4*pow(10,8) / (p * (T * 287.058)) * PI * pow(75*pow(10,-9) ,5) * exp(0/2)*pow(log(1.5),2)

Percentage Accurate: 88.3% → 99.6%
Time: 7.0s
Alternatives: 5
Speedup: 2.9×

Specification

?
\[\left(0 \leq p \land p \leq 1000000000\right) \land \left(0 \leq T \land T \leq 1000\right)\]
\[\begin{array}{l} \\ \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2} \end{array} \]
(FPCore (p T)
 :precision binary64
 (*
  (*
   (*
    (* (/ (* 4.0 (pow 10.0 8.0)) (* p (* T 287.058))) (PI))
    (pow (* 75.0 (pow 10.0 (- 9.0))) 5.0))
   (exp (/ 0.0 2.0)))
  (pow (log 1.5) 2.0)))
\begin{array}{l}

\\
\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2}
\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 5 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: 88.3% accurate, 1.0× speedup?

\[\begin{array}{l} \\ \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2} \end{array} \]
(FPCore (p T)
 :precision binary64
 (*
  (*
   (*
    (* (/ (* 4.0 (pow 10.0 8.0)) (* p (* T 287.058))) (PI))
    (pow (* 75.0 (pow 10.0 (- 9.0))) 5.0))
   (exp (/ 0.0 2.0)))
  (pow (log 1.5) 2.0)))
\begin{array}{l}

\\
\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2}
\end{array}

Alternative 1: 99.6% accurate, 2.9× speedup?

\[\begin{array}{l} [p, T] = \mathsf{sort}([p, T])\\ \\ \frac{\frac{\mathsf{PI}\left(\right)}{p} \cdot \left(3.306714148360262 \cdot 10^{-30} \cdot {\log 1.5}^{2}\right)}{T} \end{array} \]
NOTE: p and T should be sorted in increasing order before calling this function.
(FPCore (p T)
 :precision binary64
 (/ (* (/ (PI) p) (* 3.306714148360262e-30 (pow (log 1.5) 2.0))) T))
\begin{array}{l}
[p, T] = \mathsf{sort}([p, T])\\
\\
\frac{\frac{\mathsf{PI}\left(\right)}{p} \cdot \left(3.306714148360262 \cdot 10^{-30} \cdot {\log 1.5}^{2}\right)}{T}
\end{array}
Derivation

  1. Initial program 89.9%

    \[\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    2. lift-/.f64N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\color{blue}{\frac{0}{2}}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    3. metadata-evalN/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\color{blue}{0}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    4. lift-exp.f64N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{e^{0}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    5. exp-0N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{1}\right) \cdot {\log \frac{3}{2}}^{2} \]

    6. *-rgt-identity89.9

      \[\leadsto \color{blue}{\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)} \cdot {\log 1.5}^{2} \]

    7. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    8. lift-*.f64N/A

      \[\leadsto \left(\color{blue}{\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right)} \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot {\log \frac{3}{2}}^{2} \]

    9. associate-*l*N/A

      \[\leadsto \color{blue}{\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)\right)} \cdot {\log \frac{3}{2}}^{2} \]

    10. *-commutativeN/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    11. lift-/.f64N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    12. clear-numN/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{\frac{1}{\frac{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}{4 \cdot {10}^{8}}}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    13. div-invN/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}}{\frac{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}{4 \cdot {10}^{8}}}} \cdot {\log \frac{3}{2}}^{2} \]

  4. Applied rewrites99.4%

    \[\leadsto \color{blue}{\frac{\frac{2.373046875 \cdot 10^{-36} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot 7.17645 \cdot 10^{-7}}} \cdot {\log 1.5}^{2} \]
  5. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2}} \]

    2. lift-/.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot \frac{315623608846123}{439804651110400000000}}} \cdot {\log \frac{3}{2}}^{2} \]

    3. lift-/.f64N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2} \]

    4. lift-*.f64N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}}{p}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2} \]

    5. associate-/l*N/A

      \[\leadsto \frac{\color{blue}{\frac{243}{102400000000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{p}}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2} \]

