Average Error: 0.2 → 0.1
Time: 12.9s
Precision: 64
\[3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)\]
↓
\[3 + \left(\left(-12\right) \cdot x + \left(9 \cdot x\right) \cdot x\right)\]
3 \cdot \left(\left(\left(x \cdot 3\right) \cdot x - x \cdot 4\right) + 1\right)
↓
3 + \left(\left(-12\right) \cdot x + \left(9 \cdot x\right) \cdot x\right)
double f(double x) {
        double r31622320 = 3.0;
        double r31622321 = x;
        double r31622322 = r31622321 * r31622320;
        double r31622323 = r31622322 * r31622321;
        double r31622324 = 4.0;
        double r31622325 = r31622321 * r31622324;
        double r31622326 = r31622323 - r31622325;
        double r31622327 = 1.0;
        double r31622328 = r31622326 + r31622327;
        double r31622329 = r31622320 * r31622328;
        return r31622329;
}
↓
double f(double x) {
        double r31622330 = 3.0;
        double r31622331 = 12.0;
        double r31622332 = -r31622331;
        double r31622333 = x;
        double r31622334 = r31622332 * r31622333;
        double r31622335 = 9.0;
        double r31622336 = r31622335 * r31622333;
        double r31622337 = r31622336 * r31622333;
        double r31622338 = r31622334 + r31622337;
        double r31622339 = r31622330 + r31622338;
        return r31622339;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original0.2
Target0.1
Herbie0.1
\[3 + \left(\left(9 \cdot x\right) \cdot x - 12 \cdot x\right)\]

Derivation

  1. Initial program 0.2

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

  2. Simplified0.2

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

  3. Taylor expanded around 0 0.1

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

  4. Simplified0.1

    \[\leadsto \color{blue}{3 + x \cdot \left(x \cdot 9 - 12\right)}\]
  5. Using strategy rm

  6. Applied sub-neg0.1

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

  7. Applied distribute-rgt-in0.1

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

  8. Final simplification0.1

    \[\leadsto 3 + \left(\left(-12\right) \cdot x + \left(9 \cdot x\right) \cdot x\right)\]

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, D"

  :herbie-target
  (+ 3.0 (- (* (* 9.0 x) x) (* 12.0 x)))

  (* 3.0 (+ (- (* (* x 3.0) x) (* x 4.0)) 1.0)))