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

Average Error: 4.5 → 4.6

Time: 19.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.634362424204118082769787721781123677329 \cdot 10^{194}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{1}{1 - z} \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r25674887 = x;
        double r25674888 = y;
        double r25674889 = z;
        double r25674890 = r25674888 / r25674889;
        double r25674891 = t;
        double r25674892 = 1.0;
        double r25674893 = r25674892 - r25674889;
        double r25674894 = r25674891 / r25674893;
        double r25674895 = r25674890 - r25674894;
        double r25674896 = r25674887 * r25674895;
        return r25674896;
}

↓

double f(double x, double y, double z, double t) {
        double r25674897 = y;
        double r25674898 = z;
        double r25674899 = r25674897 / r25674898;
        double r25674900 = t;
        double r25674901 = 1.0;
        double r25674902 = r25674901 - r25674898;
        double r25674903 = r25674900 / r25674902;
        double r25674904 = r25674899 - r25674903;
        double r25674905 = 7.634362424204118e+194;
        bool r25674906 = r25674904 <= r25674905;
        double r25674907 = 1.0;
        double r25674908 = r25674907 / r25674902;
        double r25674909 = r25674908 * r25674900;
        double r25674910 = r25674899 - r25674909;
        double r25674911 = x;
        double r25674912 = r25674910 * r25674911;
        double r25674913 = r25674897 * r25674911;
        double r25674914 = r25674913 / r25674898;
        double r25674915 = r25674906 ? r25674912 : r25674914;
        return r25674915;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.5
Target	4.3
Herbie	4.6

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

Derivation

1. Split input into 2 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < 7.634362424204118e+194

   1. Initial program 3.1

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

   3. Applied div-inv 3.1

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

   if 7.634362424204118e+194 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 19.3

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

   3. Applied clear-num 19.3

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

   4. Taylor expanded around 0 19.4

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

4. Final simplification 4.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le 7.634362424204118082769787721781123677329 \cdot 10^{194}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{1}{1 - z} \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Reproduce

herbie shell --seed 2019171 
(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 (/ 1.0 (- 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 (/ 1.0 (- 1.0 z)))))))

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