Average Error: 29.0 → 0.4
Time: 5.3s
Precision: 64
\[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]
↓
\[0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - \frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y}}\right)\]
\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}
↓
0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - \frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y}}\right)
double f(double x, double y, double z) {
        double r598460 = x;
        double r598461 = r598460 * r598460;
        double r598462 = y;
        double r598463 = r598462 * r598462;
        double r598464 = r598461 + r598463;
        double r598465 = z;
        double r598466 = r598465 * r598465;
        double r598467 = r598464 - r598466;
        double r598468 = 2.0;
        double r598469 = r598462 * r598468;
        double r598470 = r598467 / r598469;
        return r598470;
}
↓
double f(double x, double y, double z) {
        double r598471 = 0.5;
        double r598472 = y;
        double r598473 = x;
        double r598474 = 1.0;
        double r598475 = pow(r598473, r598474);
        double r598476 = r598472 / r598473;
        double r598477 = r598475 / r598476;
        double r598478 = r598472 + r598477;
        double r598479 = z;
        double r598480 = cbrt(r598472);
        double r598481 = r598480 * r598480;
        double r598482 = r598479 / r598481;
        double r598483 = r598479 / r598480;
        double r598484 = r598482 * r598483;
        double r598485 = r598478 - r598484;
        double r598486 = r598471 * r598485;
        return r598486;
}

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

Original29.0
Target0.2
Herbie0.4
\[y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right)\]

Derivation

  1. Initial program 29.0

    \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2}\]

  2. Taylor expanded around 0 12.9

    \[\leadsto \color{blue}{\left(0.5 \cdot y + 0.5 \cdot \frac{{x}^{2}}{y}\right) - 0.5 \cdot \frac{{z}^{2}}{y}}\]

  3. Simplified12.9

    \[\leadsto \color{blue}{0.5 \cdot \left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)}\]
  4. Using strategy rm

  5. Applied sqr-pow12.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{\color{blue}{{x}^{\left(\frac{2}{2}\right)} \cdot {x}^{\left(\frac{2}{2}\right)}}}{y}\right) - \frac{{z}^{2}}{y}\right)\]

  6. Applied associate-/l*7.1

    \[\leadsto 0.5 \cdot \left(\left(y + \color{blue}{\frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{{x}^{\left(\frac{2}{2}\right)}}}}\right) - \frac{{z}^{2}}{y}\right)\]

  7. Simplified7.1

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\color{blue}{\frac{y}{x}}}\right) - \frac{{z}^{2}}{y}\right)\]
  8. Using strategy rm

  9. Applied add-cube-cbrt7.3

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{x}}\right) - \frac{{z}^{2}}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}}\right)\]

  10. Applied add-sqr-sqrt35.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{x}}\right) - \frac{{\color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)}}^{2}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right)\]

  11. Applied unpow-prod-down35.9

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{x}}\right) - \frac{\color{blue}{{\left(\sqrt{z}\right)}^{2} \cdot {\left(\sqrt{z}\right)}^{2}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}\right)\]

  12. Applied times-frac32.5

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{x}}\right) - \color{blue}{\frac{{\left(\sqrt{z}\right)}^{2}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{{\left(\sqrt{z}\right)}^{2}}{\sqrt[3]{y}}}\right)\]

  13. Simplified32.5

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{x}}\right) - \color{blue}{\frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \frac{{\left(\sqrt{z}\right)}^{2}}{\sqrt[3]{y}}\right)\]

  14. Simplified0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{\left(\frac{2}{2}\right)}}{\frac{y}{x}}\right) - \frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \color{blue}{\frac{z}{\sqrt[3]{y}}}\right)\]

  15. Final simplification0.4

    \[\leadsto 0.5 \cdot \left(\left(y + \frac{{x}^{1}}{\frac{y}{x}}\right) - \frac{z}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{z}{\sqrt[3]{y}}\right)\]

Reproduce

herbie shell --seed 2020039 +o rules:numerics
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2)))