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

Average Error: 24.0 → 10.6

Time: 6.6s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.4075801034894206 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}}}\right) \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{z - t}}}} + x\\ \mathbf{elif}\;a \le 3.1595740263796063 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r623498 = x;
        double r623499 = y;
        double r623500 = r623499 - r623498;
        double r623501 = z;
        double r623502 = t;
        double r623503 = r623501 - r623502;
        double r623504 = r623500 * r623503;
        double r623505 = a;
        double r623506 = r623505 - r623502;
        double r623507 = r623504 / r623506;
        double r623508 = r623498 + r623507;
        return r623508;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r623509 = a;
        double r623510 = -1.4075801034894206e-160;
        bool r623511 = r623509 <= r623510;
        double r623512 = y;
        double r623513 = x;
        double r623514 = r623512 - r623513;
        double r623515 = z;
        double r623516 = t;
        double r623517 = r623515 - r623516;
        double r623518 = cbrt(r623517);
        double r623519 = r623509 - r623516;
        double r623520 = cbrt(r623519);
        double r623521 = r623520 * r623520;
        double r623522 = r623518 * r623518;
        double r623523 = cbrt(r623522);
        double r623524 = r623521 / r623523;
        double r623525 = r623518 / r623524;
        double r623526 = r623514 * r623525;
        double r623527 = cbrt(r623518);
        double r623528 = r623520 / r623527;
        double r623529 = r623518 / r623528;
        double r623530 = r623526 * r623529;
        double r623531 = r623530 + r623513;
        double r623532 = 3.1595740263796063e-93;
        bool r623533 = r623509 <= r623532;
        double r623534 = r623513 / r623516;
        double r623535 = r623515 * r623512;
        double r623536 = r623535 / r623516;
        double r623537 = r623512 - r623536;
        double r623538 = fma(r623534, r623515, r623537);
        double r623539 = r623517 / r623519;
        double r623540 = r623514 * r623539;
        double r623541 = r623540 + r623513;
        double r623542 = r623533 ? r623538 : r623541;
        double r623543 = r623511 ? r623531 : r623542;
        return r623543;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.0
Target	9.2
Herbie	10.6

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.7744031700831742 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if a < -1.4075801034894206e-160

   1. Initial program 22.8

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

   2. Simplified 11.8

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

   4. Applied fma-udef 11.8

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

   6. Applied div-inv 11.9

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

   7. Applied associate-*l* 9.4

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

   8. Simplified 9.3

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

   10. Applied add-cube-cbrt 10.0

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

   11. Applied associate-/l* 10.0

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

   13. Applied add-cube-cbrt 10.0

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

   14. Applied cbrt-prod 10.0

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

   15. Applied add-cube-cbrt 10.0

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

   16. Applied times-frac 10.0

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

   17. Applied times-frac 10.0

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

   18. Applied associate-*r* 9.6

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

   if -1.4075801034894206e-160 < a < 3.1595740263796063e-93

   1. Initial program 28.8

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

   2. Simplified 24.7

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

   4. Applied fma-udef 24.7

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

   6. Applied div-inv 24.8

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

   7. Applied associate-*l* 19.4

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

   8. Simplified 19.4

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

   9. Taylor expanded around inf 15.3

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

   10. Simplified 14.9

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

   if 3.1595740263796063e-93 < a

   1. Initial program 22.0

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

   2. Simplified 10.9

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

   4. Applied fma-udef 10.9

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

   6. Applied div-inv 11.0

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

   7. Applied associate-*l* 8.6

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

   8. Simplified 8.5

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.4075801034894206 \cdot 10^{-160}:\\ \;\;\;\;\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{z - t} \cdot \sqrt[3]{z - t}}}}\right) \cdot \frac{\sqrt[3]{z - t}}{\frac{\sqrt[3]{a - t}}{\sqrt[3]{\sqrt[3]{z - t}}}} + x\\ \mathbf{elif}\;a \le 3.1595740263796063 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{z - t}{a - t} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +o rules:numerics
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< a -1.6153062845442575e-142) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t))))))

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