Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Percentage Accurate: 99.7% → 99.8%
Time: 4.8s
Alternatives: 6
Speedup: TODO×

Specification

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

Local Percentage Accuracy vs ?

The average percentage accuracy by input value. Horizontal axis shows value of an input variable; the variable is choosen in the title. Vertical axis is accuracy; higher is better. Red represent the original program, while blue represents Herbie's suggestion. These can be toggled with buttons below the plot. The line is an average while dots represent individual samples.

Accuracy vs Speed?

Herbie found 6 alternatives:

AlternativeAccuracySpeedup
The accuracy (vertical axis) and speed (horizontal axis) of each alternatives. Up and to the right is better. The red square shows the initial program, and each blue circle shows an alternative.The line shows the best available speed-accuracy tradeoffs.

Target

Original99.7%
Target99.7%
Herbie99.8%
\[6 \cdot x - 9 \cdot \left(x \cdot x\right) \]

Alternative 1?

\[\mathsf{fma}\left(x, 6, x \cdot \left(x \cdot -9\right)\right) \]
Derivation

  1. Initial program 99.7%

    \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
  2. Step-by-step derivation

    1. *-commutative99.7%

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

    2. sub-neg99.7%

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

    3. distribute-rgt-in99.7%

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

    4. metadata-eval99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(-x \cdot 3\right) \cdot 3\right) \]

    5. distribute-rgt-neg-in99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

    6. associate-*l*99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

    7. metadata-eval99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

    8. metadata-eval99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

  3. Simplified99.7%

    \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
  4. Step-by-step derivation

    1. distribute-lft-in99.7%

      \[\leadsto \color{blue}{x \cdot 6 + x \cdot \left(x \cdot -9\right)} \]

    2. fma-def99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, x \cdot \left(x \cdot -9\right)\right)} \]

  5. Applied egg-rr99.8%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, x \cdot \left(x \cdot -9\right)\right)} \]

  6. Final simplification99.8%

    \[\leadsto \mathsf{fma}\left(x, 6, x \cdot \left(x \cdot -9\right)\right) \]

Alternative 2?

\[\begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]
Derivation
  1. Split input into 2 regimes

  2. if x < -0.660000000000000031 or 0.650000000000000022 < x

    1. Initial program 99.6%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
    2. Step-by-step derivation

      1. associate-*l*99.7%

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

      2. *-commutative99.7%

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

    3. Simplified99.7%

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

    4. Taylor expanded in x around inf 96.0%

      \[\leadsto \color{blue}{-9 \cdot {x}^{2}} \]
    5. Step-by-step derivation

      1. unpow296.0%

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

    6. Simplified96.0%

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

    if -0.660000000000000031 < x < 0.650000000000000022

    1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
    2. Step-by-step derivation

      1. *-commutative99.7%

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

      2. sub-neg99.7%

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

      3. distribute-rgt-in99.7%

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

      4. metadata-eval99.7%

        \[\leadsto x \cdot \left(\color{blue}{6} + \left(-x \cdot 3\right) \cdot 3\right) \]

      5. distribute-rgt-neg-in99.7%

        \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      6. associate-*l*99.7%

        \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      7. metadata-eval99.7%

        \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      8. metadata-eval99.7%

        \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

    3. Simplified99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

    4. Taylor expanded in x around 0 96.6%

      \[\leadsto x \cdot \color{blue}{6} \]
  3. Recombined 2 regimes into one program.

  4. Final simplification96.3%

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.66 \lor \neg \left(x \leq 0.65\right):\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]

Alternative 3?

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

  1. Initial program 99.7%

    \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
  2. Step-by-step derivation

    1. associate-*l*99.7%

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

    2. *-commutative99.7%

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

  3. Simplified99.7%

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

  4. Final simplification99.7%

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

Alternative 4?

\[x \cdot \left(6 + x \cdot -9\right) \]
Derivation

  1. Initial program 99.7%

    \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
  2. Step-by-step derivation

    1. *-commutative99.7%

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

    2. sub-neg99.7%

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

    3. distribute-rgt-in99.7%

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

    4. metadata-eval99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(-x \cdot 3\right) \cdot 3\right) \]

    5. distribute-rgt-neg-in99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

    6. associate-*l*99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

    7. metadata-eval99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

    8. metadata-eval99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

  3. Simplified99.7%

    \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

  4. Final simplification99.7%

    \[\leadsto x \cdot \left(6 + x \cdot -9\right) \]

Alternative 5?

\[x \cdot 6 \]
Derivation

  1. Initial program 99.7%

    \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
  2. Step-by-step derivation

    1. *-commutative99.7%

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

    2. sub-neg99.7%

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

    3. distribute-rgt-in99.7%

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

    4. metadata-eval99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(-x \cdot 3\right) \cdot 3\right) \]

    5. distribute-rgt-neg-in99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

    6. associate-*l*99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

    7. metadata-eval99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

    8. metadata-eval99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

  3. Simplified99.7%

    \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

  4. Taylor expanded in x around 0 46.1%

    \[\leadsto x \cdot \color{blue}{6} \]

  5. Final simplification46.1%

    \[\leadsto x \cdot 6 \]

Alternative 6?

\[4 \]
Derivation

  1. Initial program 99.7%

    \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
  2. Step-by-step derivation

    1. *-commutative99.7%

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

    2. sub-neg99.7%

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

    3. distribute-rgt-in99.7%

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

    4. metadata-eval99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(-x \cdot 3\right) \cdot 3\right) \]

    5. distribute-rgt-neg-in99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

    6. associate-*l*99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

    7. metadata-eval99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

    8. metadata-eval99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

  3. Simplified99.7%

    \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
  4. Step-by-step derivation

    1. *-commutative99.7%

      \[\leadsto \color{blue}{\left(6 + x \cdot -9\right) \cdot x} \]

    2. flip-+99.7%

      \[\leadsto \color{blue}{\frac{6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}{6 - x \cdot -9}} \cdot x \]

    3. associate-*l/91.3%

      \[\leadsto \color{blue}{\frac{\left(6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9}} \]

    4. metadata-eval91.3%

      \[\leadsto \frac{\left(\color{blue}{36} - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right) \cdot x}{6 - x \cdot -9} \]

    5. pow291.3%

      \[\leadsto \frac{\left(36 - \color{blue}{{\left(x \cdot -9\right)}^{2}}\right) \cdot x}{6 - x \cdot -9} \]

  5. Applied egg-rr91.3%

    \[\leadsto \color{blue}{\frac{\left(36 - {\left(x \cdot -9\right)}^{2}\right) \cdot x}{6 - x \cdot -9}} \]

  6. Taylor expanded in x around 0 45.2%

    \[\leadsto \frac{\color{blue}{36 \cdot x}}{6 - x \cdot -9} \]
  7. Step-by-step derivation

    1. *-commutative45.2%

      \[\leadsto \frac{\color{blue}{x \cdot 36}}{6 - x \cdot -9} \]

  8. Simplified45.2%

    \[\leadsto \frac{\color{blue}{x \cdot 36}}{6 - x \cdot -9} \]

  9. Taylor expanded in x around inf 2.2%

    \[\leadsto \color{blue}{4} \]

  10. Final simplification2.2%

    \[\leadsto 4 \]

Reproduce

?
herbie shell --seed 2023166 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6.0 x) (* 9.0 (* x x)))

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