Average Error: 12.2 → 1.1
Time: 11.8s
Precision: 64
\[\frac{x \cdot \left(y + z\right)}{z}\]
↓
\[\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} + x\]
\frac{x \cdot \left(y + z\right)}{z}
↓
\left(x \cdot \frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{\sqrt[3]{z} \cdot \sqrt[3]{z}}\right) \cdot \frac{\sqrt[3]{y}}{\sqrt[3]{z}} + x
double f(double x, double y, double z) {
        double r474169 = x;
        double r474170 = y;
        double r474171 = z;
        double r474172 = r474170 + r474171;
        double r474173 = r474169 * r474172;
        double r474174 = r474173 / r474171;
        return r474174;
}
↓
double f(double x, double y, double z) {
        double r474175 = x;
        double r474176 = y;
        double r474177 = cbrt(r474176);
        double r474178 = r474177 * r474177;
        double r474179 = z;
        double r474180 = cbrt(r474179);
        double r474181 = r474180 * r474180;
        double r474182 = r474178 / r474181;
        double r474183 = r474175 * r474182;
        double r474184 = r474177 / r474180;
        double r474185 = r474183 * r474184;
        double r474186 = r474185 + r474175;
        return r474186;
}

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

Original12.2
Target3.0
Herbie1.1
\[\frac{x}{\frac{z}{y + z}}\]

Derivation

  1. Initial program 12.2

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

  2. Simplified3.2

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)}\]
  3. Using strategy rm

  4. Applied fma-udef3.2

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

  5. Simplified3.2

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

  7. Applied add-cube-cbrt3.6

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

  8. Applied add-cube-cbrt3.7

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

  9. Applied times-frac3.7

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

  10. Applied associate-*r*1.1

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

  11. Final simplification1.1

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

Reproduce

herbie shell --seed 2019350 +o rules:numerics
(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))