Average Error: 14.5 → 2.1
Time: 25.0s
Precision: 64
\[x \cdot \frac{\frac{y}{z} \cdot t}{t}\]
↓
\[\frac{\frac{y}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\frac{\sqrt[3]{z}}{\sqrt[3]{x}}}\]
x \cdot \frac{\frac{y}{z} \cdot t}{t}
↓
\frac{\frac{y}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\frac{\sqrt[3]{z}}{\sqrt[3]{x}}}
double f(double x, double y, double z, double t) {
        double r65729 = x;
        double r65730 = y;
        double r65731 = z;
        double r65732 = r65730 / r65731;
        double r65733 = t;
        double r65734 = r65732 * r65733;
        double r65735 = r65734 / r65733;
        double r65736 = r65729 * r65735;
        return r65736;
}
↓
double f(double x, double y, double z, double __attribute__((unused)) t) {
        double r65737 = y;
        double r65738 = z;
        double r65739 = cbrt(r65738);
        double r65740 = r65739 * r65739;
        double r65741 = x;
        double r65742 = cbrt(r65741);
        double r65743 = r65742 * r65742;
        double r65744 = r65740 / r65743;
        double r65745 = r65737 / r65744;
        double r65746 = r65739 / r65742;
        double r65747 = r65745 / r65746;
        return r65747;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

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Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.5

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

  2. Simplified6.2

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

  4. Applied div-inv6.2

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

  5. Applied associate-*l*6.2

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

  6. Simplified6.1

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

  8. Applied associate-*r/6.2

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

  10. Applied associate-/l*6.1

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

  12. Applied add-cube-cbrt6.9

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

  13. Applied add-cube-cbrt7.0

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

  14. Applied times-frac7.0

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

  15. Applied associate-/r*2.1

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

  16. Final simplification2.1

    \[\leadsto \frac{\frac{y}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{x} \cdot \sqrt[3]{x}}}}{\frac{\sqrt[3]{z}}{\sqrt[3]{x}}}\]

Reproduce

herbie shell --seed 2019304 +o rules:numerics
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1"
  :precision binary64
  (* x (/ (* (/ y z) t) t)))