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

Average Error: 4.2 → 0.7

Time: 4.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -8.1624099695663258 \cdot 10^{133}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -8.59090119726614977 \cdot 10^{-199}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\left(t \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 - z}\right)\right)\right) \cdot 1\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.38430042432179337 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.75750237438848121 \cdot 10^{196}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\left(t \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 - z}\right)\right)\right) \cdot 1\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r368604 = x;
        double r368605 = y;
        double r368606 = z;
        double r368607 = r368605 / r368606;
        double r368608 = t;
        double r368609 = 1.0;
        double r368610 = r368609 - r368606;
        double r368611 = r368608 / r368610;
        double r368612 = r368607 - r368611;
        double r368613 = r368604 * r368612;
        return r368613;
}

↓

double f(double x, double y, double z, double t) {
        double r368614 = y;
        double r368615 = z;
        double r368616 = r368614 / r368615;
        double r368617 = t;
        double r368618 = 1.0;
        double r368619 = r368618 - r368615;
        double r368620 = r368617 / r368619;
        double r368621 = r368616 - r368620;
        double r368622 = -8.162409969566326e+133;
        bool r368623 = r368621 <= r368622;
        double r368624 = x;
        double r368625 = r368624 * r368614;
        double r368626 = r368625 / r368615;
        double r368627 = -r368624;
        double r368628 = r368627 * r368620;
        double r368629 = r368626 + r368628;
        double r368630 = 1.0;
        double r368631 = -r368630;
        double r368632 = r368631 + r368630;
        double r368633 = r368620 * r368632;
        double r368634 = r368624 * r368633;
        double r368635 = r368629 + r368634;
        double r368636 = -8.59090119726615e-199;
        bool r368637 = r368621 <= r368636;
        double r368638 = r368630 / r368615;
        double r368639 = r368630 / r368619;
        double r368640 = log1p(r368639);
        double r368641 = expm1(r368640);
        double r368642 = r368617 * r368641;
        double r368643 = r368642 * r368630;
        double r368644 = -r368643;
        double r368645 = fma(r368614, r368638, r368644);
        double r368646 = r368624 * r368645;
        double r368647 = r368646 + r368634;
        double r368648 = 1.3843004243217934e-257;
        bool r368649 = r368621 <= r368648;
        double r368650 = r368624 / r368615;
        double r368651 = r368617 * r368624;
        double r368652 = 2.0;
        double r368653 = pow(r368615, r368652);
        double r368654 = r368651 / r368653;
        double r368655 = r368651 / r368615;
        double r368656 = fma(r368618, r368654, r368655);
        double r368657 = fma(r368614, r368650, r368656);
        double r368658 = r368657 + r368634;
        double r368659 = 8.757502374388481e+196;
        bool r368660 = r368621 <= r368659;
        double r368661 = r368660 ? r368647 : r368635;
        double r368662 = r368649 ? r368658 : r368661;
        double r368663 = r368637 ? r368647 : r368662;
        double r368664 = r368623 ? r368635 : r368663;
        return r368664;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.2
Target	4.1
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

2. if (- (/ y z) (/ t (- 1.0 z))) < -8.162409969566326e+133 or 8.757502374388481e+196 < (- (/ y z) (/ t (- 1.0 z)))

   1. Initial program 13.9

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

   3. Applied add-cube-cbrt 14.3

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

   4. Applied div-inv 14.3

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

   5. Applied prod-diff 14.3

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

   6. Applied distribute-lft-in 14.3

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

   7. Simplified 14.0

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

   8. Simplified 13.9

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

   10. Applied div-inv 13.9

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

   12. Applied fma-udef 14.0

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

   13. Applied distribute-lft-in 14.0

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

   14. Simplified 1.8

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

   15. Simplified 1.8

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

   if -8.162409969566326e+133 < (- (/ y z) (/ t (- 1.0 z))) < -8.59090119726615e-199 or 1.3843004243217934e-257 < (- (/ y z) (/ t (- 1.0 z))) < 8.757502374388481e+196

   1. Initial program 0.2

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

   3. Applied add-cube-cbrt 0.7

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

   4. Applied div-inv 0.8

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

   5. Applied prod-diff 0.8

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

   6. Applied distribute-lft-in 0.8

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

   7. Simplified 0.3

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

   8. Simplified 0.3

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

   10. Applied div-inv 0.3

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

   12. Applied expm1-log1p-u 0.3

       \[\leadsto x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\left(t \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 - z}\right)\right)}\right) \cdot 1\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\]

   if -8.59090119726615e-199 < (- (/ y z) (/ t (- 1.0 z))) < 1.3843004243217934e-257

   1. Initial program 9.0

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

   3. Applied add-cube-cbrt 9.2

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

   4. Applied div-inv 9.2

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

   5. Applied prod-diff 9.2

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

   6. Applied distribute-lft-in 9.2

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

   7. Simplified 9.1

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

   8. Simplified 9.1

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

   9. Taylor expanded around inf 1.2

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

   10. Simplified 1.1

       \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right)} + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\]
3. Recombined 3 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \le -8.1624099695663258 \cdot 10^{133}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le -8.59090119726614977 \cdot 10^{-199}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\left(t \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 - z}\right)\right)\right) \cdot 1\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 1.38430042432179337 \cdot 10^{-257}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{x}{z}, \mathsf{fma}\left(1, \frac{t \cdot x}{{z}^{2}}, \frac{t \cdot x}{z}\right)\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \le 8.75750237438848121 \cdot 10^{196}:\\ \;\;\;\;x \cdot \mathsf{fma}\left(y, \frac{1}{z}, -\left(t \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 - z}\right)\right)\right) \cdot 1\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + \left(-x\right) \cdot \frac{t}{1 - z}\right) + x \cdot \left(\frac{t}{1 - z} \cdot \left(\left(-1\right) + 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(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)))))