Average Error: 0.0 → 0.2
Time: 2.6s
Precision: 64
\[x - \frac{y}{200}\]
↓
\[\left(x - \frac{y}{\sqrt{200}} \cdot \frac{1}{\sqrt{200}}\right) + \frac{y}{\sqrt{200}} \cdot \left(\left(-\frac{1}{\sqrt{200}}\right) + \frac{1}{\sqrt{200}}\right)\]
x - \frac{y}{200}
↓
\left(x - \frac{y}{\sqrt{200}} \cdot \frac{1}{\sqrt{200}}\right) + \frac{y}{\sqrt{200}} \cdot \left(\left(-\frac{1}{\sqrt{200}}\right) + \frac{1}{\sqrt{200}}\right)
double f(double x, double y) {
        double r250000 = x;
        double r250001 = y;
        double r250002 = 200.0;
        double r250003 = r250001 / r250002;
        double r250004 = r250000 - r250003;
        return r250004;
}
↓
double f(double x, double y) {
        double r250005 = x;
        double r250006 = y;
        double r250007 = 200.0;
        double r250008 = sqrt(r250007);
        double r250009 = r250006 / r250008;
        double r250010 = 1.0;
        double r250011 = r250010 / r250008;
        double r250012 = r250009 * r250011;
        double r250013 = r250005 - r250012;
        double r250014 = -r250011;
        double r250015 = r250014 + r250011;
        double r250016 = r250009 * r250015;
        double r250017 = r250013 + r250016;
        return r250017;
}

Error

Bits error versus x

Bits error versus y

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Results

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Derivation

  1. Initial program 0.0

    \[x - \frac{y}{200}\]
  2. Using strategy rm

  3. Applied add-sqr-sqrt0.4

    \[\leadsto x - \frac{y}{\color{blue}{\sqrt{200} \cdot \sqrt{200}}}\]

  4. Applied add-cube-cbrt0.7

    \[\leadsto x - \frac{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}{\sqrt{200} \cdot \sqrt{200}}\]

  5. Applied times-frac0.7

    \[\leadsto x - \color{blue}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{200}} \cdot \frac{\sqrt[3]{y}}{\sqrt{200}}}\]

  6. Applied add-sqr-sqrt32.2

    \[\leadsto \color{blue}{\sqrt{x} \cdot \sqrt{x}} - \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{200}} \cdot \frac{\sqrt[3]{y}}{\sqrt{200}}\]

  7. Applied prod-diff32.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{x}, \sqrt{x}, -\frac{\sqrt[3]{y}}{\sqrt{200}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{200}}\right) + \mathsf{fma}\left(-\frac{\sqrt[3]{y}}{\sqrt{200}}, \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{200}}, \frac{\sqrt[3]{y}}{\sqrt{200}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{200}}\right)}\]

  8. Simplified0.2

    \[\leadsto \color{blue}{\left(x - \frac{y}{\sqrt{200}} \cdot \frac{1}{\sqrt{200}}\right)} + \mathsf{fma}\left(-\frac{\sqrt[3]{y}}{\sqrt{200}}, \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{200}}, \frac{\sqrt[3]{y}}{\sqrt{200}} \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt{200}}\right)\]

  9. Simplified0.2

    \[\leadsto \left(x - \frac{y}{\sqrt{200}} \cdot \frac{1}{\sqrt{200}}\right) + \color{blue}{\frac{y}{\sqrt{200}} \cdot \left(\left(-\frac{1}{\sqrt{200}}\right) + \frac{1}{\sqrt{200}}\right)}\]

  10. Final simplification0.2

    \[\leadsto \left(x - \frac{y}{\sqrt{200}} \cdot \frac{1}{\sqrt{200}}\right) + \frac{y}{\sqrt{200}} \cdot \left(\left(-\frac{1}{\sqrt{200}}\right) + \frac{1}{\sqrt{200}}\right)\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y)
  :name "Data.Colour.CIE:cieLAB from colour-2.3.3, D"
  :precision binary64
  (- x (/ y 200)))