Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2

Average Error: 15.1 → 0.4

Time: 17.2s

Precision: 64

Math

\[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \cdot y = -\infty \lor \neg \left(x \cdot y \le -7.653303258863243928721603622295786194715 \cdot 10^{-284} \lor \neg \left(x \cdot y \le 1.137508288798888459719974341598011528755 \cdot 10^{-219}\right) \land x \cdot y \le 1.528832091076191138986851859282284286275 \cdot 10^{175}\right):\\ \;\;\;\;\frac{x}{\left|z\right|} \cdot \frac{\frac{y}{\left|z\right|}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot y}{z}}{z}}{z + 1}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r253374 = x;
        double r253375 = y;
        double r253376 = r253374 * r253375;
        double r253377 = z;
        double r253378 = r253377 * r253377;
        double r253379 = 1.0;
        double r253380 = r253377 + r253379;
        double r253381 = r253378 * r253380;
        double r253382 = r253376 / r253381;
        return r253382;
}

↓

double f(double x, double y, double z) {
        double r253383 = x;
        double r253384 = y;
        double r253385 = r253383 * r253384;
        double r253386 = -inf.0;
        bool r253387 = r253385 <= r253386;
        double r253388 = -7.653303258863244e-284;
        bool r253389 = r253385 <= r253388;
        double r253390 = 1.1375082887988885e-219;
        bool r253391 = r253385 <= r253390;
        double r253392 = !r253391;
        double r253393 = 1.5288320910761911e+175;
        bool r253394 = r253385 <= r253393;
        bool r253395 = r253392 && r253394;
        bool r253396 = r253389 || r253395;
        double r253397 = !r253396;
        bool r253398 = r253387 || r253397;
        double r253399 = z;
        double r253400 = fabs(r253399);
        double r253401 = r253383 / r253400;
        double r253402 = r253384 / r253400;
        double r253403 = 1.0;
        double r253404 = r253399 + r253403;
        double r253405 = r253402 / r253404;
        double r253406 = r253401 * r253405;
        double r253407 = r253385 / r253399;
        double r253408 = r253407 / r253399;
        double r253409 = r253408 / r253404;
        double r253410 = r253398 ? r253406 : r253409;
        return r253410;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	15.1
Target	4.0
Herbie	0.4

\[\begin{array}{l} \mathbf{if}\;z \lt 249.6182814532307077115547144785523414612:\\ \;\;\;\;\frac{y \cdot \frac{x}{z}}{z + z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{y}{z}}{1 + z} \cdot x}{z}\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (* x y) < -inf.0 or -7.653303258863244e-284 < (* x y) < 1.1375082887988885e-219 or 1.5288320910761911e+175 < (* x y)

   1. Initial program 30.3

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
   2. Using strategy rm

   3. Applied associate-/r* 29.0

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}}\]

   4. Simplified 29.0

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{{z}^{2}}}}{z + 1}\]
   5. Using strategy rm

   6. Applied add-sqr-sqrt 29.0

      \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{\sqrt{{z}^{2}} \cdot \sqrt{{z}^{2}}}}}{z + 1}\]

   7. Applied times-frac 16.1

      \[\leadsto \frac{\color{blue}{\frac{x}{\sqrt{{z}^{2}}} \cdot \frac{y}{\sqrt{{z}^{2}}}}}{z + 1}\]

   8. Simplified 16.1

      \[\leadsto \frac{\color{blue}{\frac{x}{\left|z\right|}} \cdot \frac{y}{\sqrt{{z}^{2}}}}{z + 1}\]

   9. Simplified 1.2

      \[\leadsto \frac{\frac{x}{\left|z\right|} \cdot \color{blue}{\frac{y}{\left|z\right|}}}{z + 1}\]
   10. Using strategy rm

   11. Applied *-un-lft-identity 1.2

       \[\leadsto \frac{\frac{x}{\left|z\right|} \cdot \frac{y}{\left|z\right|}}{\color{blue}{1 \cdot \left(z + 1\right)}}\]

   12. Applied times-frac 0.7

       \[\leadsto \color{blue}{\frac{\frac{x}{\left|z\right|}}{1} \cdot \frac{\frac{y}{\left|z\right|}}{z + 1}}\]

   13. Simplified 0.7

       \[\leadsto \color{blue}{\frac{x}{\left|z\right|}} \cdot \frac{\frac{y}{\left|z\right|}}{z + 1}\]

   if -inf.0 < (* x y) < -7.653303258863244e-284 or 1.1375082887988885e-219 < (* x y) < 1.5288320910761911e+175

   1. Initial program 6.6

      \[\frac{x \cdot y}{\left(z \cdot z\right) \cdot \left(z + 1\right)}\]
   2. Using strategy rm

   3. Applied associate-/r* 4.6

      \[\leadsto \color{blue}{\frac{\frac{x \cdot y}{z \cdot z}}{z + 1}}\]

   4. Simplified 4.6

      \[\leadsto \frac{\color{blue}{\frac{x \cdot y}{{z}^{2}}}}{z + 1}\]
   5. Using strategy rm

   6. Applied unpow2 4.6

      \[\leadsto \frac{\frac{x \cdot y}{\color{blue}{z \cdot z}}}{z + 1}\]

   7. Applied associate-/r* 0.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{x \cdot y}{z}}{z}}}{z + 1}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y = -\infty \lor \neg \left(x \cdot y \le -7.653303258863243928721603622295786194715 \cdot 10^{-284} \lor \neg \left(x \cdot y \le 1.137508288798888459719974341598011528755 \cdot 10^{-219}\right) \land x \cdot y \le 1.528832091076191138986851859282284286275 \cdot 10^{175}\right):\\ \;\;\;\;\frac{x}{\left|z\right|} \cdot \frac{\frac{y}{\left|z\right|}}{z + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\frac{x \cdot y}{z}}{z}}{z + 1}\\ \end{array}\]

Reproduce

herbie shell --seed 2019212 +o rules:numerics
(FPCore (x y z)
  :name "Statistics.Distribution.Beta:$cvariance from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< z 249.618281453230708) (/ (* y (/ x z)) (+ z (* z z))) (/ (* (/ (/ y z) (+ 1 z)) x) z))

  (/ (* x y) (* (* z z) (+ z 1))))