Average Error: 0.1 → 0.1
Time: 2.3s
Precision: binary64
\[[x, y]=\mathsf{sort}([x, y])\]
\[\left(x \cdot 3\right) \cdot y - z\]
↓
\[x \cdot \left(y \cdot 3\right) - z\]
\left(x \cdot 3\right) \cdot y - z
↓
x \cdot \left(y \cdot 3\right) - z
(FPCore (x y z) :precision binary64 (- (* (* x 3.0) y) z))
↓
(FPCore (x y z) :precision binary64 (- (* x (* y 3.0)) z))
double code(double x, double y, double z) {
	return ((x * 3.0) * y) - z;
}
↓
double code(double x, double y, double z) {
	return (x * (y * 3.0)) - z;
}

Error

Bits error versus x

Bits error versus y

Bits error versus z

Try it out

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Results

Enter valid numbers for all inputs

Target

Original0.1
Target0.1
Herbie0.1
\[x \cdot \left(3 \cdot y\right) - z\]

Derivation

  1. Initial program 0.1

    \[\left(x \cdot 3\right) \cdot y - z\]
  2. Using strategy rm

  3. Applied pow1_binary640.1

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

  4. Applied pow1_binary640.1

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

  5. Applied pow1_binary640.1

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

  6. Applied pow-prod-down_binary640.1

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

  7. Applied pow-prod-down_binary640.1

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

  8. Simplified0.1

    \[\leadsto {\color{blue}{\left(3 \cdot \left(x \cdot y\right)\right)}}^{1} - z\]
  9. Using strategy rm

  10. Applied add-cube-cbrt_binary640.1

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

  11. Applied associate-*l*_binary640.2

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

  12. Simplified0.2

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

  13. Taylor expanded around 0 0.4

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

  14. Simplified0.1

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

  15. Final simplification0.1

    \[\leadsto x \cdot \left(y \cdot 3\right) - z\]

Alternatives

Reproduce

herbie shell --seed 2021118 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (* x (* 3.0 y)) z)

  (- (* (* x 3.0) y) z))