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

Average Error: 25.0 → 7.5

Time: 6.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -6.156727181680058362159255815336745142488 \cdot 10^{-230}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.508571022062013089185934498746472468243 \cdot 10^{205}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r678708 = x;
        double r678709 = y;
        double r678710 = r678709 - r678708;
        double r678711 = z;
        double r678712 = t;
        double r678713 = r678711 - r678712;
        double r678714 = r678710 * r678713;
        double r678715 = a;
        double r678716 = r678715 - r678712;
        double r678717 = r678714 / r678716;
        double r678718 = r678708 + r678717;
        return r678718;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r678719 = x;
        double r678720 = y;
        double r678721 = r678720 - r678719;
        double r678722 = z;
        double r678723 = t;
        double r678724 = r678722 - r678723;
        double r678725 = r678721 * r678724;
        double r678726 = a;
        double r678727 = r678726 - r678723;
        double r678728 = r678725 / r678727;
        double r678729 = r678719 + r678728;
        double r678730 = -inf.0;
        bool r678731 = r678729 <= r678730;
        double r678732 = r678727 / r678721;
        double r678733 = r678722 / r678732;
        double r678734 = r678723 / r678732;
        double r678735 = r678734 - r678719;
        double r678736 = r678733 - r678735;
        double r678737 = -6.156727181680058e-230;
        bool r678738 = r678729 <= r678737;
        double r678739 = 0.0;
        bool r678740 = r678729 <= r678739;
        double r678741 = r678719 / r678723;
        double r678742 = r678722 * r678720;
        double r678743 = r678742 / r678723;
        double r678744 = r678720 - r678743;
        double r678745 = fma(r678741, r678722, r678744);
        double r678746 = 1.508571022062013e+205;
        bool r678747 = r678729 <= r678746;
        double r678748 = r678747 ? r678729 : r678736;
        double r678749 = r678740 ? r678745 : r678748;
        double r678750 = r678738 ? r678729 : r678749;
        double r678751 = r678731 ? r678736 : r678750;
        return r678751;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	25.0
Target	9.6
Herbie	7.5

\[\begin{array}{l} \mathbf{if}\;a \lt -1.615306284544257464183904494091872805513 \cdot 10^{-142}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;a \lt 3.774403170083174201868024161554637965035 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \end{array}\]

Derivation

1. Split input into 3 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -inf.0 or 1.508571022062013e+205 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 55.2

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

   2. Simplified 16.4

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

   4. Applied clear-num 16.5

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

   6. Applied fma-udef 16.6

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

   7. Simplified 16.4

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

   9. Applied div-sub 16.4

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

   10. Applied associate-+l- 12.2

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

   if -inf.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < -6.156727181680058e-230 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 1.508571022062013e+205

   1. Initial program 1.9

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

   if -6.156727181680058e-230 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 54.4

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

   2. Simplified 55.1

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

   4. Applied clear-num 55.1

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

   6. Applied fma-udef 55.3

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

   7. Simplified 55.3

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

   8. Taylor expanded around inf 22.4

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

   9. Simplified 24.4

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

4. Final simplification 7.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} = -\infty:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -6.156727181680058362159255815336745142488 \cdot 10^{-230}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - \frac{z \cdot y}{t}\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 1.508571022062013089185934498746472468243 \cdot 10^{205}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{\frac{a - t}{y - x}} - \left(\frac{t}{\frac{a - t}{y - x}} - x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 +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))))