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

Average Error: 24.1 → 9.0

Time: 24.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.403564582318585211719020373880695846914 \cdot 10^{-264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(z - t\right), y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - y \cdot \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(z - t\right), y - x, x\right)\\ \end{array}\]

C

double f(double x, double y, double z, double t, double a) {
        double r24845701 = x;
        double r24845702 = y;
        double r24845703 = r24845702 - r24845701;
        double r24845704 = z;
        double r24845705 = t;
        double r24845706 = r24845704 - r24845705;
        double r24845707 = r24845703 * r24845706;
        double r24845708 = a;
        double r24845709 = r24845708 - r24845705;
        double r24845710 = r24845707 / r24845709;
        double r24845711 = r24845701 + r24845710;
        return r24845711;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r24845712 = x;
        double r24845713 = y;
        double r24845714 = r24845713 - r24845712;
        double r24845715 = z;
        double r24845716 = t;
        double r24845717 = r24845715 - r24845716;
        double r24845718 = r24845714 * r24845717;
        double r24845719 = a;
        double r24845720 = r24845719 - r24845716;
        double r24845721 = r24845718 / r24845720;
        double r24845722 = r24845712 + r24845721;
        double r24845723 = -1.4035645823185852e-264;
        bool r24845724 = r24845722 <= r24845723;
        double r24845725 = 1.0;
        double r24845726 = r24845725 / r24845720;
        double r24845727 = r24845726 * r24845717;
        double r24845728 = fma(r24845727, r24845714, r24845712);
        double r24845729 = 0.0;
        bool r24845730 = r24845722 <= r24845729;
        double r24845731 = r24845712 / r24845716;
        double r24845732 = r24845715 / r24845716;
        double r24845733 = r24845713 * r24845732;
        double r24845734 = r24845713 - r24845733;
        double r24845735 = fma(r24845731, r24845715, r24845734);
        double r24845736 = r24845730 ? r24845735 : r24845728;
        double r24845737 = r24845724 ? r24845728 : r24845736;
        return r24845737;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.1
Target	9.5
Herbie	9.0

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

Derivation

1. Split input into 2 regimes

2. if (+ x (/ (* (- y x) (- z t)) (- a t))) < -1.4035645823185852e-264 or 0.0 < (+ x (/ (* (- y x) (- z t)) (- a t)))

   1. Initial program 21.1

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

   2. Simplified 7.8

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

   4. Applied div-inv 7.9

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

   if -1.4035645823185852e-264 < (+ x (/ (* (- y x) (- z t)) (- a t))) < 0.0

   1. Initial program 57.6

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

   2. Simplified 57.7

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

   4. Applied div-inv 57.6

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

   6. Applied add-cube-cbrt 57.6

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

   7. Applied associate-*r* 57.6

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

   8. Taylor expanded around inf 19.6

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

   9. Simplified 22.4

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

4. Final simplification 9.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le -1.403564582318585211719020373880695846914 \cdot 10^{-264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(z - t\right), y - x, x\right)\\ \mathbf{elif}\;x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t} \le 0.0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z, y - y \cdot \frac{z}{t}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t} \cdot \left(z - t\right), y - x, x\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(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) 1.0) (/ (- z t) (- a t)))) (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))

  (+ x (/ (* (- y x) (- z t)) (- a t))))