(4/3)*tan(pi/8)

Percentage Accurate: 99.5% → 99.7%
Time: 2.3s
Alternatives: 7
Speedup: 1.1×

Specification

?
\[3 \leq pi \land pi \leq 4\]
\[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]
(FPCore (pi)
  :precision binary64
  (* (/ 4.0 3.0) (tan (/ pi 8.0))))
double code(double pi) {
	return (4.0 / 3.0) * tan((pi / 8.0));
}

real(8) function code(pi)
    real(8), intent (in) :: pi
    code = (4.0d0 / 3.0d0) * tan((pi / 8.0d0))
end function

public static double code(double pi) {
	return (4.0 / 3.0) * Math.tan((pi / 8.0));
}

def code(pi):
	return (4.0 / 3.0) * math.tan((pi / 8.0))

function code(pi)
	return Float64(Float64(4.0 / 3.0) * tan(Float64(pi / 8.0)))
end

function tmp = code(pi)
	tmp = (4.0 / 3.0) * tan((pi / 8.0));
end

code[pi_] := N[(N[(4.0 / 3.0), $MachinePrecision] * N[Tan[N[(pi / 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right)

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 7 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: 99.5% accurate, 1.0× speedup?

\[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]
(FPCore (pi)
  :precision binary64
  (* (/ 4.0 3.0) (tan (/ pi 8.0))))
double code(double pi) {
	return (4.0 / 3.0) * tan((pi / 8.0));
}

real(8) function code(pi)
    real(8), intent (in) :: pi
    code = (4.0d0 / 3.0d0) * tan((pi / 8.0d0))
end function

public static double code(double pi) {
	return (4.0 / 3.0) * Math.tan((pi / 8.0));
}

def code(pi):
	return (4.0 / 3.0) * math.tan((pi / 8.0))

function code(pi)
	return Float64(Float64(4.0 / 3.0) * tan(Float64(pi / 8.0)))
end

function tmp = code(pi)
	tmp = (4.0 / 3.0) * tan((pi / 8.0));
end

code[pi_] := N[(N[(4.0 / 3.0), $MachinePrecision] * N[Tan[N[(pi / 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right)

Alternative 1: 99.7% accurate, 1.1× speedup?

\[\frac{\tan \left(0.125 \cdot pi\right)}{0.75} \]
(FPCore (pi)
  :precision binary64
  (/ (tan (* 0.125 pi)) 0.75))
double code(double pi) {
	return tan((0.125 * pi)) / 0.75;
}

real(8) function code(pi)
    real(8), intent (in) :: pi
    code = tan((0.125d0 * pi)) / 0.75d0
end function

public static double code(double pi) {
	return Math.tan((0.125 * pi)) / 0.75;
}

def code(pi):
	return math.tan((0.125 * pi)) / 0.75

function code(pi)
	return Float64(tan(Float64(0.125 * pi)) / 0.75)
end

function tmp = code(pi)
	tmp = tan((0.125 * pi)) / 0.75;
end

code[pi_] := N[(N[Tan[N[(0.125 * pi), $MachinePrecision]], $MachinePrecision] / 0.75), $MachinePrecision]

\frac{\tan \left(0.125 \cdot pi\right)}{0.75}
Derivation

  1. Initial program 99.5%

    \[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]
  2. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right)} \]

    2. lift-tan.f64N/A

      \[\leadsto \frac{4}{3} \cdot \color{blue}{\tan \left(\frac{pi}{8}\right)} \]

    3. tan-quotN/A

      \[\leadsto \frac{4}{3} \cdot \color{blue}{\frac{\sin \left(\frac{pi}{8}\right)}{\cos \left(\frac{pi}{8}\right)}} \]

    4. associate-*r/N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}{\cos \left(\frac{pi}{8}\right)}} \]

    5. lower-/.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}{\cos \left(\frac{pi}{8}\right)}} \]

    6. *-commutativeN/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{pi}{8}\right) \cdot \frac{4}{3}}}{\cos \left(\frac{pi}{8}\right)} \]

