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

Average Error: 24.9 → 11.4

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;a \le -1.9761177729831929 \cdot 10^{-99} \lor \neg \left(a \le 3.1844700299462174 \cdot 10^{-74}\right):\\ \;\;\;\;\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y - x}}{\sqrt[3]{a - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r734396 = x;
        double r734397 = y;
        double r734398 = r734397 - r734396;
        double r734399 = z;
        double r734400 = t;
        double r734401 = r734399 - r734400;
        double r734402 = r734398 * r734401;
        double r734403 = a;
        double r734404 = r734403 - r734400;
        double r734405 = r734402 / r734404;
        double r734406 = r734396 + r734405;
        return r734406;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r734407 = a;
        double r734408 = -1.976117772983193e-99;
        bool r734409 = r734407 <= r734408;
        double r734410 = 3.1844700299462174e-74;
        bool r734411 = r734407 <= r734410;
        double r734412 = !r734411;
        bool r734413 = r734409 || r734412;
        double r734414 = z;
        double r734415 = t;
        double r734416 = r734414 - r734415;
        double r734417 = y;
        double r734418 = x;
        double r734419 = r734417 - r734418;
        double r734420 = cbrt(r734419);
        double r734421 = r734420 * r734420;
        double r734422 = r734407 - r734415;
        double r734423 = cbrt(r734422);
        double r734424 = r734423 * r734423;
        double r734425 = r734421 / r734424;
        double r734426 = r734416 * r734425;
        double r734427 = r734420 / r734423;
        double r734428 = r734426 * r734427;
        double r734429 = r734428 + r734418;
        double r734430 = r734418 / r734415;
        double r734431 = r734414 * r734417;
        double r734432 = r734431 / r734415;
        double r734433 = r734417 - r734432;
        double r734434 = fma(r734430, r734414, r734433);
        double r734435 = r734413 ? r734429 : r734434;
        return r734435;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.9
Target	9.8
Herbie	11.4

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

Derivation

1. Split input into 2 regimes

2. if a < -1.976117772983193e-99 or 3.1844700299462174e-74 < a

   1. Initial program 22.5

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

   2. Simplified 10.5

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

   4. Applied div-inv 10.6

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

   6. Applied fma-udef 10.7

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

   7. Simplified 10.6

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

   9. Applied add-cube-cbrt 11.1

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

   10. Applied add-cube-cbrt 11.2

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

   11. Applied times-frac 11.2

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

   12. Applied associate-*r* 8.9

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

   if -1.976117772983193e-99 < a < 3.1844700299462174e-74

   1. Initial program 29.8

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

   2. Simplified 25.3

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

   4. Applied div-inv 25.4

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

   6. Applied fma-udef 25.5

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

   7. Simplified 25.4

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

   8. Taylor expanded around inf 17.2

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

   9. Simplified 16.6

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

4. Final simplification 11.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \le -1.9761177729831929 \cdot 10^{-99} \lor \neg \left(a \le 3.1844700299462174 \cdot 10^{-74}\right):\\ \;\;\;\;\left(\left(z - t\right) \cdot \frac{\sqrt[3]{y - x} \cdot \sqrt[3]{y - x}}{\sqrt[3]{a - t} \cdot \sqrt[3]{a - t}}\right) \cdot \frac{\sqrt[3]{y - x}}{\sqrt[3]{a - t}} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020057 +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))))