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

Average Error: 4.6 → 1.3

Time: 4.1s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -6.28902280085759557 \cdot 10^{184}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.2136975206029316 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(\sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}} \cdot \sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}}\right) \cdot \sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.9453220644851443 \cdot 10^{-279}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 6.56399033386167465 \cdot 10^{218}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z}\right) + \left(-x\right) \cdot \frac{t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r575290 = x;
        double r575291 = y;
        double r575292 = z;
        double r575293 = r575291 / r575292;
        double r575294 = t;
        double r575295 = 1.0;
        double r575296 = r575295 - r575292;
        double r575297 = r575294 / r575296;
        double r575298 = r575293 - r575297;
        double r575299 = r575290 * r575298;
        return r575299;
}

↓

double f(double x, double y, double z, double t) {
        double r575300 = y;
        double r575301 = z;
        double r575302 = r575300 / r575301;
        double r575303 = t;
        double r575304 = 1.0;
        double r575305 = r575304 - r575301;
        double r575306 = r575303 / r575305;
        double r575307 = r575302 - r575306;
        double r575308 = -6.2890228008575956e+184;
        bool r575309 = r575307 <= r575308;
        double r575310 = x;
        double r575311 = r575310 * r575300;
        double r575312 = 1.0;
        double r575313 = r575312 / r575301;
        double r575314 = r575311 * r575313;
        double r575315 = -r575310;
        double r575316 = r575315 * r575306;
        double r575317 = r575314 + r575316;
        double r575318 = -1.2136975206029316e-125;
        bool r575319 = r575307 <= r575318;
        double r575320 = r575310 * r575302;
        double r575321 = cbrt(r575316);
        double r575322 = r575321 * r575321;
        double r575323 = r575322 * r575321;
        double r575324 = r575320 + r575323;
        double r575325 = 2.9453220644851443e-279;
        bool r575326 = r575307 <= r575325;
        double r575327 = r575311 / r575301;
        double r575328 = r575303 * r575310;
        double r575329 = 2.0;
        double r575330 = pow(r575301, r575329);
        double r575331 = r575328 / r575330;
        double r575332 = r575304 * r575331;
        double r575333 = r575328 / r575301;
        double r575334 = r575332 + r575333;
        double r575335 = r575327 + r575334;
        double r575336 = 6.563990333861675e+218;
        bool r575337 = r575307 <= r575336;
        double r575338 = cbrt(r575310);
        double r575339 = r575338 * r575338;
        double r575340 = r575338 * r575302;
        double r575341 = r575339 * r575340;
        double r575342 = r575341 + r575316;
        double r575343 = r575337 ? r575342 : r575317;
        double r575344 = r575326 ? r575335 : r575343;
        double r575345 = r575319 ? r575324 : r575344;
        double r575346 = r575309 ? r575317 : r575345;
        return r575346;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.6
Target	4.2
Herbie	1.3

\[\begin{array}{l} \mathbf{if}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt -7.62322630331204244 \cdot 10^{-196}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \mathbf{elif}\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \lt 1.41339449277023022 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ \end{array}\]

Derivation

1. Split input into 4 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -6.2890228008575956e+184 or 6.563990333861675e+218 < (- (/ 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 div-inv 19.3

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

   5. Applied sub-neg 19.3

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

   6. Applied distribute-lft-in 19.3

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

   7. Simplified 19.3

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

   9. Applied div-inv 19.3

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

   10. Applied associate-*r* 0.9

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

   if -6.2890228008575956e+184 < (- (/ y z) (/ t (- 1.0 z))) < -1.2136975206029316e-125

   1. Initial program 0.2

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

   3. Applied div-inv 0.2

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

   5. Applied sub-neg 0.2

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

   6. Applied distribute-lft-in 0.2

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

   7. Simplified 0.2

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

   9. Applied add-cube-cbrt 0.7

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

   if -1.2136975206029316e-125 < (- (/ y z) (/ t (- 1.0 z))) < 2.9453220644851443e-279

   1. Initial program 7.9

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

   3. Applied div-inv 8.0

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

   4. Taylor expanded around inf 4.2

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

   if 2.9453220644851443e-279 < (- (/ y z) (/ t (- 1.0 z))) < 6.563990333861675e+218

   1. Initial program 0.2

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

   3. Applied div-inv 0.2

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

   5. Applied sub-neg 0.2

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

   6. Applied distribute-lft-in 0.2

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

   7. Simplified 0.2

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

   9. Applied add-cube-cbrt 0.8

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

   10. Applied associate-*l* 0.8

       \[\leadsto \color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z}\right)} + \left(-x\right) \cdot \frac{t}{1 - z}\]
3. Recombined 4 regimes into one program.

4. Final simplification 1.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -6.28902280085759557 \cdot 10^{184}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -1.2136975206029316 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \frac{y}{z} + \left(\sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}} \cdot \sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}}\right) \cdot \sqrt[3]{\left(-x\right) \cdot \frac{t}{1 - z}}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 2.9453220644851443 \cdot 10^{-279}:\\ \;\;\;\;\frac{x \cdot y}{z} + \left(1 \cdot \frac{t \cdot x}{{z}^{2}} + \frac{t \cdot x}{z}\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 6.56399033386167465 \cdot 10^{218}:\\ \;\;\;\;\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \left(\sqrt[3]{x} \cdot \frac{y}{z}\right) + \left(-x\right) \cdot \frac{t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 
(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)))))