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

Average Error: 4.3 → 0.3

Time: 16.7s

Precision: 64

Math

\[x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)\]

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1.0 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) - \frac{t}{\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}} \cdot z\right)}{z \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le -5.604253807165862 \cdot 10^{-286}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le 0.0:\\ \;\;\;\;\frac{x \cdot t}{z} \cdot \left(\frac{1.0}{z} + 1\right) + \frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le 2.55312552975251 \cdot 10^{+287}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) - \frac{t}{\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}} \cdot z\right)}{z \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right)}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r20524965 = x;
        double r20524966 = y;
        double r20524967 = z;
        double r20524968 = r20524966 / r20524967;
        double r20524969 = t;
        double r20524970 = 1.0;
        double r20524971 = r20524970 - r20524967;
        double r20524972 = r20524969 / r20524971;
        double r20524973 = r20524968 - r20524972;
        double r20524974 = r20524965 * r20524973;
        return r20524974;
}

↓

double f(double x, double y, double z, double t) {
        double r20524975 = y;
        double r20524976 = z;
        double r20524977 = r20524975 / r20524976;
        double r20524978 = t;
        double r20524979 = 1.0;
        double r20524980 = r20524979 - r20524976;
        double r20524981 = r20524978 / r20524980;
        double r20524982 = r20524977 - r20524981;
        double r20524983 = -inf.0;
        bool r20524984 = r20524982 <= r20524983;
        double r20524985 = x;
        double r20524986 = cbrt(r20524980);
        double r20524987 = cbrt(r20524986);
        double r20524988 = r20524987 * r20524987;
        double r20524989 = r20524988 * r20524987;
        double r20524990 = r20524975 * r20524989;
        double r20524991 = r20524986 * r20524986;
        double r20524992 = r20524978 / r20524991;
        double r20524993 = r20524992 * r20524976;
        double r20524994 = r20524990 - r20524993;
        double r20524995 = r20524985 * r20524994;
        double r20524996 = r20524976 * r20524989;
        double r20524997 = r20524995 / r20524996;
        double r20524998 = -5.604253807165862e-286;
        bool r20524999 = r20524982 <= r20524998;
        double r20525000 = r20524982 * r20524985;
        double r20525001 = 0.0;
        bool r20525002 = r20524982 <= r20525001;
        double r20525003 = r20524985 * r20524978;
        double r20525004 = r20525003 / r20524976;
        double r20525005 = r20524979 / r20524976;
        double r20525006 = 1.0;
        double r20525007 = r20525005 + r20525006;
        double r20525008 = r20525004 * r20525007;
        double r20525009 = r20524975 * r20524985;
        double r20525010 = r20525009 / r20524976;
        double r20525011 = r20525008 + r20525010;
        double r20525012 = 2.55312552975251e+287;
        bool r20525013 = r20524982 <= r20525012;
        double r20525014 = r20525013 ? r20525000 : r20524997;
        double r20525015 = r20525002 ? r20525011 : r20525014;
        double r20525016 = r20524999 ? r20525000 : r20525015;
        double r20525017 = r20524984 ? r20524997 : r20525016;
        return r20525017;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.3
Target	3.9
Herbie	0.3

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \lt -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1.0 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \lt 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1.0 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1.0 - z}\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -inf.0 or 2.55312552975251e+287 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 50.0

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)\]
   2. Using strategy rm

   3. Applied add-cube-cbrt 50.0

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{t}{\color{blue}{\left(\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}\right) \cdot \sqrt[3]{1.0 - z}}}\right)\]

   4. Applied associate-/r* 50.0

      \[\leadsto x \cdot \left(\frac{y}{z} - \color{blue}{\frac{\frac{t}{\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}}}{\sqrt[3]{1.0 - z}}}\right)\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 50.0

      \[\leadsto x \cdot \left(\frac{y}{z} - \frac{\frac{t}{\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}}}{\color{blue}{\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}}}\right)\]
   7. Using strategy rm

   8. Applied frac-sub 50.0

      \[\leadsto x \cdot \color{blue}{\frac{y \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) - z \cdot \frac{t}{\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}}}{z \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right)}}\]

   9. Applied associate-*r/ 0.3

      \[\leadsto \color{blue}{\frac{x \cdot \left(y \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) - z \cdot \frac{t}{\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}}\right)}{z \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right)}}\]

   if -inf.0 < (- (/ y z) (/ t (- 1.0 z))) < -5.604253807165862e-286 or 0.0 < (- (/ y z) (/ t (- 1.0 z))) < 2.55312552975251e+287

   1. Initial program 0.3

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)\]
   2. Using strategy rm

   3. Applied *-commutative 0.3

      \[\leadsto \color{blue}{\left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \cdot x}\]

   if -5.604253807165862e-286 < (- (/ y z) (/ t (- 1.0 z))) < 0.0

   1. Initial program 19.7

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1.0 - z}\right)\]
   2. Using strategy rm

   3. Applied *-commutative 19.7

      \[\leadsto \color{blue}{\left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \cdot x}\]

   4. Taylor expanded around inf 0.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(1.0 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)}\]

   5. Simplified 0.1

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(\frac{1.0}{z} + 1\right) \cdot \frac{t \cdot x}{z}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1.0 - z} = -\infty:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) - \frac{t}{\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}} \cdot z\right)}{z \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right)}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le -5.604253807165862 \cdot 10^{-286}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le 0.0:\\ \;\;\;\;\frac{x \cdot t}{z} \cdot \left(\frac{1.0}{z} + 1\right) + \frac{y \cdot x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1.0 - z} \le 2.55312552975251 \cdot 10^{+287}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1.0 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) - \frac{t}{\sqrt[3]{1.0 - z} \cdot \sqrt[3]{1.0 - z}} \cdot z\right)}{z \cdot \left(\left(\sqrt[3]{\sqrt[3]{1.0 - z}} \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right) \cdot \sqrt[3]{\sqrt[3]{1.0 - z}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))