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

Average Error: 24.3 → 10.9

Time: 30.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -6.143232131925929682871142551410290774749 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{elif}\;a \le 1.987670566207135403433863883298285206393 \cdot 10^{-115}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r444422 = x;
        double r444423 = y;
        double r444424 = z;
        double r444425 = r444423 - r444424;
        double r444426 = t;
        double r444427 = r444426 - r444422;
        double r444428 = r444425 * r444427;
        double r444429 = a;
        double r444430 = r444429 - r444424;
        double r444431 = r444428 / r444430;
        double r444432 = r444422 + r444431;
        return r444432;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r444433 = a;
        double r444434 = -6.14323213192593e-28;
        bool r444435 = r444433 <= r444434;
        double r444436 = x;
        double r444437 = y;
        double r444438 = z;
        double r444439 = r444437 - r444438;
        double r444440 = r444433 - r444438;
        double r444441 = cbrt(r444440);
        double r444442 = r444441 * r444441;
        double r444443 = r444439 / r444442;
        double r444444 = cbrt(r444441);
        double r444445 = r444444 * r444444;
        double r444446 = r444443 / r444445;
        double r444447 = t;
        double r444448 = r444447 - r444436;
        double r444449 = r444448 / r444444;
        double r444450 = r444446 * r444449;
        double r444451 = r444436 + r444450;
        double r444452 = 1.9876705662071354e-115;
        bool r444453 = r444433 <= r444452;
        double r444454 = r444436 / r444438;
        double r444455 = r444447 / r444438;
        double r444456 = r444454 - r444455;
        double r444457 = r444437 * r444456;
        double r444458 = r444447 + r444457;
        double r444459 = cbrt(r444439);
        double r444460 = cbrt(r444442);
        double r444461 = r444459 / r444460;
        double r444462 = r444459 / r444444;
        double r444463 = r444459 / r444441;
        double r444464 = r444448 / r444441;
        double r444465 = r444463 * r444464;
        double r444466 = r444462 * r444465;
        double r444467 = r444461 * r444466;
        double r444468 = r444436 + r444467;
        double r444469 = r444453 ? r444458 : r444468;
        double r444470 = r444435 ? r444451 : r444469;
        return r444470;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.3
Target	11.7
Herbie	10.9

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

Derivation

1. Split input into 3 regimes

2. if a < -6.14323213192593e-28

   1. Initial program 22.7

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

   3. Applied add-cube-cbrt 23.0

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

   4. Applied times-frac 7.7

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

   6. Applied add-cube-cbrt 7.9

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

   7. Applied *-un-lft-identity 7.9

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

   8. Applied times-frac 7.9

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

   9. Applied associate-*r* 7.9

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

   10. Simplified 7.9

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

   if -6.14323213192593e-28 < a < 1.9876705662071354e-115

   1. Initial program 27.7

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

   3. Applied add-cube-cbrt 28.4

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

   4. Applied times-frac 20.7

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

   6. Applied add-cube-cbrt 20.4

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

   7. Applied times-frac 20.4

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

   8. Applied associate-*l* 20.0

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

   9. Taylor expanded around inf 17.2

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

   10. Simplified 15.4

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

   if 1.9876705662071354e-115 < a

   1. Initial program 22.3

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

   3. Applied add-cube-cbrt 22.7

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

   4. Applied times-frac 9.2

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

   6. Applied add-cube-cbrt 9.1

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

   7. Applied times-frac 9.1

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

   8. Applied associate-*l* 9.0

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

   10. Applied add-cube-cbrt 9.0

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

   11. Applied cbrt-prod 9.1

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

   12. Applied times-frac 9.1

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

   13. Applied associate-*l* 9.1

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

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -6.143232131925929682871142551410290774749 \cdot 10^{-28}:\\ \;\;\;\;x + \frac{\frac{y - z}{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}}{\sqrt[3]{\sqrt[3]{a - z}} \cdot \sqrt[3]{\sqrt[3]{a - z}}} \cdot \frac{t - x}{\sqrt[3]{\sqrt[3]{a - z}}}\\ \mathbf{elif}\;a \le 1.987670566207135403433863883298285206393 \cdot 10^{-115}:\\ \;\;\;\;t + y \cdot \left(\frac{x}{z} - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sqrt[3]{y - z}}{\sqrt[3]{\sqrt[3]{a - z} \cdot \sqrt[3]{a - z}}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{\sqrt[3]{a - z}}} \cdot \left(\frac{\sqrt[3]{y - z}}{\sqrt[3]{a - z}} \cdot \frac{t - x}{\sqrt[3]{a - z}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))