Average Error: 12.6 → 1.9
Time: 2.8s
Precision: binary64
\[\frac{x \cdot \left(y - z\right)}{y} \]
↓
\[\begin{array}{l} t_0 := \sqrt[3]{y - z}\\ \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_0 \cdot t_0}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y}}{t_0}} \end{array} \]
\frac{x \cdot \left(y - z\right)}{y}
↓
\begin{array}{l}
t_0 := \sqrt[3]{y - z}\\
\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\frac{\sqrt[3]{y} \cdot \sqrt[3]{y}}{t_0 \cdot t_0}} \cdot \frac{\sqrt[3]{x}}{\frac{\sqrt[3]{y}}{t_0}}
\end{array}

(FPCore (x y z) :precision binary64 (/ (* x (- y z)) y))
↓
(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (cbrt (- y z))))
   (*
    (/ (* (cbrt x) (cbrt x)) (/ (* (cbrt y) (cbrt y)) (* t_0 t_0)))
    (/ (cbrt x) (/ (cbrt y) t_0)))))
double code(double x, double y, double z) {
	return (x * (y - z)) / y;
}
↓
double code(double x, double y, double z) {
	double t_0 = cbrt(y - z);
	return ((cbrt(x) * cbrt(x)) / ((cbrt(y) * cbrt(y)) / (t_0 * t_0))) * (cbrt(x) / (cbrt(y) / t_0));
}

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

Original12.6
Target3.1
Herbie1.9
\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \]

Derivation

  1. Initial program 12.6

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

  2. Applied associate-/l*_binary642.9

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

  3. Applied add-cube-cbrt_binary644.1

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

  4. Applied add-cube-cbrt_binary643.4

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

  5. Applied times-frac_binary643.4

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

  6. Applied add-cube-cbrt_binary644.3

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

  7. Applied times-frac_binary641.9

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

  8. Final simplification1.9

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

Reproduce

herbie shell --seed 2022081 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

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