0.5 * (1 + (x - 0.5) / 0.5 + (1 / PI) * sin((x - 0.5) / 0.5 * PI))

Percentage Accurate: 85.2% → 85.3%
Time: 7.7s
Alternatives: 10
Speedup: 1.1×

Specification

?
\[0.001 \leq x \land x \leq 1\]
\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 0.5}{0.5}\\ 0.5 \cdot \left(\left(1 + t\_0\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(t\_0 \cdot \mathsf{PI}\left(\right)\right)\right) \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- x 0.5) 0.5)))
   (* 0.5 (+ (+ 1.0 t_0) (* (/ 1.0 (PI)) (sin (* t_0 (PI))))))))
\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{x - 0.5}{0.5}\\
0.5 \cdot \left(\left(1 + t\_0\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(t\_0 \cdot \mathsf{PI}\left(\right)\right)\right)
\end{array}
\end{array}

Sampling outcomes in binary64 precision:

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 10 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.

Initial Program: 85.2% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - 0.5}{0.5}\\ 0.5 \cdot \left(\left(1 + t\_0\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(t\_0 \cdot \mathsf{PI}\left(\right)\right)\right) \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- x 0.5) 0.5)))
   (* 0.5 (+ (+ 1.0 t_0) (* (/ 1.0 (PI)) (sin (* t_0 (PI))))))))
\begin{array}{l}

\\
\begin{array}{l}
t_0 := \frac{x - 0.5}{0.5}\\
0.5 \cdot \left(\left(1 + t\_0\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(t\_0 \cdot \mathsf{PI}\left(\right)\right)\right)
\end{array}
\end{array}

Alternative 1: 85.3% accurate, 0.3× speedup?

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot x\\ \mathsf{fma}\left(\mathsf{fma}\left(\sin t\_0, \cos \mathsf{PI}\left(\right), \cos t\_0 \cdot \left(-\sin \mathsf{PI}\left(\right)\right)\right), \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \end{array} \end{array} \]
(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (PI) 2.0) x)))
   (fma
    (fma (sin t_0) (cos (PI)) (* (cos t_0) (- (sin (PI)))))
    (/ 0.5 (PI))
    (fma (fma x 2.0 -1.0) 0.5 0.5))))
\begin{array}{l}

\\
\begin{array}{l}
t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot x\\
\mathsf{fma}\left(\mathsf{fma}\left(\sin t\_0, \cos \mathsf{PI}\left(\right), \cos t\_0 \cdot \left(-\sin \mathsf{PI}\left(\right)\right)\right), \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right)
\end{array}
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    2. lift-+.f64N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    3. distribute-rgt-inN/A

      \[\leadsto \color{blue}{\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} + \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}} \]

    4. +-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2}} \]

    5. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    6. lift-/.f64N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    7. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    8. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{\left(1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}}{\mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    9. *-lft-identityN/A

      \[\leadsto \frac{\left(1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    10. times-fracN/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)}{1} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

  4. Applied rewrites85.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}{1}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right)} \]
  5. Step-by-step derivation

    1. lift-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}{1}}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    2. /-rgt-identity85.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]

    3. lift-sin.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    4. lift-*.f64N/A

      \[\leadsto \mathsf{fma}\left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    5. lift-fma.f64N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(x \cdot 2 + -1\right)}\right), \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    6. distribute-lft-inN/A

      \[\leadsto \mathsf{fma}\left(\sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(x \cdot 2\right) + \mathsf{PI}\left(\right) \cdot -1\right)}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    7. sin-sumN/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot \left(x \cdot 2\right)\right) \cdot \cos \left(\mathsf{PI}\left(\right) \cdot -1\right) + \cos \left(\mathsf{PI}\left(\right) \cdot \left(x \cdot 2\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot -1\right)}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    8. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\sin \color{blue}{\left(\left(x \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \cos \left(\mathsf{PI}\left(\right) \cdot -1\right) + \cos \left(\mathsf{PI}\left(\right) \cdot \left(x \cdot 2\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot -1\right), \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    9. lower-fma.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sin \left(\left(x \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right), \cos \left(\mathsf{PI}\left(\right) \cdot -1\right), \cos \left(\mathsf{PI}\left(\right) \cdot \left(x \cdot 2\right)\right) \cdot \sin \left(\mathsf{PI}\left(\right) \cdot -1\right)\right)}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

