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

Average Error: 24.0 → 8.7

Time: 18.3s

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 r33178551 = x;
        double r33178552 = y;
        double r33178553 = r33178552 - r33178551;
        double r33178554 = z;
        double r33178555 = t;
        double r33178556 = r33178554 - r33178555;
        double r33178557 = r33178553 * r33178556;
        double r33178558 = a;
        double r33178559 = r33178558 - r33178555;
        double r33178560 = r33178557 / r33178559;
        double r33178561 = r33178551 + r33178560;
        return r33178561;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r33178562 = x;
        double r33178563 = y;
        double r33178564 = r33178563 - r33178562;
        double r33178565 = z;
        double r33178566 = t;
        double r33178567 = r33178565 - r33178566;
        double r33178568 = r33178564 * r33178567;
        double r33178569 = a;
        double r33178570 = r33178569 - r33178566;
        double r33178571 = r33178568 / r33178570;
        double r33178572 = r33178562 + r33178571;
        double r33178573 = -1.3435650229451346e-276;
        bool r33178574 = r33178572 <= r33178573;
        double r33178575 = r33178569 / r33178567;
        double r33178576 = 1.0;
        double r33178577 = r33178567 / r33178566;
        double r33178578 = r33178576 / r33178577;
        double r33178579 = r33178575 - r33178578;
        double r33178580 = r33178564 / r33178579;
        double r33178581 = r33178580 + r33178562;
        double r33178582 = 0.0;
        bool r33178583 = r33178572 <= r33178582;
        double r33178584 = r33178565 * r33178562;
        double r33178585 = r33178584 / r33178566;
        double r33178586 = r33178563 + r33178585;
        double r33178587 = r33178565 * r33178563;
        double r33178588 = r33178587 / r33178566;
        double r33178589 = r33178586 - r33178588;
        double r33178590 = r33178566 / r33178567;
        double r33178591 = r33178575 - r33178590;
        double r33178592 = r33178591 / r33178564;
        double r33178593 = r33178576 / r33178592;
        double r33178594 = r33178593 + r33178562;
        double r33178595 = r33178583 ? r33178589 : r33178594;
        double r33178596 = r33178574 ? r33178581 : r33178595;
        return r33178596;
}

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))))