Average Error: 7.8 → 2.6
Time: 3.4s
Precision: binary64
\[\frac{x + y}{1 - \frac{y}{z}}\]
↓
\[\frac{1}{\frac{1}{x + y} - \frac{y}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\frac{1}{z}}{\sqrt[3]{x + y}}}\]
\frac{x + y}{1 - \frac{y}{z}}
↓
\frac{1}{\frac{1}{x + y} - \frac{y}{\sqrt[3]{x + y} \cdot \sqrt[3]{x + y}} \cdot \frac{\frac{1}{z}}{\sqrt[3]{x + y}}}
double code(double x, double y, double z) {
	return ((double) (((double) (x + y)) / ((double) (1.0 - ((double) (y / z))))));
}
↓
double code(double x, double y, double z) {
	return ((double) (1.0 / ((double) (((double) (1.0 / ((double) (x + y)))) - ((double) (((double) (y / ((double) (((double) cbrt(((double) (x + y)))) * ((double) cbrt(((double) (x + y)))))))) * ((double) (((double) (1.0 / z)) / ((double) cbrt(((double) (x + y))))))))))));
}

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

Original7.8
Target4.3
Herbie2.6
\[\begin{array}{l} \mathbf{if}\;y \lt -3.74293107626898565 \cdot 10^{171}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \mathbf{elif}\;y \lt 3.55346624560867344 \cdot 10^{168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{-y} \cdot z\\ \end{array}\]

Derivation

  1. Initial program 7.8

    \[\frac{x + y}{1 - \frac{y}{z}}\]
  2. Using strategy rm

  3. Applied clear-num8.0

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

  5. Applied div-sub8.0

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

  7. Applied add-cube-cbrt8.4

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

  8. Applied div-inv8.4

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

  9. Applied times-frac2.6

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

  10. Final simplification2.6

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

Reproduce

herbie shell --seed 2020175 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :herbie-target
  (if (< y -3.7429310762689856e+171) (* (/ (+ y x) (neg y)) z) (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) (* (/ (+ y x) (neg y)) z)))

  (/ (+ x y) (- 1.0 (/ y z))))