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

Average Error: 24.7 → 9.9

Time: 32.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -5.220186090722651365771827959279614297882 \cdot 10^{-153} \lor \neg \left(a \le 1.35412616605907257317759566489239423007 \cdot 10^{-114}\right):\\ \;\;\;\;x + \left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{\frac{y - z}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r421496 = x;
        double r421497 = y;
        double r421498 = z;
        double r421499 = r421497 - r421498;
        double r421500 = t;
        double r421501 = r421500 - r421496;
        double r421502 = r421499 * r421501;
        double r421503 = a;
        double r421504 = r421503 - r421498;
        double r421505 = r421502 / r421504;
        double r421506 = r421496 + r421505;
        return r421506;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r421507 = a;
        double r421508 = -5.220186090722651e-153;
        bool r421509 = r421507 <= r421508;
        double r421510 = 1.3541261660590726e-114;
        bool r421511 = r421507 <= r421510;
        double r421512 = !r421511;
        bool r421513 = r421509 || r421512;
        double r421514 = x;
        double r421515 = t;
        double r421516 = r421515 - r421514;
        double r421517 = cbrt(r421516);
        double r421518 = r421517 * r421517;
        double r421519 = z;
        double r421520 = r421507 - r421519;
        double r421521 = cbrt(r421520);
        double r421522 = r421521 * r421521;
        double r421523 = cbrt(r421522);
        double r421524 = r421518 / r421523;
        double r421525 = y;
        double r421526 = r421525 - r421519;
        double r421527 = r421526 / r421521;
        double r421528 = cbrt(r421521);
        double r421529 = r421527 / r421528;
        double r421530 = r421524 * r421529;
        double r421531 = r421517 / r421521;
        double r421532 = r421530 * r421531;
        double r421533 = r421514 + r421532;
        double r421534 = r421514 / r421519;
        double r421535 = r421515 / r421519;
        double r421536 = r421534 - r421535;
        double r421537 = r421525 * r421536;
        double r421538 = r421515 + r421537;
        double r421539 = r421513 ? r421533 : r421538;
        return r421539;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.7
Target	11.9
Herbie	9.9

\[\begin{array}{l} \mathbf{if}\;z \lt -1.253613105609503593846459977496550767343 \cdot 10^{188}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{elif}\;z \lt 4.446702369113811028051510715777703865332 \cdot 10^{64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y}{z} \cdot \left(t - x\right)\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if a < -5.220186090722651e-153 or 1.3541261660590726e-114 < a

   1. Initial program 22.9

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

   3. Applied add-cube-cbrt 23.3

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

   4. Applied times-frac 9.6

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

   6. Applied *-un-lft-identity 9.6

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

   7. Applied cbrt-prod 9.6

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

   8. Applied add-cube-cbrt 9.8

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

   9. Applied times-frac 9.8

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

   10. Applied associate-*r* 9.3

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

   11. Simplified 9.3

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

   13. Applied add-cube-cbrt 9.3

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

   14. Applied cbrt-prod 9.4

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

   15. Applied *-un-lft-identity 9.4

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

   16. Applied cbrt-prod 9.4

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

   17. Applied *-un-lft-identity 9.4

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

   18. Applied times-frac 9.4

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

   19. Applied times-frac 9.4

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

   20. Applied associate-*r* 9.3

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

   21. Simplified 9.3

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

   if -5.220186090722651e-153 < a < 1.3541261660590726e-114

   1. Initial program 30.0

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

   3. Applied add-cube-cbrt 30.6

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

   4. Applied times-frac 22.1

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

   6. Applied *-un-lft-identity 22.1

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

   7. Applied cbrt-prod 22.1

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

   8. Applied add-cube-cbrt 22.3

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

   9. Applied times-frac 22.3

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

   10. Applied associate-*r* 21.1

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

   11. Simplified 21.1

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

   13. Applied add-cube-cbrt 21.1

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

   14. Applied cbrt-prod 21.1

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

   15. Applied add-cube-cbrt 21.2

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

   16. Applied cbrt-prod 21.3

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

   17. Applied add-cube-cbrt 21.2

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

   18. Applied times-frac 21.2

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

   19. Applied times-frac 21.2

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

   20. Applied associate-*r* 20.6

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

   21. Taylor expanded around inf 13.6

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

   22. Simplified 11.6

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

4. Final simplification 9.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -5.220186090722651365771827959279614297882 \cdot 10^{-153} \lor \neg \left(a \le 1.35412616605907257317759566489239423007 \cdot 10^{-114}\right):\\ \;\;\;\;x + \left(\frac{\sqrt[3]{t - x} \cdot \sqrt[3]{t - x}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \frac{\frac{y - z}{\sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}}}\right) \cdot \frac{\sqrt[3]{t - x}}{\sqrt[3]{a - z}}\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< z -1.25361310560950359e188) (- t (* (/ y z) (- t x))) (if (< z 4.44670236911381103e64) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x)))))

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