus a

Bits error versus b

Bits error versus eps

Target

Original	58.7
Target	14.5
Herbie	2.9

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

Derivation

1. Split input into 2 regimes

2. if eps < -1.5467614379857191e-65 or 7.3489842320315626e-65 < eps

   1. Initial program 52.3

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

   2. Simplified 6.5

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}}{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}\]
   3. Using strategy rm

   4. Applied add-cube-cbrt 7.3

      \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}}{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}}{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}\right) \cdot \sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}}{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}}\]

   if -1.5467614379857191e-65 < eps < 7.3489842320315626e-65

   1. Initial program 60.1

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

   2. Simplified 40.6

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(\varepsilon \cdot a\right)\right)}}{\mathsf{expm1}\left(\left(\varepsilon \cdot b\right)\right)}}\]

   3. Taylor expanded around 0 2.0

      \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -1.5467614379857191 \cdot 10^{-65}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}} \cdot \left(\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}}\right)\\ \mathbf{elif}\;\varepsilon \le 7.3489842320315626 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}} \cdot \left(\sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}} \cdot \sqrt[3]{\frac{\frac{\mathsf{expm1}\left(\left(\left(a + b\right) \cdot \varepsilon\right)\right) \cdot \varepsilon}{\mathsf{expm1}\left(\left(a \cdot \varepsilon\right)\right)}}{\mathsf{expm1}\left(\left(b \cdot \varepsilon\right)\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019125 +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))))