    7. lower-*.f64N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{pi}{8}\right) \cdot \frac{4}{3}}}{\cos \left(\frac{pi}{8}\right)} \]

    8. lower-sin.f64N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{pi}{8}\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    9. lift-/.f64N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{pi}{8}\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    10. mult-flipN/A

      \[\leadsto \frac{\sin \color{blue}{\left(pi \cdot \frac{1}{8}\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    11. *-commutativeN/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{1}{8} \cdot pi\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    12. lower-*.f64N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{1}{8} \cdot pi\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    13. metadata-evalN/A

      \[\leadsto \frac{\sin \left(\color{blue}{\frac{1}{8}} \cdot pi\right) \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    14. lift-/.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \color{blue}{\frac{4}{3}}}{\cos \left(\frac{pi}{8}\right)} \]

    15. metadata-evalN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \color{blue}{\frac{4}{3}}}{\cos \left(\frac{pi}{8}\right)} \]

    16. lift-/.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{pi}{8}\right)}} \]

    17. frac-2negN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(pi\right)}{\mathsf{neg}\left(8\right)}\right)}} \]

    18. distribute-frac-neg2N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)\right)}} \]

    19. cos-negN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\cos \left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)}} \]

    20. lower-cos.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\cos \left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)}} \]

    21. frac-2negN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(pi\right)\right)\right)}{\mathsf{neg}\left(8\right)}\right)}} \]

    22. remove-double-negN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \left(\frac{\color{blue}{pi}}{\mathsf{neg}\left(8\right)}\right)} \]

    23. mult-flipN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(pi \cdot \frac{1}{\mathsf{neg}\left(8\right)}\right)}} \]

    24. *-commutativeN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{1}{\mathsf{neg}\left(8\right)} \cdot pi\right)}} \]

    25. lower-*.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{1}{\mathsf{neg}\left(8\right)} \cdot pi\right)}} \]

    26. metadata-evalN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \left(\frac{1}{\color{blue}{-8}} \cdot pi\right)} \]

    27. metadata-eval99.3%

      \[\leadsto \frac{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}{\cos \left(\color{blue}{-0.125} \cdot pi\right)} \]

  3. Applied rewrites99.3%

    \[\leadsto \color{blue}{\frac{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}{\cos \left(-0.125 \cdot pi\right)}} \]
  4. Step-by-step derivation

    1. lift-cos.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\cos \left(\frac{-1}{8} \cdot pi\right)}} \]

    2. sin-+PI/2-revN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\sin \left(\frac{-1}{8} \cdot pi + \frac{\mathsf{PI}\left(\right)}{2}\right)}} \]

    3. lower-sin.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\sin \left(\frac{-1}{8} \cdot pi + \frac{\mathsf{PI}\left(\right)}{2}\right)}} \]

    4. lift-*.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \left(\color{blue}{\frac{-1}{8} \cdot pi} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

    5. lower-fma.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{8}, pi, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \]

    6. lift-PI.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \left(\mathsf{fma}\left(\frac{-1}{8}, pi, \frac{\color{blue}{\pi}}{2}\right)\right)} \]

    7. mult-flipN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \left(\mathsf{fma}\left(\frac{-1}{8}, pi, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right)} \]

    8. lower-*.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \left(\mathsf{fma}\left(\frac{-1}{8}, pi, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right)} \]

    9. metadata-eval99.4%

      \[\leadsto \frac{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}{\sin \left(\mathsf{fma}\left(-0.125, pi, \pi \cdot \color{blue}{0.5}\right)\right)} \]

  5. Applied rewrites99.4%

    \[\leadsto \frac{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}{\color{blue}{\sin \left(\mathsf{fma}\left(-0.125, pi, \pi \cdot 0.5\right)\right)}} \]

  7. Applied rewrites99.4%

    \[\leadsto \color{blue}{\frac{1.3333333333333333}{\frac{1}{\tan \left(pi \cdot 0.125\right)}}} \]
  8. Step-by-step derivation

