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

Average Error: 4.7 → 2.9

Time: 19.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.609192483439543839031923140622339800609 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \le 2.673551738340748240588302325440591673285 \cdot 10^{49}:\\ \;\;\;\;\left(x \cdot t\right) \cdot \frac{-1}{1 - z} + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r383635 = x;
        double r383636 = y;
        double r383637 = z;
        double r383638 = r383636 / r383637;
        double r383639 = t;
        double r383640 = 1.0;
        double r383641 = r383640 - r383637;
        double r383642 = r383639 / r383641;
        double r383643 = r383638 - r383642;
        double r383644 = r383635 * r383643;
        return r383644;
}

↓

double f(double x, double y, double z, double t) {
        double r383645 = x;
        double r383646 = -1.6091924834395438e-32;
        bool r383647 = r383645 <= r383646;
        double r383648 = z;
        double r383649 = y;
        double r383650 = r383648 / r383649;
        double r383651 = r383645 / r383650;
        double r383652 = t;
        double r383653 = 1.0;
        double r383654 = r383653 - r383648;
        double r383655 = r383652 / r383654;
        double r383656 = -r383655;
        double r383657 = r383645 * r383656;
        double r383658 = r383651 + r383657;
        double r383659 = 2.673551738340748e+49;
        bool r383660 = r383645 <= r383659;
        double r383661 = r383645 * r383652;
        double r383662 = -1.0;
        double r383663 = r383662 / r383654;
        double r383664 = r383661 * r383663;
        double r383665 = r383645 * r383649;
        double r383666 = r383665 / r383648;
        double r383667 = r383664 + r383666;
        double r383668 = r383649 / r383648;
        double r383669 = 1.0;
        double r383670 = r383654 / r383652;
        double r383671 = r383669 / r383670;
        double r383672 = r383668 - r383671;
        double r383673 = r383645 * r383672;
        double r383674 = r383660 ? r383667 : r383673;
        double r383675 = r383647 ? r383658 : r383674;
        return r383675;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.7
Target	4.6
Herbie	2.9

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

Derivation

1. Split input into 3 regimes

2. if x < -1.6091924834395438e-32

   1. Initial program 2.5

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

   3. Applied sub-neg 2.5

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

   4. Applied distribute-lft-in 2.5

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

   5. Simplified 8.7

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

   7. Applied associate-/l* 2.6

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

   if -1.6091924834395438e-32 < x < 2.673551738340748e+49

   1. Initial program 5.9

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

   3. Applied sub-neg 5.9

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

   4. Applied distribute-lft-in 5.9

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

   5. Simplified 2.8

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

   7. Applied div-inv 2.9

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

   8. Applied distribute-rgt-neg-in 2.9

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

   9. Applied associate-*r* 2.9

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

   if 2.673551738340748e+49 < x

   1. Initial program 3.1

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

   3. Applied clear-num 3.5

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

4. Final simplification 2.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.609192483439543839031923140622339800609 \cdot 10^{-32}:\\ \;\;\;\;\frac{x}{\frac{z}{y}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;x \le 2.673551738340748240588302325440591673285 \cdot 10^{49}:\\ \;\;\;\;\left(x \cdot t\right) \cdot \frac{-1}{1 - z} + \frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{1}{\frac{1 - z}{t}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019306 +o rules:numerics
(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)))) -7.62322630331204244e-196) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 1.41339449277023022e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z)))))))

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