Average Error: 7.6 → 1.5
Time: 20.6s
Precision: 64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\frac{1}{\frac{1}{y + x} - \frac{\frac{y}{y + x}}{z}}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\frac{1}{\frac{1}{y + x} - \frac{\frac{y}{y + x}}{z}}
double f(double x, double y, double z) {
        double r318267 = x;
        double r318268 = y;
        double r318269 = r318267 + r318268;
        double r318270 = 1.0;
        double r318271 = z;
        double r318272 = r318268 / r318271;
        double r318273 = r318270 - r318272;
        double r318274 = r318269 / r318273;
        return r318274;
}
↓
double f(double x, double y, double z) {
        double r318275 = 1.0;
        double r318276 = 1.0;
        double r318277 = y;
        double r318278 = x;
        double r318279 = r318277 + r318278;
        double r318280 = r318276 / r318279;
        double r318281 = r318277 / r318279;
        double r318282 = z;
        double r318283 = r318281 / r318282;
        double r318284 = r318280 - r318283;
        double r318285 = r318275 / r318284;
        return r318285;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original7.6
Target4.1
Herbie1.5
\[\begin{array}{l} \mathbf{if}\;y \lt -3.742931076268985646434612946949172132145 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.553466245608673435460441960303815115662 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 7.6

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

  3. Applied clear-num7.8

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

  5. Applied div-sub7.8

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

  6. Simplified7.8

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

  7. Simplified1.5

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

  9. Applied pow11.5

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

  10. Final simplification1.5

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

Reproduce

herbie shell --seed 2019322 +o rules:numerics
(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))))