    6. lift-*.f64N/A

      \[\leadsto \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{p}}{\color{blue}{T \cdot \frac{315623608846123}{439804651110400000000}}} \cdot {\log \frac{3}{2}}^{2} \]

    7. *-commutativeN/A

      \[\leadsto \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{p}}{\color{blue}{\frac{315623608846123}{439804651110400000000} \cdot T}} \cdot {\log \frac{3}{2}}^{2} \]

    8. times-fracN/A

      \[\leadsto \color{blue}{\left(\frac{\frac{243}{102400000000000000000000000000000000000}}{\frac{315623608846123}{439804651110400000000}} \cdot \frac{\frac{\mathsf{PI}\left(\right)}{p}}{T}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    9. metadata-evalN/A

      \[\leadsto \left(\color{blue}{\frac{7776}{2351579135999999940395355224609375}} \cdot \frac{\frac{\mathsf{PI}\left(\right)}{p}}{T}\right) \cdot {\log \frac{3}{2}}^{2} \]

    10. associate-/l/N/A

      \[\leadsto \left(\frac{7776}{2351579135999999940395355224609375} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{T \cdot p}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    11. lift-*.f64N/A

      \[\leadsto \left(\frac{7776}{2351579135999999940395355224609375} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{T \cdot p}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    12. lift-/.f64N/A

      \[\leadsto \left(\frac{7776}{2351579135999999940395355224609375} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{T \cdot p}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    13. *-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{T \cdot p} \cdot \frac{7776}{2351579135999999940395355224609375}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    14. associate-*l*N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{T \cdot p} \cdot \left(\frac{7776}{2351579135999999940395355224609375} \cdot {\log \frac{3}{2}}^{2}\right)} \]

    15. lift-/.f64N/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right)}{T \cdot p}} \cdot \left(\frac{7776}{2351579135999999940395355224609375} \cdot {\log \frac{3}{2}}^{2}\right) \]

    16. lift-*.f64N/A

      \[\leadsto \frac{\mathsf{PI}\left(\right)}{\color{blue}{T \cdot p}} \cdot \left(\frac{7776}{2351579135999999940395355224609375} \cdot {\log \frac{3}{2}}^{2}\right) \]

    17. associate-/l/N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{p}}{T}} \cdot \left(\frac{7776}{2351579135999999940395355224609375} \cdot {\log \frac{3}{2}}^{2}\right) \]

    18. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{p} \cdot \left(\frac{7776}{2351579135999999940395355224609375} \cdot {\log \frac{3}{2}}^{2}\right)}{T}} \]

    19. lower-/.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{p} \cdot \left(\frac{7776}{2351579135999999940395355224609375} \cdot {\log \frac{3}{2}}^{2}\right)}{T}} \]

  6. Applied rewrites99.6%

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{PI}\left(\right)}{p} \cdot \left(3.306714148360262 \cdot 10^{-30} \cdot {\log 1.5}^{2}\right)}{T}} \]
  7. Add Preprocessing

Alternative 2: 94.8% accurate, 2.9× speedup?

\[\begin{array}{l} [p, T] = \mathsf{sort}([p, T])\\ \\ \frac{{\log 1.5}^{2} \cdot \left(9.4921875 \cdot 10^{-28} \cdot \mathsf{PI}\left(\right)\right)}{\left(287.058 \cdot T\right) \cdot p} \end{array} \]
NOTE: p and T should be sorted in increasing order before calling this function.
(FPCore (p T)
 :precision binary64
 (/ (* (pow (log 1.5) 2.0) (* 9.4921875e-28 (PI))) (* (* 287.058 T) p)))
\begin{array}{l}
[p, T] = \mathsf{sort}([p, T])\\
\\
\frac{{\log 1.5}^{2} \cdot \left(9.4921875 \cdot 10^{-28} \cdot \mathsf{PI}\left(\right)\right)}{\left(287.058 \cdot T\right) \cdot p}
\end{array}
Derivation

  1. Initial program 89.9%

    \[\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2} \]
  2. Add Preprocessing

  4. Applied rewrites95.0%

    \[\leadsto \color{blue}{\frac{{\log 1.5}^{2} \cdot \left(9.4921875 \cdot 10^{-28} \cdot \mathsf{PI}\left(\right)\right)}{\left(287.058 \cdot T\right) \cdot p}} \]
  5. Add Preprocessing