  6. Applied rewrites85.1%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot x\right), \cos \mathsf{PI}\left(\right), \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot x\right) \cdot \left(-\sin \mathsf{PI}\left(\right)\right)\right)}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]
  7. Add Preprocessing

Alternative 2: 85.2% accurate, 0.5× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left({\left({\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, x, -1\right)\right)}^{-1}\right)}^{-1}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (fma
  (pow (pow (sin (* (PI) (fma 2.0 x -1.0))) -1.0) -1.0)
  (/ 0.5 (PI))
  (fma (fma x 2.0 -1.0) 0.5 0.5)))
\begin{array}{l}

\\
\mathsf{fma}\left({\left({\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, x, -1\right)\right)}^{-1}\right)}^{-1}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right)
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    2. lift-+.f64N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    3. distribute-rgt-inN/A

      \[\leadsto \color{blue}{\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} + \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}} \]

    4. +-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2}} \]

    5. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    6. lift-/.f64N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    7. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    8. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{\left(1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}}{\mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    9. *-lft-identityN/A

      \[\leadsto \frac{\left(1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    10. times-fracN/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)}{1} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

  4. Applied rewrites85.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}{1}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right)} \]
  5. Step-by-step derivation

    1. lift-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}{1}}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    2. clear-numN/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{1}{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}}}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    3. lower-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{\frac{1}{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}}}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    4. inv-powN/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}^{-1}}}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    5. lower-pow.f6485.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}^{-1}}}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]

    6. lift-fma.f64N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{{\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(x \cdot 2 + -1\right)}\right)}^{-1}}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    7. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{{\sin \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{2 \cdot x} + -1\right)\right)}^{-1}}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    8. lower-fma.f6485.0

      \[\leadsto \mathsf{fma}\left(\frac{1}{{\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(2, x, -1\right)}\right)}^{-1}}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]

  6. Applied rewrites85.0%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, x, -1\right)\right)}^{-1}}}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]

  7. Final simplification85.0%

    \[\leadsto \mathsf{fma}\left({\left({\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, x, -1\right)\right)}^{-1}\right)}^{-1}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]
  8. Add Preprocessing

Alternative 3: 31.8% accurate, 1.0× speedup?

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + {\mathsf{PI}\left(\right)}^{-1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x, 1.3333333333333333, -2\right)\right) \cdot x\right)\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  0.5
  (+
   (+ 1.0 (/ (- x 0.5) 0.5))
   (*
    (pow (PI) -1.0)
    (* (* (PI) (fma (* (* (* (PI) x) (PI)) x) 1.3333333333333333 -2.0)) x)))))
\begin{array}{l}

\\
0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + {\mathsf{PI}\left(\right)}^{-1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x, 1.3333333333333333, -2\right)\right) \cdot x\right)\right)
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \frac{1}{2} \cdot \left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\sin \left(-1 \cdot \mathsf{PI}\left(\right)\right) + x \cdot \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right) + x \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-4}{3} \cdot \left(x \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)}\right) \]

  5. Applied rewrites31.9%

    \[\leadsto 0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x, 1.3333333333333333, -2\right)\right) \cdot x\right)}\right) \]

  6. Final simplification31.9%

    \[\leadsto 0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + {\mathsf{PI}\left(\right)}^{-1} \cdot \left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x, 1.3333333333333333, -2\right)\right) \cdot x\right)\right) \]
  7. Add Preprocessing

Alternative 4: 85.2% accurate, 1.1× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, x, -1\right)\right), \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (fma
  (sin (* (PI) (fma 2.0 x -1.0)))
  (/ 0.5 (PI))
  (fma (fma x 2.0 -1.0) 0.5 0.5)))
\begin{array}{l}

\\
\mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, x, -1\right)\right), \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right)
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    2. lift-+.f64N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    3. distribute-rgt-inN/A

      \[\leadsto \color{blue}{\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} + \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}} \]

    4. +-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2}} \]

    5. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    6. lift-/.f64N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    7. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    8. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{\left(1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}}{\mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    9. *-lft-identityN/A

      \[\leadsto \frac{\left(1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    10. times-fracN/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)}{1} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

  4. Applied rewrites85.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}{1}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right)} \]
  5. Step-by-step derivation

    1. lift-/.f64N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}{1}}, \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    2. /-rgt-identity85.0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]

    3. lift-fma.f64N/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(x \cdot 2 + -1\right)}\right), \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    4. *-commutativeN/A

      \[\leadsto \mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \left(\color{blue}{2 \cdot x} + -1\right)\right), \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), \frac{1}{2}, \frac{1}{2}\right)\right) \]

