Average Error: 12.8 → 1.1
Time: 15.4s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\frac{1}{\frac{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}}{\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{z + y}}}}}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\frac{1}{\frac{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}}{\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{z + y}}}}}
double f(double x, double y, double z) {
        double r16639026 = x;
        double r16639027 = y;
        double r16639028 = z;
        double r16639029 = r16639027 + r16639028;
        double r16639030 = r16639026 * r16639029;
        double r16639031 = r16639030 / r16639028;
        return r16639031;
}
↓
double f(double x, double y, double z) {
        double r16639032 = 1.0;
        double r16639033 = z;
        double r16639034 = cbrt(r16639033);
        double r16639035 = r16639034 * r16639034;
        double r16639036 = y;
        double r16639037 = r16639033 + r16639036;
        double r16639038 = cbrt(r16639037);
        double r16639039 = r16639038 * r16639038;
        double r16639040 = r16639035 / r16639039;
        double r16639041 = x;
        double r16639042 = r16639034 / r16639038;
        double r16639043 = r16639041 / r16639042;
        double r16639044 = r16639040 / r16639043;
        double r16639045 = r16639032 / r16639044;
        return r16639045;
}

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

Original12.8
Target3.1
Herbie1.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.8

    \[\frac{x \cdot \left(y + z\right)}{z}\]
  2. Using strategy rm

  3. Applied associate-/l*3.1

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

  5. Applied clear-num3.2

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

  7. Applied add-cube-cbrt4.3

    \[\leadsto \frac{1}{\frac{\frac{z}{\color{blue}{\left(\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}\right) \cdot \sqrt[3]{y + z}}}}{x}}\]

  8. Applied add-cube-cbrt3.7

    \[\leadsto \frac{1}{\frac{\frac{\color{blue}{\left(\sqrt[3]{z} \cdot \sqrt[3]{z}\right) \cdot \sqrt[3]{z}}}{\left(\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}\right) \cdot \sqrt[3]{y + z}}}{x}}\]

  9. Applied times-frac3.7

    \[\leadsto \frac{1}{\frac{\color{blue}{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}} \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{y + z}}}}{x}}\]

  10. Applied associate-/l*1.1

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}}{\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{y + z}}}}}}\]

  11. Final simplification1.1

    \[\leadsto \frac{1}{\frac{\frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{z + y} \cdot \sqrt[3]{z + y}}}{\frac{x}{\frac{\sqrt[3]{z}}{\sqrt[3]{z + y}}}}}\]

Reproduce

herbie shell --seed 2019171 +o rules:numerics
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"

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

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