Alternative 3: 94.6% accurate, 2.9× speedup?

\[\begin{array}{l} [p, T] = \mathsf{sort}([p, T])\\ \\ \frac{9.4921875 \cdot 10^{-28} \cdot \mathsf{PI}\left(\right)}{\left(287.058 \cdot T\right) \cdot p} \cdot {\log 1.5}^{2} \end{array} \]
NOTE: p and T should be sorted in increasing order before calling this function.
(FPCore (p T)
 :precision binary64
 (* (/ (* 9.4921875e-28 (PI)) (* (* 287.058 T) p)) (pow (log 1.5) 2.0)))
\begin{array}{l}
[p, T] = \mathsf{sort}([p, T])\\
\\
\frac{9.4921875 \cdot 10^{-28} \cdot \mathsf{PI}\left(\right)}{\left(287.058 \cdot T\right) \cdot p} \cdot {\log 1.5}^{2}
\end{array}
Derivation

  1. Initial program 89.9%

    \[\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    2. lift-/.f64N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\color{blue}{\frac{0}{2}}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    3. metadata-evalN/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\color{blue}{0}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    4. lift-exp.f64N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{e^{0}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    5. exp-0N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{1}\right) \cdot {\log \frac{3}{2}}^{2} \]

    6. *-rgt-identity89.9

      \[\leadsto \color{blue}{\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)} \cdot {\log 1.5}^{2} \]

    7. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    8. lift-*.f64N/A

      \[\leadsto \left(\color{blue}{\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right)} \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot {\log \frac{3}{2}}^{2} \]

    9. associate-*l*N/A

      \[\leadsto \color{blue}{\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)\right)} \cdot {\log \frac{3}{2}}^{2} \]

    10. lift-/.f64N/A

      \[\leadsto \left(\color{blue}{\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)\right) \cdot {\log \frac{3}{2}}^{2} \]

    11. associate-*l/N/A

      \[\leadsto \color{blue}{\frac{\left(4 \cdot {10}^{8}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}} \cdot {\log \frac{3}{2}}^{2} \]

    12. lower-/.f64N/A

      \[\leadsto \color{blue}{\frac{\left(4 \cdot {10}^{8}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}} \cdot {\log \frac{3}{2}}^{2} \]

  4. Applied rewrites94.8%

    \[\leadsto \color{blue}{\frac{9.4921875 \cdot 10^{-28} \cdot \mathsf{PI}\left(\right)}{\left(287.058 \cdot T\right) \cdot p}} \cdot {\log 1.5}^{2} \]
  5. Add Preprocessing

Alternative 4: 94.1% accurate, 2.9× speedup?

\[\begin{array}{l} [p, T] = \mathsf{sort}([p, T])\\ \\ \frac{3.306714148360262 \cdot 10^{-30}}{T \cdot p} \cdot \left({\log 1.5}^{2} \cdot \mathsf{PI}\left(\right)\right) \end{array} \]
NOTE: p and T should be sorted in increasing order before calling this function.
(FPCore (p T)
 :precision binary64
 (* (/ 3.306714148360262e-30 (* T p)) (* (pow (log 1.5) 2.0) (PI))))
\begin{array}{l}
[p, T] = \mathsf{sort}([p, T])\\
\\
\frac{3.306714148360262 \cdot 10^{-30}}{T \cdot p} \cdot \left({\log 1.5}^{2} \cdot \mathsf{PI}\left(\right)\right)
\end{array}
Derivation

  1. Initial program 89.9%

    \[\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    2. lift-/.f64N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\color{blue}{\frac{0}{2}}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    3. metadata-evalN/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\color{blue}{0}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    4. lift-exp.f64N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{e^{0}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    5. exp-0N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{1}\right) \cdot {\log \frac{3}{2}}^{2} \]

    6. *-rgt-identity89.9

      \[\leadsto \color{blue}{\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)} \cdot {\log 1.5}^{2} \]

    7. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    8. lift-*.f64N/A

      \[\leadsto \left(\color{blue}{\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right)} \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot {\log \frac{3}{2}}^{2} \]

