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

Average Error: 4.5 → 3.1

Time: 16.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1848377431932672203757011904118128640:\\ \;\;\;\;\frac{1 \cdot t}{\frac{z \cdot z}{x}} + \left(\frac{x}{\frac{z}{y}} + \frac{t}{z} \cdot x\right)\\ \mathbf{elif}\;z \le 1.714214484199184448434438031445311204718 \cdot 10^{-214}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(-\frac{x}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r24078227 = x;
        double r24078228 = y;
        double r24078229 = z;
        double r24078230 = r24078228 / r24078229;
        double r24078231 = t;
        double r24078232 = 1.0;
        double r24078233 = r24078232 - r24078229;
        double r24078234 = r24078231 / r24078233;
        double r24078235 = r24078230 - r24078234;
        double r24078236 = r24078227 * r24078235;
        return r24078236;
}

↓

double f(double x, double y, double z, double t) {
        double r24078237 = z;
        double r24078238 = -1.8483774319326722e+36;
        bool r24078239 = r24078237 <= r24078238;
        double r24078240 = 1.0;
        double r24078241 = t;
        double r24078242 = r24078240 * r24078241;
        double r24078243 = r24078237 * r24078237;
        double r24078244 = x;
        double r24078245 = r24078243 / r24078244;
        double r24078246 = r24078242 / r24078245;
        double r24078247 = y;
        double r24078248 = r24078237 / r24078247;
        double r24078249 = r24078244 / r24078248;
        double r24078250 = r24078241 / r24078237;
        double r24078251 = r24078250 * r24078244;
        double r24078252 = r24078249 + r24078251;
        double r24078253 = r24078246 + r24078252;
        double r24078254 = 1.7142144841991844e-214;
        bool r24078255 = r24078237 <= r24078254;
        double r24078256 = r24078244 * r24078247;
        double r24078257 = r24078256 / r24078237;
        double r24078258 = r24078240 - r24078237;
        double r24078259 = r24078258 / r24078241;
        double r24078260 = r24078244 / r24078259;
        double r24078261 = -r24078260;
        double r24078262 = r24078257 + r24078261;
        double r24078263 = r24078247 / r24078237;
        double r24078264 = 1.0;
        double r24078265 = r24078264 / r24078258;
        double r24078266 = r24078241 * r24078265;
        double r24078267 = r24078263 - r24078266;
        double r24078268 = r24078244 * r24078267;
        double r24078269 = r24078255 ? r24078262 : r24078268;
        double r24078270 = r24078239 ? r24078253 : r24078269;
        return r24078270;
}

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	3.1

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

Derivation

1. Split input into 3 regimes

2. if z < -1.8483774319326722e+36

   1. Initial program 2.3

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

   2. Taylor expanded around inf 10.0

      \[\leadsto \color{blue}{\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)}\]

   3. Simplified 2.4

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

   if -1.8483774319326722e+36 < z < 1.7142144841991844e-214

   1. Initial program 8.9

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

   3. Applied clear-num 9.0

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

   5. Applied sub-neg 9.0

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

   6. Applied distribute-rgt-in 9.0

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

   7. Simplified 9.0

      \[\leadsto \frac{y}{z} \cdot x + \color{blue}{\frac{x \cdot -1}{\frac{1 - z}{t}}}\]
   8. Using strategy rm

   9. Applied associate-*l/ 3.9

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

   if 1.7142144841991844e-214 < z

   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

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

4. Final simplification 3.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1848377431932672203757011904118128640:\\ \;\;\;\;\frac{1 \cdot t}{\frac{z \cdot z}{x}} + \left(\frac{x}{\frac{z}{y}} + \frac{t}{z} \cdot x\right)\\ \mathbf{elif}\;z \le 1.714214484199184448434438031445311204718 \cdot 10^{-214}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(-\frac{x}{\frac{1 - z}{t}}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

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)))))