Average Error: 0.1 → 0.1
Time: 2.0s
Precision: binary64
\[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]
↓
\[x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right) \]
\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right)
↓
x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)

(FPCore (x) :precision binary64 (* (* x x) (- 3.0 (* x 2.0))))
↓
(FPCore (x) :precision binary64 (* x (* x (fma x -2.0 3.0))))
double code(double x) {
	return (x * x) * (3.0 - (x * 2.0));
}
↓
double code(double x) {
	return x * (x * fma(x, -2.0, 3.0));
}

Error

Bits error versus x

Target

Original0.1
Target0.1
Herbie0.1
\[x \cdot \left(x \cdot \left(3 - x \cdot 2\right)\right) \]

Derivation

  1. Initial program 0.1

    \[\left(x \cdot x\right) \cdot \left(3 - x \cdot 2\right) \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, -2, 3\right)} \]

  3. Applied add-cube-cbrt_binary640.6

    \[\leadsto \color{blue}{\left(\sqrt[3]{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, -2, 3\right)} \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, -2, 3\right)}\right) \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, -2, 3\right)}} \]

  4. Simplified0.6

    \[\leadsto \color{blue}{\left(\sqrt[3]{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)} \cdot \sqrt[3]{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)}\right)} \cdot \sqrt[3]{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, -2, 3\right)} \]

  5. Simplified0.6

    \[\leadsto \left(\sqrt[3]{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)} \cdot \sqrt[3]{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)}\right) \cdot \color{blue}{\sqrt[3]{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)}} \]

  6. Applied pow1/3_binary6417.5

    \[\leadsto \left(\sqrt[3]{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)} \cdot \sqrt[3]{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)}\right) \cdot \color{blue}{{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{0.3333333333333333}} \]

  7. Applied pow1/3_binary647.5

    \[\leadsto \left(\sqrt[3]{x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)} \cdot \color{blue}{{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{0.3333333333333333}}\right) \cdot {\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{0.3333333333333333} \]

  8. Applied pow1/3_binary6418.0

    \[\leadsto \left(\color{blue}{{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{0.3333333333333333}} \cdot {\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{0.3333333333333333}\right) \cdot {\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{0.3333333333333333} \]

  9. Applied pow-sqr_binary6418.0

    \[\leadsto \color{blue}{{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{\left(2 \cdot 0.3333333333333333\right)}} \cdot {\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{0.3333333333333333} \]

  10. Applied pow-prod-up_binary640.1

    \[\leadsto \color{blue}{{\left(x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right)\right)}^{\left(2 \cdot 0.3333333333333333 + 0.3333333333333333\right)}} \]

  11. Final simplification0.1

    \[\leadsto x \cdot \left(x \cdot \mathsf{fma}\left(x, -2, 3\right)\right) \]

Reproduce

herbie shell --seed 2022088 
(FPCore (x)
  :name "Data.Spline.Key:interpolateKeys from smoothie-0.4.0.2"
  :precision binary64

  :herbie-target
  (* x (* x (- 3.0 (* x 2.0))))

  (* (* x x) (- 3.0 (* x 2.0))))