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

Average Error: 7.8 → 6.7

Time: 40.4s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.693280280124389469819150824952027040683 \cdot 10^{-271} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

C

double f(double x, double y, double z) {
        double r446682 = x;
        double r446683 = y;
        double r446684 = r446682 + r446683;
        double r446685 = 1.0;
        double r446686 = z;
        double r446687 = r446683 / r446686;
        double r446688 = r446685 - r446687;
        double r446689 = r446684 / r446688;
        return r446689;
}

↓

double f(double x, double y, double z) {
        double r446690 = x;
        double r446691 = y;
        double r446692 = r446690 + r446691;
        double r446693 = 1.0;
        double r446694 = z;
        double r446695 = r446691 / r446694;
        double r446696 = r446693 - r446695;
        double r446697 = r446692 / r446696;
        double r446698 = -7.693280280124389e-271;
        bool r446699 = r446697 <= r446698;
        double r446700 = -0.0;
        bool r446701 = r446697 <= r446700;
        double r446702 = !r446701;
        bool r446703 = r446699 || r446702;
        double r446704 = sqrt(r446693);
        double r446705 = sqrt(r446691);
        double r446706 = sqrt(r446694);
        double r446707 = r446705 / r446706;
        double r446708 = r446704 + r446707;
        double r446709 = r446692 / r446708;
        double r446710 = r446704 - r446707;
        double r446711 = r446709 / r446710;
        double r446712 = r446703 ? r446697 : r446711;
        return r446712;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	7.8
Target	4.0
Herbie	6.7

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

Derivation

1. Split input into 2 regimes

2. if (/ (+ x y) (- 1.0 (/ y z))) < -7.693280280124389e-271 or -0.0 < (/ (+ x y) (- 1.0 (/ y z)))

   1. Initial program 4.3

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

   if -7.693280280124389e-271 < (/ (+ x y) (- 1.0 (/ y z))) < -0.0

   1. Initial program 54.4

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

   3. Applied add-sqr-sqrt 55.5

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

   4. Applied add-sqr-sqrt 59.7

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

   5. Applied times-frac 59.7

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

   6. Applied add-sqr-sqrt 59.7

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

   7. Applied difference-of-squares 59.7

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

   8. Applied associate-/r* 37.1

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

4. Final simplification 6.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y}{1 - \frac{y}{z}} \le -7.693280280124389469819150824952027040683 \cdot 10^{-271} \lor \neg \left(\frac{x + y}{1 - \frac{y}{z}} \le -0.0\right):\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x + y}{\sqrt{1} + \frac{\sqrt{y}}{\sqrt{z}}}}{\sqrt{1} - \frac{\sqrt{y}}{\sqrt{z}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(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))))