    9. associate-*l*N/A

      \[\leadsto \color{blue}{\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)\right)} \cdot {\log \frac{3}{2}}^{2} \]

    10. *-commutativeN/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    11. lift-/.f64N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    12. clear-numN/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{\frac{1}{\frac{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}{4 \cdot {10}^{8}}}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    13. div-invN/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}}{\frac{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}{4 \cdot {10}^{8}}}} \cdot {\log \frac{3}{2}}^{2} \]

  4. Applied rewrites99.4%

    \[\leadsto \color{blue}{\frac{\frac{2.373046875 \cdot 10^{-36} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot 7.17645 \cdot 10^{-7}}} \cdot {\log 1.5}^{2} \]
  5. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2}} \]

    2. lift-/.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot \frac{315623608846123}{439804651110400000000}}} \cdot {\log \frac{3}{2}}^{2} \]

    3. clear-numN/A

      \[\leadsto \color{blue}{\frac{1}{\frac{T \cdot \frac{315623608846123}{439804651110400000000}}{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}}} \cdot {\log \frac{3}{2}}^{2} \]

    4. associate-/r/N/A

      \[\leadsto \color{blue}{\left(\frac{1}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    5. lift-*.f64N/A

      \[\leadsto \left(\frac{1}{\color{blue}{T \cdot \frac{315623608846123}{439804651110400000000}}} \cdot \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}\right) \cdot {\log \frac{3}{2}}^{2} \]

    6. *-commutativeN/A

      \[\leadsto \left(\frac{1}{\color{blue}{\frac{315623608846123}{439804651110400000000} \cdot T}} \cdot \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}\right) \cdot {\log \frac{3}{2}}^{2} \]

    7. associate-/r*N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{1}{\frac{315623608846123}{439804651110400000000}}}{T}} \cdot \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}\right) \cdot {\log \frac{3}{2}}^{2} \]

    8. metadata-evalN/A

      \[\leadsto \left(\frac{\color{blue}{\frac{439804651110400000000}{315623608846123}}}{T} \cdot \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}\right) \cdot {\log \frac{3}{2}}^{2} \]

    9. lift-/.f64N/A

      \[\leadsto \left(\frac{\frac{439804651110400000000}{315623608846123}}{T} \cdot \color{blue}{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    10. frac-timesN/A

      \[\leadsto \color{blue}{\frac{\frac{439804651110400000000}{315623608846123} \cdot \left(\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)\right)}{T \cdot p}} \cdot {\log \frac{3}{2}}^{2} \]

    11. lift-*.f64N/A

      \[\leadsto \frac{\frac{439804651110400000000}{315623608846123} \cdot \left(\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)\right)}{\color{blue}{T \cdot p}} \cdot {\log \frac{3}{2}}^{2} \]

    12. associate-*l/N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{439804651110400000000}{315623608846123}}{T \cdot p} \cdot \left(\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot {\log \frac{3}{2}}^{2} \]

    13. lift-/.f64N/A

      \[\leadsto \left(\color{blue}{\frac{\frac{439804651110400000000}{315623608846123}}{T \cdot p}} \cdot \left(\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)\right)\right) \cdot {\log \frac{3}{2}}^{2} \]

    14. associate-*r*N/A

      \[\leadsto \color{blue}{\frac{\frac{439804651110400000000}{315623608846123}}{T \cdot p} \cdot \left(\left(\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)\right) \cdot {\log \frac{3}{2}}^{2}\right)} \]

    15. lift-*.f64N/A

      \[\leadsto \frac{\frac{439804651110400000000}{315623608846123}}{T \cdot p} \cdot \left(\color{blue}{\left(\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)\right)} \cdot {\log \frac{3}{2}}^{2}\right) \]

    16. associate-*l*N/A

      \[\leadsto \frac{\frac{439804651110400000000}{315623608846123}}{T \cdot p} \cdot \color{blue}{\left(\frac{243}{102400000000000000000000000000000000000} \cdot \left(\mathsf{PI}\left(\right) \cdot {\log \frac{3}{2}}^{2}\right)\right)} \]

