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

Average Error: 24.8 → 11.0

Time: 5.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -2.0356593442717544 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y - x}}{\sqrt[3]{a - t}}, x\right)\\ \mathbf{elif}\;a \le 8.42218082085157541 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y - x}{\sqrt[3]{a - t}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r605506 = x;
        double r605507 = y;
        double r605508 = r605507 - r605506;
        double r605509 = z;
        double r605510 = t;
        double r605511 = r605509 - r605510;
        double r605512 = r605508 * r605511;
        double r605513 = a;
        double r605514 = r605513 - r605510;
        double r605515 = r605512 / r605514;
        double r605516 = r605506 + r605515;
        return r605516;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r605517 = a;
        double r605518 = -2.0356593442717544e-150;
        bool r605519 = r605517 <= r605518;
        double r605520 = z;
        double r605521 = t;
        double r605522 = r605520 - r605521;
        double r605523 = y;
        double r605524 = x;
        double r605525 = r605523 - r605524;
        double r605526 = cbrt(r605525);
        double r605527 = r605526 * r605526;
        double r605528 = r605517 - r605521;
        double r605529 = cbrt(r605528);
        double r605530 = r605529 * r605529;
        double r605531 = r605527 / r605530;
        double r605532 = r605522 * r605531;
        double r605533 = r605526 / r605529;
        double r605534 = fma(r605532, r605533, r605524);
        double r605535 = 8.422180820851575e-190;
        bool r605536 = r605517 <= r605535;
        double r605537 = r605524 / r605521;
        double r605538 = r605520 * r605523;
        double r605539 = r605538 / r605521;
        double r605540 = r605523 - r605539;
        double r605541 = fma(r605537, r605520, r605540);
        double r605542 = r605522 / r605530;
        double r605543 = r605525 / r605529;
        double r605544 = r605542 * r605543;
        double r605545 = r605544 + r605524;
        double r605546 = r605536 ? r605541 : r605545;
        double r605547 = r605519 ? r605534 : r605546;
        return r605547;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.8
Target	9.6
Herbie	11.0

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

Derivation

1. Split input into 3 regimes

2. if a < -2.0356593442717544e-150

   1. Initial program 24.3

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

   2. Simplified 12.7

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

   4. Applied div-inv 12.8

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

   6. Applied fma-udef 12.8

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

   7. Simplified 12.8

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

   9. Applied add-cube-cbrt 13.3

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

   10. Applied add-cube-cbrt 13.4

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

   11. Applied times-frac 13.4

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

   12. Applied associate-*r* 10.7

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

   14. Applied fma-def 10.7

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

   if -2.0356593442717544e-150 < a < 8.422180820851575e-190

   1. Initial program 29.9

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

   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 div-inv 24.8

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

   6. Applied fma-udef 24.8

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

   7. Simplified 24.7

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

   8. Taylor expanded around inf 12.6

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

   9. Simplified 12.1

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

   if 8.422180820851575e-190 < a

   1. Initial program 22.8

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

   2. Simplified 12.7

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

   4. Applied div-inv 12.8

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

   6. Applied fma-udef 12.8

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

   7. Simplified 12.8

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

   9. Applied add-cube-cbrt 13.2

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

   10. Applied *-un-lft-identity 13.2

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

   11. Applied times-frac 13.3

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

   12. Applied associate-*r* 10.9

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

   13. Simplified 10.9

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

4. Final simplification 11.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -2.0356593442717544 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}, \frac{\sqrt[3]{y - x}}{\sqrt[3]{a - t}}, x\right)\\ \mathbf{elif}\;a \le 8.42218082085157541 \cdot 10^{-190}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - t}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}} \cdot \frac{y - x}{\sqrt[3]{a - t}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 +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))))