Average Error: 1.5 → 0.5
Time: 2.3min
Precision: binary64
Cost: 2176
\[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
↓
\[\left|\frac{x + 4}{y} - \left(\left|\sqrt[3]{x}\right| \cdot \sqrt{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right|\]
\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|
↓
\left|\frac{x + 4}{y} - \left(\left|\sqrt[3]{x}\right| \cdot \sqrt{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right|
(FPCore (x y z) :precision binary64 (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))
↓
(FPCore (x y z)
 :precision binary64
 (fabs
  (-
   (/ (+ x 4.0) y)
   (*
    (* (fabs (cbrt x)) (sqrt (/ 1.0 (* (cbrt y) (cbrt y)))))
    (* (* z (/ (cbrt x) (cbrt y))) (fabs (/ (cbrt x) (cbrt y))))))))
double code(double x, double y, double z) {
	return fabs(((x + 4.0) / y) - ((x / y) * z));
}
↓
double code(double x, double y, double z) {
	return fabs(((x + 4.0) / y) - ((fabs(cbrt(x)) * sqrt(1.0 / (cbrt(y) * cbrt(y)))) * ((z * (cbrt(x) / cbrt(y))) * fabs(cbrt(x) / cbrt(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
Alternative 1
Accuracy0.6
Cost2752
\[\left|\frac{x + 4}{y} - \left(\left|\sqrt[3]{\sqrt[3]{x}} \cdot \left(\sqrt[3]{\sqrt[3]{x}} \cdot \sqrt[3]{\sqrt[3]{x}}\right)\right| \cdot \sqrt{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right|\]
Alternative 2
Accuracy0.6
Cost2752
\[\left|\frac{x + 4}{y} - \left(\left|\sqrt[3]{x}\right| \cdot \sqrt{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{\sqrt[3]{y}} \cdot \left(\sqrt[3]{\sqrt[3]{y}} \cdot \sqrt[3]{\sqrt[3]{y}}\right)}\right|\right)\right|\]
Alternative 3
Accuracy0.7
Cost2432
\[\left|\frac{x + 4}{y} - \sqrt{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{-x} \cdot \sqrt[3]{-1}}{\sqrt[3]{y}}\right|\right)\right|\]
Alternative 4
Accuracy1.2
Cost1920
\[\left|\frac{x + 4}{y} - e^{\log \left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)} \cdot \left(\left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right| \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)\right)\right|\]

Derivation

  1. Initial program 1.5

    \[\left|\frac{x + 4}{y} - \frac{x}{y} \cdot z\right|\]
  2. Using strategy rm

  3. Applied add-cube-cbrt_binary64_4541.8

    \[\leadsto \left|\frac{x + 4}{y} - \frac{x}{\color{blue}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}}} \cdot z\right|\]

  4. Applied add-cube-cbrt_binary64_4541.9

    \[\leadsto \left|\frac{x + 4}{y} - \frac{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}}}{\left(\sqrt[3]{y} \cdot \sqrt[3]{y}\right) \cdot \sqrt[3]{y}} \cdot z\right|\]

  5. Applied times-frac_binary64_4251.9

    \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} \cdot z\right|\]

  6. Applied associate-*l*_binary64_3600.7

    \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot z\right)}\right|\]

  7. Simplified0.7

    \[\leadsto \left|\frac{x + 4}{y} - \frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}} \cdot \color{blue}{\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)}\right|\]
  8. Using strategy rm

  9. Applied add-sqr-sqrt_binary64_4410.7

    \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(\sqrt{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \sqrt{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right)} \cdot \left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right|\]

  10. Applied associate-*l*_binary64_3600.7

    \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\sqrt{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \left(\sqrt{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)\right)}\right|\]

  11. Simplified0.7

    \[\leadsto \left|\frac{x + 4}{y} - \sqrt{\frac{\sqrt[3]{x} \cdot \sqrt[3]{x}}{\sqrt[3]{y} \cdot \sqrt[3]{y}}} \cdot \color{blue}{\left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)}\right|\]
  12. Using strategy rm

  13. Applied div-inv_binary64_4160.7

    \[\leadsto \left|\frac{x + 4}{y} - \sqrt{\color{blue}{\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}} \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right|\]

  14. Applied sqrt-prod_binary64_4350.5

    \[\leadsto \left|\frac{x + 4}{y} - \color{blue}{\left(\sqrt{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \sqrt{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right)} \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right|\]

  15. Simplified0.5

    \[\leadsto \left|\frac{x + 4}{y} - \left(\color{blue}{\left|\sqrt[3]{x}\right|} \cdot \sqrt{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right|\]
  16. Using strategy rm

  17. Applied pow1_binary64_4800.5

    \[\leadsto \left|\frac{x + 4}{y} - \left(\left|\sqrt[3]{x}\right| \cdot \sqrt{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\color{blue}{{\left(\sqrt[3]{y}\right)}^{1}}}\right|\right)\right|\]

  18. Final simplification0.5

    \[\leadsto \left|\frac{x + 4}{y} - \left(\left|\sqrt[3]{x}\right| \cdot \sqrt{\frac{1}{\sqrt[3]{y} \cdot \sqrt[3]{y}}}\right) \cdot \left(\left(z \cdot \frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot \left|\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right|\right)\right|\]

Reproduce

herbie shell --seed 2020322 
(FPCore (x y z)
  :name "fabs fraction 1"
  :precision binary64
  (fabs (- (/ (+ x 4.0) y) (* (/ x y) z))))