    17. associate-*r*N/A

      \[\leadsto \color{blue}{\left(\frac{\frac{439804651110400000000}{315623608846123}}{T \cdot p} \cdot \frac{243}{102400000000000000000000000000000000000}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot {\log \frac{3}{2}}^{2}\right)} \]

  6. Applied rewrites94.5%

    \[\leadsto \color{blue}{\frac{3.306714148360262 \cdot 10^{-30}}{T \cdot p} \cdot \left({\log 1.5}^{2} \cdot \mathsf{PI}\left(\right)\right)} \]
  7. Add Preprocessing

Alternative 5: 91.5% accurate, 2.9× speedup?

\[\begin{array}{l} [p, T] = \mathsf{sort}([p, T])\\ \\ \left(\frac{\mathsf{PI}\left(\right)}{T \cdot p} \cdot {\log 1.5}^{2}\right) \cdot 3.306714148360262 \cdot 10^{-30} \end{array} \]
NOTE: p and T should be sorted in increasing order before calling this function.
(FPCore (p T)
 :precision binary64
 (* (* (/ (PI) (* T p)) (pow (log 1.5) 2.0)) 3.306714148360262e-30))
\begin{array}{l}
[p, T] = \mathsf{sort}([p, T])\\
\\
\left(\frac{\mathsf{PI}\left(\right)}{T \cdot p} \cdot {\log 1.5}^{2}\right) \cdot 3.306714148360262 \cdot 10^{-30}
\end{array}
Derivation

  1. Initial program 89.9%

    \[\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right) \cdot {\log 1.5}^{2} \]
  2. Add Preprocessing
  3. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\frac{0}{2}}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    2. lift-/.f64N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\color{blue}{\frac{0}{2}}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    3. metadata-evalN/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot e^{\color{blue}{0}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    4. lift-exp.f64N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{e^{0}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    5. exp-0N/A

      \[\leadsto \left(\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{1}\right) \cdot {\log \frac{3}{2}}^{2} \]

    6. *-rgt-identity89.9

      \[\leadsto \color{blue}{\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot 287.058\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)} \cdot {\log 1.5}^{2} \]

    7. lift-*.f64N/A

      \[\leadsto \color{blue}{\left(\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    8. lift-*.f64N/A

      \[\leadsto \left(\color{blue}{\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \mathsf{PI}\left(\right)\right)} \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot {\log \frac{3}{2}}^{2} \]

    9. associate-*l*N/A

      \[\leadsto \color{blue}{\left(\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)} \cdot \left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right)\right)} \cdot {\log \frac{3}{2}}^{2} \]

    10. *-commutativeN/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    11. lift-/.f64N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{\frac{4 \cdot {10}^{8}}{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    12. clear-numN/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}\right) \cdot \color{blue}{\frac{1}{\frac{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}{4 \cdot {10}^{8}}}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    13. div-invN/A

      \[\leadsto \color{blue}{\frac{\mathsf{PI}\left(\right) \cdot {\left(75 \cdot {10}^{\left(-9\right)}\right)}^{5}}{\frac{p \cdot \left(T \cdot \frac{315623608846123}{1099511627776}\right)}{4 \cdot {10}^{8}}}} \cdot {\log \frac{3}{2}}^{2} \]

  4. Applied rewrites99.4%

    \[\leadsto \color{blue}{\frac{\frac{2.373046875 \cdot 10^{-36} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot 7.17645 \cdot 10^{-7}}} \cdot {\log 1.5}^{2} \]
  5. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2}} \]

    2. lift-/.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}{T \cdot \frac{315623608846123}{439804651110400000000}}} \cdot {\log \frac{3}{2}}^{2} \]

    3. lift-/.f64N/A

      \[\leadsto \frac{\color{blue}{\frac{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}{p}}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2} \]

    4. lift-*.f64N/A

      \[\leadsto \frac{\frac{\color{blue}{\frac{243}{102400000000000000000000000000000000000} \cdot \mathsf{PI}\left(\right)}}{p}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2} \]

    5. associate-/l*N/A

      \[\leadsto \frac{\color{blue}{\frac{243}{102400000000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{p}}}{T \cdot \frac{315623608846123}{439804651110400000000}} \cdot {\log \frac{3}{2}}^{2} \]

