Average Error: 6.4 → 0.9
Time: 23.7s
Precision: 64
\[x + \frac{\left(y - x\right) \cdot z}{t}\]
↓
\[\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}} + x\]
x + \frac{\left(y - x\right) \cdot z}{t}
↓
\left(\left(y - x\right) \cdot \frac{\sqrt[3]{z} \cdot \sqrt[3]{z}}{\sqrt[3]{t} \cdot \sqrt[3]{t}}\right) \cdot \frac{\sqrt[3]{z}}{\sqrt[3]{t}} + x
double f(double x, double y, double z, double t) {
        double r291949 = x;
        double r291950 = y;
        double r291951 = r291950 - r291949;
        double r291952 = z;
        double r291953 = r291951 * r291952;
        double r291954 = t;
        double r291955 = r291953 / r291954;
        double r291956 = r291949 + r291955;
        return r291956;
}
↓
double f(double x, double y, double z, double t) {
        double r291957 = y;
        double r291958 = x;
        double r291959 = r291957 - r291958;
        double r291960 = z;
        double r291961 = cbrt(r291960);
        double r291962 = r291961 * r291961;
        double r291963 = t;
        double r291964 = cbrt(r291963);
        double r291965 = r291964 * r291964;
        double r291966 = r291962 / r291965;
        double r291967 = r291959 * r291966;
        double r291968 = r291961 / r291964;
        double r291969 = r291967 * r291968;
        double r291970 = r291969 + r291958;
        return r291970;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original6.4
Target2.0
Herbie0.9
\[\begin{array}{l} \mathbf{if}\;x \lt -9.025511195533004570453352523209034680317 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x \lt 4.275032163700714748507147332551979944314 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array}\]

Derivation

  1. Initial program 6.4

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

  2. Simplified6.2

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

  4. Applied div-inv6.2

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - x\right) \cdot \frac{1}{t}}, z, x\right)\]
  5. Using strategy rm

  6. Applied fma-udef6.2

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

  7. Simplified2.2

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

  9. Applied add-cube-cbrt2.8

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

  10. Applied add-cube-cbrt2.9

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

  11. Applied times-frac2.9

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

  12. Applied associate-*r*0.9

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

  13. Final simplification0.9

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

Reproduce

herbie shell --seed 2019347 +o rules:numerics
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (if (< x -9.025511195533005e-135) (- x (* (/ z t) (- x y))) (if (< x 4.275032163700715e-250) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z)))))

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