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

Average Error: 7.6 → 0.2

Time: 15.8s

Precision: 64

Math

\[\frac{x + y}{1 - \frac{y}{z}}\]

↓

\[\begin{array}{l} \mathbf{if}\;y \le -164807406.9940239489078521728515625 \lor \neg \left(y \le 21140046859011661824211317635153590227040000\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{1}{\frac{x + y}{y}}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r481642 = x;
        double r481643 = y;
        double r481644 = r481642 + r481643;
        double r481645 = 1.0;
        double r481646 = z;
        double r481647 = r481643 / r481646;
        double r481648 = r481645 - r481647;
        double r481649 = r481644 / r481648;
        return r481649;
}

↓

double f(double x, double y, double z) {
        double r481650 = y;
        double r481651 = -164807406.99402395;
        bool r481652 = r481650 <= r481651;
        double r481653 = 2.1140046859011662e+43;
        bool r481654 = r481650 <= r481653;
        double r481655 = !r481654;
        bool r481656 = r481652 || r481655;
        double r481657 = 1.0;
        double r481658 = 1.0;
        double r481659 = x;
        double r481660 = r481659 + r481650;
        double r481661 = r481658 / r481660;
        double r481662 = r481660 / r481650;
        double r481663 = r481657 / r481662;
        double r481664 = z;
        double r481665 = r481663 / r481664;
        double r481666 = r481661 - r481665;
        double r481667 = r481657 / r481666;
        double r481668 = r481650 / r481664;
        double r481669 = r481658 - r481668;
        double r481670 = r481660 / r481669;
        double r481671 = r481656 ? r481667 : r481670;
        return r481671;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.6
Target	4.2
Herbie	0.2

\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

1. Split input into 2 regimes

2. if y < -164807406.99402395 or 2.1140046859011662e+43 < y

   1. Initial program 16.5

      \[\frac{x + y}{1 - \frac{y}{z}}\]
   2. Using strategy rm

   3. Applied clear-num 16.6

      \[\leadsto \color{blue}{\frac{1}{\frac{1 - \frac{y}{z}}{x + y}}}\]
   4. Using strategy rm

   5. Applied div-sub 16.6

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x + y} - \frac{\frac{y}{z}}{x + y}}}\]

   6. Simplified 0.2

      \[\leadsto \frac{1}{\frac{1}{x + y} - \color{blue}{\frac{\frac{y}{x + y}}{z}}}\]
   7. Using strategy rm

   8. Applied clear-num 0.2

      \[\leadsto \frac{1}{\frac{1}{x + y} - \frac{\color{blue}{\frac{1}{\frac{x + y}{y}}}}{z}}\]

   if -164807406.99402395 < y < 2.1140046859011662e+43

   1. Initial program 0.2

      \[\frac{x + y}{1 - \frac{y}{z}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \le -164807406.9940239489078521728515625 \lor \neg \left(y \le 21140046859011661824211317635153590227040000\right):\\ \;\;\;\;\frac{1}{\frac{1}{x + y} - \frac{\frac{1}{\frac{x + y}{y}}}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (- y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z)))

  (/ (+ x y) (- 1 (/ y z))))