Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B

Average Error: 14.3 → 1.4

Time: 4.4s

Precision: 64

Math

\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

↓

\[\left(\left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]

C

double f(double x, double y, double z, double t) {
        double r626302 = x;
        double r626303 = y;
        double r626304 = z;
        double r626305 = r626303 / r626304;
        double r626306 = t;
        double r626307 = r626305 * r626306;
        double r626308 = r626307 / r626306;
        double r626309 = r626302 * r626308;
        return r626309;
}

↓

double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r626310 = x;
        double r626311 = y;
        double r626312 = cbrt(r626311);
        double r626313 = z;
        double r626314 = cbrt(r626313);
        double r626315 = r626312 / r626314;
        double r626316 = r626310 * r626315;
        double r626317 = r626316 * r626315;
        double r626318 = r626317 * r626315;
        return r626318;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Target

Original	14.3
Target	1.4
Herbie	1.4

\[\begin{array}{l} \mathbf{if}\;\frac{\frac{y}{z} \cdot t}{t} \lt -1.206722051230450047215521150762600712224 \cdot 10^{245}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt -5.90752223693390632993316700759382836344 \cdot 10^{-275}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 5.658954423153415216825328199697215652986 \cdot 10^{-65}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;\frac{\frac{y}{z} \cdot t}{t} \lt 2.008718050240713347941382056648619307142 \cdot 10^{217}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array}\]

Derivation

1. Initial program 14.3

   \[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]

2. Simplified 6.2

   \[\leadsto \color{blue}{x \cdot \frac{y}{z}}\]
3. Using strategy rm

4. Applied add-cube-cbrt 7.0

   \[\leadsto x \cdot \frac{y}{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}\]

5. Applied add-cube-cbrt 7.2

   \[\leadsto x \cdot \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}\]

6. Applied times-frac 7.2

   \[\leadsto x \cdot \color{blue}{\left(\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)}\]

7. Applied associate-*r* 2.0

   \[\leadsto \color{blue}{\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}}\]
8. Using strategy rm

9. Applied times-frac 2.0

   \[\leadsto \left(x \cdot \color{blue}{\left(\frac{\sqrt[3]{y}}{\sqrt[3]{z}} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]

10. Applied associate-*r* 1.4

    \[\leadsto \color{blue}{\left(\left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right)} \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]

11. Final simplification 1.4

    \[\leadsto \left(\left(x \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}}\]

Reproduce

herbie shell --seed 2020002 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, B"
  :precision binary64

  :herbie-target
  (if (< (/ (* (/ y z) t) t) -1.20672205123045e+245) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) -5.907522236933906e-275) (* x (/ y z)) (if (< (/ (* (/ y z) t) t) 5.658954423153415e-65) (/ y (/ z x)) (if (< (/ (* (/ y z) t) t) 2.0087180502407133e+217) (* x (/ y z)) (/ (* y x) z)))))

  (* x (/ (* (/ y z) t) t)))