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

Average Error: 4.7 → 3.7

Time: 7.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;z \le -1.07919684621796776 \cdot 10^{42}:\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;z \le 2.8051310836370026 \cdot 10^{-193}:\\ \;\;\;\;1 \cdot \frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t) {
        double r427652 = x;
        double r427653 = y;
        double r427654 = z;
        double r427655 = r427653 / r427654;
        double r427656 = t;
        double r427657 = 1.0;
        double r427658 = r427657 - r427654;
        double r427659 = r427656 / r427658;
        double r427660 = r427655 - r427659;
        double r427661 = r427652 * r427660;
        return r427661;
}

↓

double f(double x, double y, double z, double t) {
        double r427662 = z;
        double r427663 = -1.0791968462179678e+42;
        bool r427664 = r427662 <= r427663;
        double r427665 = x;
        double r427666 = cbrt(r427662);
        double r427667 = r427666 * r427666;
        double r427668 = r427665 / r427667;
        double r427669 = y;
        double r427670 = r427669 / r427666;
        double r427671 = r427668 * r427670;
        double r427672 = t;
        double r427673 = 1.0;
        double r427674 = r427673 - r427662;
        double r427675 = r427672 / r427674;
        double r427676 = -r427675;
        double r427677 = r427665 * r427676;
        double r427678 = r427671 + r427677;
        double r427679 = 2.8051310836370026e-193;
        bool r427680 = r427662 <= r427679;
        double r427681 = 1.0;
        double r427682 = r427665 * r427669;
        double r427683 = r427682 / r427662;
        double r427684 = r427681 * r427683;
        double r427685 = r427684 + r427677;
        double r427686 = sqrt(r427662);
        double r427687 = r427665 / r427686;
        double r427688 = r427669 / r427686;
        double r427689 = r427687 * r427688;
        double r427690 = r427689 + r427677;
        double r427691 = r427680 ? r427685 : r427690;
        double r427692 = r427664 ? r427678 : r427691;
        return r427692;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	4.7
Target	4.3
Herbie	3.7

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

Derivation

1. Split input into 3 regimes

2. if z < -1.0791968462179678e+42

   1. Initial program 2.7

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

   3. Applied sub-neg 2.7

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

   4. Applied distribute-lft-in 2.7

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

   6. Applied add-cube-cbrt 3.1

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

   7. Applied *-un-lft-identity 3.1

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

   8. Applied times-frac 3.1

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

   9. Applied associate-*r* 4.3

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

   10. Simplified 4.3

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

   if -1.0791968462179678e+42 < z < 2.8051310836370026e-193

   1. Initial program 8.4

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

   3. Applied sub-neg 8.4

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

   4. Applied distribute-lft-in 8.4

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

   6. Applied add-cube-cbrt 9.0

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

   7. Applied *-un-lft-identity 9.0

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

   8. Applied times-frac 9.0

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

   9. Applied associate-*r* 5.9

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

   10. Simplified 5.9

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

   12. Applied *-un-lft-identity 5.9

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

   13. Applied associate-*l* 5.9

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

   14. Simplified 4.0

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

   if 2.8051310836370026e-193 < z

   1. Initial program 3.2

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

   3. Applied sub-neg 3.2

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

   4. Applied distribute-lft-in 3.2

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

   6. Applied add-sqr-sqrt 3.4

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

   7. Applied *-un-lft-identity 3.4

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

   8. Applied times-frac 3.4

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

   9. Applied associate-*r* 3.1

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

   10. Simplified 3.1

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

4. Final simplification 3.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \le -1.07919684621796776 \cdot 10^{42}:\\ \;\;\;\;\frac{x}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{y}{\sqrt[3]{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{elif}\;z \le 2.8051310836370026 \cdot 10^{-193}:\\ \;\;\;\;1 \cdot \frac{x \cdot y}{z} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\sqrt{z}} \cdot \frac{y}{\sqrt{z}} + x \cdot \left(-\frac{t}{1 - z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))))