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

Average Error: 24.2 → 8.5

Time: 19.4s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

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

C

double f(double x, double y, double z, double t, double a) {
        double r649703 = x;
        double r649704 = y;
        double r649705 = r649704 - r649703;
        double r649706 = z;
        double r649707 = t;
        double r649708 = r649706 - r649707;
        double r649709 = r649705 * r649708;
        double r649710 = a;
        double r649711 = r649710 - r649707;
        double r649712 = r649709 / r649711;
        double r649713 = r649703 + r649712;
        return r649713;
}

↓

double f(double x, double y, double z, double t, double a) {
        double r649714 = z;
        double r649715 = a;
        double r649716 = t;
        double r649717 = r649715 - r649716;
        double r649718 = r649714 / r649717;
        double r649719 = y;
        double r649720 = x;
        double r649721 = r649719 - r649720;
        double r649722 = r649718 * r649721;
        double r649723 = r649716 / r649717;
        double r649724 = -r649723;
        double r649725 = fma(r649721, r649724, r649720);
        double r649726 = r649722 + r649725;
        return r649726;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	24.2
Target	9.3
Herbie	8.5

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

Derivation

1. Initial program 24.2

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

2. Simplified 14.6

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

4. Applied fma-udef 14.7

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

6. Applied div-inv 14.7

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

7. Applied associate-*l* 11.7

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

8. Simplified 11.6

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

10. Applied div-sub 11.6

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

12. Applied sub-neg 11.6

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

13. Applied distribute-rgt-in 11.6

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

14. Applied associate-+l+ 8.6

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

15. Simplified 8.5

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

16. Final simplification 8.5

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

Reproduce

herbie shell --seed 2020045 +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))))