Average Error: 0.1 → 0.1
Time: 1.9s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[{x}^{2} \cdot 9 + \left(3 - 12 \cdot x\right)\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
{x}^{2} \cdot 9 + \left(3 - 12 \cdot x\right)
double code(double x) {
	return ((double) (3.0 * ((double) (((double) (((double) (((double) (x * 3.0)) * x)) - ((double) (x * 4.0)))) + 1.0))));
}
↓
double code(double x) {
	return ((double) (((double) (((double) pow(x, 2.0)) * 9.0)) + ((double) (3.0 - ((double) (12.0 * x))))));
}

Error

Bits error versus x

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Results

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Target

Original0.1
Target0.1
Herbie0.1
\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)\]

Derivation

  1. Initial program 0.1

    \[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot 3 - 4, 1\right) \cdot 3}\]

  3. Taylor expanded around 0 0.1

    \[\leadsto \color{blue}{\left(9 \cdot {x}^{2} + 3\right) - 12 \cdot x}\]

  4. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, 9, 3 - 12 \cdot x\right)}\]
  5. Using strategy rm

  6. Applied fma-udef0.1

    \[\leadsto \color{blue}{{x}^{2} \cdot 9 + \left(3 - 12 \cdot x\right)}\]

  7. Final simplification0.1

    \[\leadsto {x}^{2} \cdot 9 + \left(3 - 12 \cdot x\right)\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"
  :precision binary64

  :herbie-target
  (+ 3 (- (* (* 9 x) x) (* 12 x)))

  (* 3 (+ (- (* (* x 3) x) (* x 4)) 1)))