Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Average Error: 24.5 → 11.0

Time: 7.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -5.42677252619702282 \cdot 10^{-135}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{1} \cdot \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \mathbf{elif}\;a \le 3.02356623104327892 \cdot 10^{-145}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \left(\sqrt[3]{\frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}} \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}}\right)\right) \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}}\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r794600 = x;
        double r794601 = y;
        double r794602 = z;
        double r794603 = r794601 - r794602;
        double r794604 = t;
        double r794605 = r794604 - r794600;
        double r794606 = r794603 * r794605;
        double r794607 = a;
        double r794608 = r794607 - r794602;
        double r794609 = r794606 / r794608;
        double r794610 = r794600 + r794609;
        return r794610;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r794611 = a;
        double r794612 = -5.426772526197023e-135;
        bool r794613 = r794611 <= r794612;
        double r794614 = x;
        double r794615 = y;
        double r794616 = z;
        double r794617 = r794615 - r794616;
        double r794618 = cbrt(r794617);
        double r794619 = r794618 * r794618;
        double r794620 = r794611 - r794616;
        double r794621 = cbrt(r794620);
        double r794622 = r794619 / r794621;
        double r794623 = 1.0;
        double r794624 = r794622 / r794623;
        double r794625 = r794618 / r794621;
        double r794626 = r794621 * r794621;
        double r794627 = cbrt(r794626);
        double r794628 = r794625 / r794627;
        double r794629 = t;
        double r794630 = r794629 - r794614;
        double r794631 = cbrt(r794621);
        double r794632 = r794630 / r794631;
        double r794633 = r794628 * r794632;
        double r794634 = r794624 * r794633;
        double r794635 = r794614 + r794634;
        double r794636 = 3.023566231043279e-145;
        bool r794637 = r794611 <= r794636;
        double r794638 = r794614 * r794615;
        double r794639 = r794638 / r794616;
        double r794640 = r794639 + r794629;
        double r794641 = r794629 * r794615;
        double r794642 = r794641 / r794616;
        double r794643 = r794640 - r794642;
        double r794644 = r794617 / r794626;
        double r794645 = r794644 / r794627;
        double r794646 = cbrt(r794632);
        double r794647 = r794646 * r794646;
        double r794648 = r794645 * r794647;
        double r794649 = r794648 * r794646;
        double r794650 = r794614 + r794649;
        double r794651 = r794637 ? r794643 : r794650;
        double r794652 = r794613 ? r794635 : r794651;
        return r794652;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.5
Target	11.7
Herbie	11.0

\[\begin{array}{l} \mathbf{if}\;z \lt -1.25361310560950359 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.44670236911381103 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -5.426772526197023e-135

   1. Initial program 22.5

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

   3. Applied add-cube-cbrt 22.9

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

   4. Applied times-frac 9.7

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 9.7

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

   7. Applied cbrt-prod 9.8

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\color{blue}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}}\]

   8. Applied *-un-lft-identity 9.8

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

   9. Applied times-frac 9.8

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

   10. Applied associate-*r* 9.5

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

   11. Simplified 9.5

       \[\leadsto x + \color{blue}{\frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\]
   12. Using strategy rm

   13. Applied *-un-lft-identity 9.5

       \[\leadsto x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\color{blue}{1 \cdot \sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\]

   14. Applied add-cube-cbrt 9.5

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

   15. Applied times-frac 9.5

       \[\leadsto x + \frac{\color{blue}{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}}{1 \cdot \sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\]

   16. Applied times-frac 9.5

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

   17. Applied associate-*l* 9.5

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

   if -5.426772526197023e-135 < a < 3.023566231043279e-145

   1. Initial program 29.9

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

   2. Taylor expanded around inf 14.1

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

   if 3.023566231043279e-145 < a

   1. Initial program 23.1

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

   3. Applied add-cube-cbrt 23.4

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

   4. Applied times-frac 10.5

      \[\leadsto x + \color{blue}{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}}\]
   5. Using strategy rm

   6. Applied add-cube-cbrt 10.6

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

   7. Applied cbrt-prod 10.6

      \[\leadsto x + \frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \frac{t - x}{\color{blue}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}}}\]

   8. Applied *-un-lft-identity 10.6

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

   9. Applied times-frac 10.6

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

   10. Applied associate-*r* 10.4

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

   11. Simplified 10.4

       \[\leadsto x + \color{blue}{\frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\]
   12. Using strategy rm

   13. Applied add-cube-cbrt 10.5

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

   14. Applied associate-*r* 10.5

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.42677252619702282 \cdot 10^{-135}:\\ \;\;\;\;x + \frac{\frac{\sqrt[3]{y - z} \cdot \sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{1} \cdot \left(\frac{\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\right)\\ \mathbf{elif}\;a \le 3.02356623104327892 \cdot 10^{-145}:\\ \;\;\;\;\left(\frac{x \cdot y}{z} + t\right) - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \left(\sqrt[3]{\frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}} \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}}\right)\right) \cdot \sqrt[3]{\frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.2536131056095036e+188) (- t (* (/ y z) (- t x))) (if (< z 4.446702369113811e+64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))