    6. lift-*.f64N/A

      \[\leadsto \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{p}}{\color{blue}{T \cdot \frac{315623608846123}{439804651110400000000}}} \cdot {\log \frac{3}{2}}^{2} \]

    7. *-commutativeN/A

      \[\leadsto \frac{\frac{243}{102400000000000000000000000000000000000} \cdot \frac{\mathsf{PI}\left(\right)}{p}}{\color{blue}{\frac{315623608846123}{439804651110400000000} \cdot T}} \cdot {\log \frac{3}{2}}^{2} \]

    8. times-fracN/A

      \[\leadsto \color{blue}{\left(\frac{\frac{243}{102400000000000000000000000000000000000}}{\frac{315623608846123}{439804651110400000000}} \cdot \frac{\frac{\mathsf{PI}\left(\right)}{p}}{T}\right)} \cdot {\log \frac{3}{2}}^{2} \]

    9. metadata-evalN/A

      \[\leadsto \left(\color{blue}{\frac{7776}{2351579135999999940395355224609375}} \cdot \frac{\frac{\mathsf{PI}\left(\right)}{p}}{T}\right) \cdot {\log \frac{3}{2}}^{2} \]

    10. associate-/l/N/A

      \[\leadsto \left(\frac{7776}{2351579135999999940395355224609375} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{T \cdot p}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    11. lift-*.f64N/A

      \[\leadsto \left(\frac{7776}{2351579135999999940395355224609375} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{T \cdot p}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    12. lift-/.f64N/A

      \[\leadsto \left(\frac{7776}{2351579135999999940395355224609375} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{T \cdot p}}\right) \cdot {\log \frac{3}{2}}^{2} \]

    13. associate-*l*N/A

      \[\leadsto \color{blue}{\frac{7776}{2351579135999999940395355224609375} \cdot \left(\frac{\mathsf{PI}\left(\right)}{T \cdot p} \cdot {\log \frac{3}{2}}^{2}\right)} \]

    14. *-commutativeN/A

      \[\leadsto \frac{7776}{2351579135999999940395355224609375} \cdot \color{blue}{\left({\log \frac{3}{2}}^{2} \cdot \frac{\mathsf{PI}\left(\right)}{T \cdot p}\right)} \]

    15. *-commutativeN/A

      \[\leadsto \color{blue}{\left({\log \frac{3}{2}}^{2} \cdot \frac{\mathsf{PI}\left(\right)}{T \cdot p}\right) \cdot \frac{7776}{2351579135999999940395355224609375}} \]

    16. lower-*.f64N/A

      \[\leadsto \color{blue}{\left({\log \frac{3}{2}}^{2} \cdot \frac{\mathsf{PI}\left(\right)}{T \cdot p}\right) \cdot \frac{7776}{2351579135999999940395355224609375}} \]

    17. *-commutativeN/A

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{T \cdot p} \cdot {\log \frac{3}{2}}^{2}\right)} \cdot \frac{7776}{2351579135999999940395355224609375} \]

    18. lower-*.f6492.3

      \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{T \cdot p} \cdot {\log 1.5}^{2}\right)} \cdot 3.306714148360262 \cdot 10^{-30} \]

  6. Applied rewrites92.3%

    \[\leadsto \color{blue}{\left(\frac{\mathsf{PI}\left(\right)}{T \cdot p} \cdot {\log 1.5}^{2}\right) \cdot 3.306714148360262 \cdot 10^{-30}} \]
  7. Add Preprocessing

Reproduce

?
herbie shell --seed 1 
(FPCore (p T)
  :name "4*pow(10,8) / (p * (T * 287.058)) * PI * pow(75*pow(10,-9) ,5) * exp(0/2)*pow(log(1.5),2)"
  :precision binary64
  :pre (and (and (<= 0.0 p) (<= p 1000000000.0)) (and (<= 0.0 T) (<= T 1000.0)))
  (* (* (* (* (/ (* 4.0 (pow 10.0 8.0)) (* p (* T 287.058))) (PI)) (pow (* 75.0 (pow 10.0 (- 9.0))) 5.0)) (exp (/ 0.0 2.0))) (pow (log 1.5) 2.0)))