Average Error: 12.5 → 1.1
Time: 2.8s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\left(x \cdot \frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y + z}}{\sqrt[3]{z}}\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\left(x \cdot \frac{\sqrt[3]{y + z} \cdot \sqrt[3]{y + z}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y + z}}{\sqrt[3]{z}}
double f(double x, double y, double z) {
        double r336085 = x;
        double r336086 = y;
        double r336087 = z;
        double r336088 = r336086 + r336087;
        double r336089 = r336085 * r336088;
        double r336090 = r336089 / r336087;
        return r336090;
}
↓
double f(double x, double y, double z) {
        double r336091 = x;
        double r336092 = y;
        double r336093 = z;
        double r336094 = r336092 + r336093;
        double r336095 = cbrt(r336094);
        double r336096 = r336095 * r336095;
        double r336097 = cbrt(r336093);
        double r336098 = r336097 * r336097;
        double r336099 = r336096 / r336098;
        double r336100 = r336091 * r336099;
        double r336101 = r336095 / r336097;
        double r336102 = r336100 * r336101;
        return r336102;
}

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

Original12.5
Target2.9
Herbie1.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.5

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

  3. Applied *-un-lft-identity12.5

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

  4. Applied times-frac3.1

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

  5. Simplified3.1

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

  7. Applied add-cube-cbrt4.3

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

  8. Applied add-cube-cbrt3.6

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

  9. Applied times-frac3.6

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

  10. Applied associate-*r*1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2020035 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

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

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