    5. lower-fma.f6485.0

      \[\leadsto \mathsf{fma}\left(\sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{fma}\left(2, x, -1\right)}\right), \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]

  6. Applied rewrites85.0%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(2, x, -1\right)\right)}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right) \]
  7. Add Preprocessing

Alternative 5: 85.0% accurate, 1.2× speedup?

\[\begin{array}{l} \\ 0.5 \cdot \mathsf{fma}\left(2, x, \frac{\sin \left(\mathsf{fma}\left(2, x, -1\right) \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (* 0.5 (fma 2.0 x (/ (sin (* (fma 2.0 x -1.0) (PI))) (PI)))))
\begin{array}{l}

\\
0.5 \cdot \mathsf{fma}\left(2, x, \frac{\sin \left(\mathsf{fma}\left(2, x, -1\right) \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}\right)
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around inf

    \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \left(2 + \frac{\sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(x - \frac{1}{2}\right)\right)\right)}{x \cdot \mathsf{PI}\left(\right)}\right)\right)} \]

  5. Applied rewrites84.9%

    \[\leadsto 0.5 \cdot \color{blue}{\mathsf{fma}\left(2, x, \frac{\sin \left(\mathsf{fma}\left(2, x, -1\right) \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}\right)} \]
  6. Add Preprocessing

Alternative 6: 85.0% accurate, 1.2× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{0.5}{\mathsf{PI}\left(\right)}, \sin \left(\mathsf{fma}\left(2, x, -1\right) \cdot \mathsf{PI}\left(\right)\right), x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (fma (/ 0.5 (PI)) (sin (* (fma 2.0 x -1.0) (PI))) x))
\begin{array}{l}

\\
\mathsf{fma}\left(\frac{0.5}{\mathsf{PI}\left(\right)}, \sin \left(\mathsf{fma}\left(2, x, -1\right) \cdot \mathsf{PI}\left(\right)\right), x\right)
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around inf

    \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{2} \cdot \frac{\sin \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot \left(x - \frac{1}{2}\right)\right)\right)}{x \cdot \mathsf{PI}\left(\right)}\right)} \]

  5. Applied rewrites84.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{\mathsf{PI}\left(\right)}, \sin \left(\mathsf{fma}\left(2, x, -1\right) \cdot \mathsf{PI}\left(\right)\right), x\right)} \]
  6. Add Preprocessing

Alternative 7: 31.8% accurate, 3.1× speedup?

\[\begin{array}{l} \\ 0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x, 1.3333333333333333, -2\right) \cdot x\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (*
  0.5
  (+
   (+ 1.0 (/ (- x 0.5) 0.5))
   (* (fma (* (* (* (PI) x) (PI)) x) 1.3333333333333333 -2.0) x))))
\begin{array}{l}

\\
0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x, 1.3333333333333333, -2\right) \cdot x\right)
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \frac{1}{2} \cdot \left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \color{blue}{\left(x \cdot \left(2 \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right) + x \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-4}{3} \cdot \left(x \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right) + \frac{\sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}\right)}\right) \]

  5. Applied rewrites31.9%

    \[\leadsto 0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \color{blue}{\mathsf{fma}\left(\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right)\right) \cdot x, 1.3333333333333333, -2\right) \cdot x}\right) \]
  6. Add Preprocessing

Alternative 8: 31.8% accurate, 5.8× speedup?

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right), 0.6666666666666666, 0\right), 0\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (fma (* x x) (fma (* (* (PI) x) (PI)) 0.6666666666666666 0.0) 0.0))
\begin{array}{l}

\\
\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right), 0.6666666666666666, 0\right), 0\right)
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    2. lift-+.f64N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \]

    3. distribute-rgt-inN/A

      \[\leadsto \color{blue}{\left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} + \left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}} \]

    4. +-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2}} \]

    5. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    6. lift-/.f64N/A

      \[\leadsto \left(\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    7. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}} \cdot \frac{1}{2} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    8. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{\left(1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}}{\mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    9. *-lft-identityN/A

      \[\leadsto \frac{\left(1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \frac{1}{2}}{\color{blue}{1 \cdot \mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

    10. times-fracN/A

      \[\leadsto \color{blue}{\frac{1 \cdot \sin \left(\frac{x - \frac{1}{2}}{\frac{1}{2}} \cdot \mathsf{PI}\left(\right)\right)}{1} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}} + \left(1 + \frac{x - \frac{1}{2}}{\frac{1}{2}}\right) \cdot \frac{1}{2} \]

