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

Average Error: 24.0 → 8.7

Time: 17.7s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.3435650229451346 \cdot 10^{-276}:\\ \;\;\;\;\frac{y - x}{\frac{a}{z - t} - \frac{1}{\frac{z - t}{t}}} + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(y + \frac{z \cdot x}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{z - t} - \frac{t}{z - t}}{y - x}} + x\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r33819530 = x;
        double r33819531 = y;
        double r33819532 = r33819531 - r33819530;
        double r33819533 = z;
        double r33819534 = t;
        double r33819535 = r33819533 - r33819534;
        double r33819536 = r33819532 * r33819535;
        double r33819537 = a;
        double r33819538 = r33819537 - r33819534;
        double r33819539 = r33819536 / r33819538;
        double r33819540 = r33819530 + r33819539;
        return r33819540;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r33819541 = x;
        double r33819542 = y;
        double r33819543 = r33819542 - r33819541;
        double r33819544 = z;
        double r33819545 = t;
        double r33819546 = r33819544 - r33819545;
        double r33819547 = r33819543 * r33819546;
        double r33819548 = a;
        double r33819549 = r33819548 - r33819545;
        double r33819550 = r33819547 / r33819549;
        double r33819551 = r33819541 + r33819550;
        double r33819552 = -1.3435650229451346e-276;
        bool r33819553 = r33819551 <= r33819552;
        double r33819554 = r33819548 / r33819546;
        double r33819555 = 1.0;
        double r33819556 = r33819546 / r33819545;
        double r33819557 = r33819555 / r33819556;
        double r33819558 = r33819554 - r33819557;
        double r33819559 = r33819543 / r33819558;
        double r33819560 = r33819559 + r33819541;
        double r33819561 = 0.0;
        bool r33819562 = r33819551 <= r33819561;
        double r33819563 = r33819544 * r33819541;
        double r33819564 = r33819563 / r33819545;
        double r33819565 = r33819542 + r33819564;
        double r33819566 = r33819544 * r33819542;
        double r33819567 = r33819566 / r33819545;
        double r33819568 = r33819565 - r33819567;
        double r33819569 = r33819545 / r33819546;
        double r33819570 = r33819554 - r33819569;
        double r33819571 = r33819570 / r33819543;
        double r33819572 = r33819555 / r33819571;
        double r33819573 = r33819572 + r33819541;
        double r33819574 = r33819562 ? r33819568 : r33819573;
        double r33819575 = r33819553 ? r33819560 : r33819574;
        return r33819575;
}

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	9.6
Herbie	8.7

\[\begin{array}{l} \mathbf{if}\;a \lt -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.3435650229451346e-276

   1. Initial program 20.3

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

   3. Applied associate-/l* 7.6

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

   5. Applied div-sub 7.6

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

   7. Applied clear-num 7.6

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

   if -1.3435650229451346e-276 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 59.5

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

   2. Taylor expanded around inf 19.8

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

   if 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 21.2

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

   3. Applied associate-/l* 7.7

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

   5. Applied div-sub 7.7

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

   7. Applied clear-num 7.8

      \[\leadsto x + \color{blue}{\frac{1}{\frac{\frac{a}{z - t} - \frac{t}{z - t}}{y - x}}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 8.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.3435650229451346 \cdot 10^{-276}:\\ \;\;\;\;\frac{y - x}{\frac{a}{z - t} - \frac{1}{\frac{z - t}{t}}} + x\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\left(y + \frac{z \cdot x}{t}\right) - \frac{z \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{a}{z - t} - \frac{t}{z - t}}{y - x}} + x\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"

  :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))))