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

Average Error: 4.4 → 1.9

Time: 30.7s

Precision: 64

Math

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

↓

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

C

double f(double x, double y, double z, double t) {
        double r269650 = x;
        double r269651 = y;
        double r269652 = z;
        double r269653 = r269651 / r269652;
        double r269654 = t;
        double r269655 = 1.0;
        double r269656 = r269655 - r269652;
        double r269657 = r269654 / r269656;
        double r269658 = r269653 - r269657;
        double r269659 = r269650 * r269658;
        return r269659;
}

↓

double f(double x, double y, double z, double t) {
        double r269660 = y;
        double r269661 = cbrt(r269660);
        double r269662 = r269661 * r269661;
        double r269663 = z;
        double r269664 = cbrt(r269663);
        double r269665 = 3.0;
        double r269666 = pow(r269664, r269665);
        double r269667 = cbrt(r269666);
        double r269668 = r269667 * r269667;
        double r269669 = r269662 / r269668;
        double r269670 = r269661 / r269664;
        double r269671 = x;
        double r269672 = r269670 * r269671;
        double r269673 = r269669 * r269672;
        double r269674 = t;
        double r269675 = 1.0;
        double r269676 = r269675 - r269663;
        double r269677 = r269674 / r269676;
        double r269678 = -r269677;
        double r269679 = r269671 * r269678;
        double r269680 = r269673 + r269679;
        return r269680;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.4
Target	4.1
Herbie	1.9

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

Derivation

1. Initial program 4.4

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

3. Applied add-cube-cbrt 4.9

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

4. Applied add-cube-cbrt 5.1

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

5. Applied times-frac 5.1

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

6. Applied fma-neg 5.1

   \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}, \frac{\sqrt[3]{y}}{\sqrt[3]{z}}, -\frac{t}{1 - z}\right)}\]
7. Using strategy rm

8. Applied fma-udef 5.1

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

9. Applied distribute-lft-in 5.1

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

10. Simplified 4.9

    \[\leadsto \color{blue}{\frac{y}{{\left(\sqrt[3]{z}\right)}^{3}} \cdot x} + x \cdot \left(-\frac{t}{1 - z}\right)\]
11. Using strategy rm

12. Applied add-cube-cbrt 5.1

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

13. Applied add-cube-cbrt 5.1

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

14. Applied times-frac 5.1

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

15. Applied associate-*l* 1.9

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

16. Simplified 1.9

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

17. Final simplification 1.9

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

Reproduce

herbie shell --seed 2019199 +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 (/ 1.0 (- 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)))))