  4. Applied rewrites85.0%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\sin \left(\mathsf{PI}\left(\right) \cdot \mathsf{fma}\left(x, 2, -1\right)\right)}{1}, \frac{0.5}{\mathsf{PI}\left(\right)}, \mathsf{fma}\left(\mathsf{fma}\left(x, 2, -1\right), 0.5, 0.5\right)\right)} \]

  5. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)} + x \cdot \left(1 + \left(\cos \left(-1 \cdot \mathsf{PI}\left(\right)\right) + x \cdot \left(-1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-2}{3} \cdot \left(x \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right)} \]

  7. Applied rewrites31.9%

    \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right), 0.6666666666666666, 0 \cdot \mathsf{PI}\left(\right)\right), \frac{0}{\mathsf{PI}\left(\right)}\right)} \]

  8. Final simplification31.9%

    \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot x\right) \cdot \mathsf{PI}\left(\right), 0.6666666666666666, 0\right), 0\right) \]
  9. Add Preprocessing

Alternative 9: 31.8% accurate, 6.2× speedup?

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(\left(\left(x \cdot 0.6666666666666666\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right) \end{array} \]
(FPCore (x)
 :precision binary64
 (* (* x x) (* (* (* x 0.6666666666666666) (PI)) (PI))))
\begin{array}{l}

\\
\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot 0.6666666666666666\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)} + x \cdot \left(\frac{1}{2} \cdot \left(2 + 2 \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right) + x \cdot \left(-1 \cdot \left(\mathsf{PI}\left(\right) \cdot \sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{-2}{3} \cdot \left(x \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)} \]

  5. Applied rewrites31.9%

    \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(\left(\left(x \cdot 0.6666666666666666\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right)\right)} \]
  6. Add Preprocessing

Alternative 10: 3.2% accurate, 161.0× speedup?

\[\begin{array}{l} \\ 0 \end{array} \]
(FPCore (x) :precision binary64 0.0)
double code(double x) {
	return 0.0;
}

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

public static double code(double x) {
	return 0.0;
}

def code(x):
	return 0.0

function code(x)
	return 0.0
end

function tmp = code(x)
	tmp = 0.0;
end

code[x_] := 0.0

\begin{array}{l}

\\
0
\end{array}
Derivation

  1. Initial program 85.0%

    \[0.5 \cdot \left(\left(1 + \frac{x - 0.5}{0.5}\right) + \frac{1}{\mathsf{PI}\left(\right)} \cdot \sin \left(\frac{x - 0.5}{0.5} \cdot \mathsf{PI}\left(\right)\right)\right) \]
  2. Add Preprocessing

  3. Taylor expanded in x around 0

    \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{\sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}} \]
  4. Step-by-step derivation

    1. associate-*r/N/A

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \sin \left(-1 \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{PI}\left(\right)}} \]

    2. *-commutativeN/A

      \[\leadsto \frac{\color{blue}{\sin \left(-1 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{2}}}{\mathsf{PI}\left(\right)} \]

    3. associate-/l*N/A

      \[\leadsto \color{blue}{\sin \left(-1 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)}} \]

    4. mul-1-negN/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{neg}\left(\mathsf{PI}\left(\right)\right)\right)} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)} \]

    5. sin-negN/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin \mathsf{PI}\left(\right)\right)\right)} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)} \]

    6. sin-PIN/A

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{0}\right)\right) \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)} \]

    7. metadata-evalN/A

      \[\leadsto \color{blue}{0} \cdot \frac{\frac{1}{2}}{\mathsf{PI}\left(\right)} \]

    8. mul0-lft3.2

      \[\leadsto \color{blue}{0} \]

  5. Applied rewrites3.2%

    \[\leadsto \color{blue}{0} \]
  6. Add Preprocessing

Reproduce

?
herbie shell --seed 1 
(FPCore (x)
  :name "0.5 * (1 + (x - 0.5) / 0.5 + (1 / PI) * sin((x - 0.5) / 0.5 * PI))"
  :precision binary64
  :pre (and (<= 0.001 x) (<= x 1.0))
  (* 0.5 (+ (+ 1.0 (/ (- x 0.5) 0.5)) (* (/ 1.0 (PI)) (sin (* (/ (- x 0.5) 0.5) (PI)))))))