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

Average Error: 4.8 → 1.8

Time: 24.5s

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}, \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{z}}{\sqrt[3]{y}}}, \left(-x\right) \cdot \frac{t}{1.0 - z}\right)\]

C

double f(double x, double y, double z, double t) {
        double r22023305 = x;
        double r22023306 = y;
        double r22023307 = z;
        double r22023308 = r22023306 / r22023307;
        double r22023309 = t;
        double r22023310 = 1.0;
        double r22023311 = r22023310 - r22023307;
        double r22023312 = r22023309 / r22023311;
        double r22023313 = r22023308 - r22023312;
        double r22023314 = r22023305 * r22023313;
        return r22023314;
}

↓

double f(double x, double y, double z, double t) {
        double r22023315 = x;
        double r22023316 = cbrt(r22023315);
        double r22023317 = r22023316 * r22023316;
        double r22023318 = z;
        double r22023319 = cbrt(r22023318);
        double r22023320 = r22023319 * r22023319;
        double r22023321 = y;
        double r22023322 = cbrt(r22023321);
        double r22023323 = r22023322 * r22023322;
        double r22023324 = r22023320 / r22023323;
        double r22023325 = r22023317 / r22023324;
        double r22023326 = r22023319 / r22023322;
        double r22023327 = r22023316 / r22023326;
        double r22023328 = -r22023315;
        double r22023329 = t;
        double r22023330 = 1.0;
        double r22023331 = r22023330 - r22023318;
        double r22023332 = r22023329 / r22023331;
        double r22023333 = r22023328 * r22023332;
        double r22023334 = fma(r22023325, r22023327, r22023333);
        return r22023334;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.8
Target	4.5
Herbie	1.8

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

Derivation

1. Initial program 4.8

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

3. Applied div-inv 4.9

   \[\leadsto x \cdot \left(\color{blue}{y \cdot \frac{1}{z}} - \frac{t}{1.0 - z}\right)\]

4. Applied fma-neg 4.9

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(y, \frac{1}{z}, -\frac{t}{1.0 - z}\right)}\]
5. Using strategy rm

6. Applied fma-udef 4.9

   \[\leadsto x \cdot \color{blue}{\left(y \cdot \frac{1}{z} + \left(-\frac{t}{1.0 - z}\right)\right)}\]

7. Applied distribute-lft-in 4.9

   \[\leadsto \color{blue}{x \cdot \left(y \cdot \frac{1}{z}\right) + x \cdot \left(-\frac{t}{1.0 - z}\right)}\]

8. Simplified 4.6

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

10. Applied add-cube-cbrt 5.0

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

11. Applied add-cube-cbrt 5.2

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

12. Applied times-frac 5.2

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

13. Applied add-cube-cbrt 5.3

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

14. Applied times-frac 1.8

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

15. Applied fma-def 1.8

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

16. Final simplification 1.8

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

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(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)))))