Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Average Error: 4.9 → 1.5

Time: 5.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 4.071156994590675814961087478639352093597 \cdot 10^{242}\right):\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) - z \cdot \left(1 \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\right)}{z \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r461607 = x;
        double r461608 = y;
        double r461609 = z;
        double r461610 = r461608 / r461609;
        double r461611 = t;
        double r461612 = 1.0;
        double r461613 = r461612 - r461609;
        double r461614 = r461611 / r461613;
        double r461615 = r461610 - r461614;
        double r461616 = r461607 * r461615;
        return r461616;
}

↓

double f(double x, double y, double z, double t) {
        double r461617 = y;
        double r461618 = z;
        double r461619 = r461617 / r461618;
        double r461620 = t;
        double r461621 = 1.0;
        double r461622 = r461621 - r461618;
        double r461623 = r461620 / r461622;
        double r461624 = r461619 - r461623;
        double r461625 = -inf.0;
        bool r461626 = r461624 <= r461625;
        double r461627 = 4.071156994590676e+242;
        bool r461628 = r461624 <= r461627;
        double r461629 = !r461628;
        bool r461630 = r461626 || r461629;
        double r461631 = x;
        double r461632 = cbrt(r461622);
        double r461633 = r461632 * r461632;
        double r461634 = r461617 * r461633;
        double r461635 = 1.0;
        double r461636 = r461620 / r461632;
        double r461637 = r461635 * r461636;
        double r461638 = r461618 * r461637;
        double r461639 = r461634 - r461638;
        double r461640 = r461631 * r461639;
        double r461641 = r461618 * r461633;
        double r461642 = r461640 / r461641;
        double r461643 = r461631 * r461624;
        double r461644 = r461630 ? r461642 : r461643;
        return r461644;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.9
Target	4.4
Herbie	1.5

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.623226303312042442144691872793570510727 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.413394492770230216018398633584271456447 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 4.071156994590676e+242 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 40.2

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

   3. Applied add-cube-cbrt 40.2

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{t}{\color{blue}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}}\right)\]

   4. Applied *-un-lft-identity 40.2

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{\color{blue}{1 \cdot t}}{\left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) \cdot \sqrt[3]{1 - z}}\right)\]

   5. Applied times-frac 40.2

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}} \cdot \frac{t}{\sqrt[3]{1 - z}}}\right)\]
   6. Using strategy rm

   7. Applied associate-*l/ 40.2

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{1 \cdot \frac{t}{\sqrt[3]{1 - z}}}{\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}}}\right)\]

   8. Applied frac-sub 41.4

      \[\leadsto x \cdot \color{blue}{\frac{y \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) - z \cdot \left(1 \cdot \frac{t}{\sqrt[3]{1 - z}}\right)}{z \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right)}}\]

   9. Applied associate-*r/ 1.6

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) - z \cdot \left(1 \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\right)}{z \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right)}}\]

   if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < 4.071156994590676e+242

   1. Initial program 1.5

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 1.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} = -\infty \lor \neg \left(\frac{y}{z} - \frac{t}{1 - z} \le 4.071156994590675814961087478639352093597 \cdot 10^{242}\right):\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right) - z \cdot \left(1 \cdot \frac{t}{\sqrt[3]{1 - z}}\right)\right)}{z \cdot \left(\sqrt[3]{1 - z} \cdot \sqrt[3]{1 - z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

  (* x (- (/ y z) (/ t (- 1 z)))))