    1. lift-/.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3}}{\frac{1}{\tan \left(pi \cdot \frac{1}{8}\right)}}} \]

    2. lift-/.f64N/A

      \[\leadsto \frac{\frac{4}{3}}{\color{blue}{\frac{1}{\tan \left(pi \cdot \frac{1}{8}\right)}}} \]

    3. associate-/r/N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3}}{1} \cdot \tan \left(pi \cdot \frac{1}{8}\right)} \]

    4. metadata-evalN/A

      \[\leadsto \color{blue}{\frac{4}{3}} \cdot \tan \left(pi \cdot \frac{1}{8}\right) \]

    5. metadata-evalN/A

      \[\leadsto \color{blue}{\frac{1}{\frac{3}{4}}} \cdot \tan \left(pi \cdot \frac{1}{8}\right) \]

    6. associate-/r/N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{3}{4}}{\tan \left(pi \cdot \frac{1}{8}\right)}}} \]

    7. div-flip-revN/A

      \[\leadsto \color{blue}{\frac{\tan \left(pi \cdot \frac{1}{8}\right)}{\frac{3}{4}}} \]

    8. lower-/.f6499.7%

      \[\leadsto \color{blue}{\frac{\tan \left(pi \cdot 0.125\right)}{0.75}} \]

    9. lift-*.f64N/A

      \[\leadsto \frac{\tan \color{blue}{\left(pi \cdot \frac{1}{8}\right)}}{\frac{3}{4}} \]

    10. *-commutativeN/A

      \[\leadsto \frac{\tan \color{blue}{\left(\frac{1}{8} \cdot pi\right)}}{\frac{3}{4}} \]

    11. lower-*.f6499.7%

      \[\leadsto \frac{\tan \color{blue}{\left(0.125 \cdot pi\right)}}{0.75} \]

  9. Applied rewrites99.7%

    \[\leadsto \color{blue}{\frac{\tan \left(0.125 \cdot pi\right)}{0.75}} \]
  10. Add Preprocessing

Alternative 2: 99.5% accurate, 1.1× speedup?

\[\tan \left(0.125 \cdot pi\right) \cdot 1.3333333333333333 \]
(FPCore (pi)
  :precision binary64
  (* (tan (* 0.125 pi)) 1.3333333333333333))
double code(double pi) {
	return tan((0.125 * pi)) * 1.3333333333333333;
}

real(8) function code(pi)
    real(8), intent (in) :: pi
    code = tan((0.125d0 * pi)) * 1.3333333333333333d0
end function

public static double code(double pi) {
	return Math.tan((0.125 * pi)) * 1.3333333333333333;
}

def code(pi):
	return math.tan((0.125 * pi)) * 1.3333333333333333

function code(pi)
	return Float64(tan(Float64(0.125 * pi)) * 1.3333333333333333)
end

function tmp = code(pi)
	tmp = tan((0.125 * pi)) * 1.3333333333333333;
end

code[pi_] := N[(N[Tan[N[(0.125 * pi), $MachinePrecision]], $MachinePrecision] * 1.3333333333333333), $MachinePrecision]

\tan \left(0.125 \cdot pi\right) \cdot 1.3333333333333333
Derivation

  1. Initial program 99.5%

    \[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]
  2. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right)} \]

    2. *-commutativeN/A

      \[\leadsto \color{blue}{\tan \left(\frac{pi}{8}\right) \cdot \frac{4}{3}} \]

    3. lower-*.f6499.5%

      \[\leadsto \color{blue}{\tan \left(\frac{pi}{8}\right) \cdot \frac{4}{3}} \]

    4. lift-/.f64N/A

      \[\leadsto \tan \color{blue}{\left(\frac{pi}{8}\right)} \cdot \frac{4}{3} \]

    5. mult-flipN/A

      \[\leadsto \tan \color{blue}{\left(pi \cdot \frac{1}{8}\right)} \cdot \frac{4}{3} \]

    6. *-commutativeN/A

      \[\leadsto \tan \color{blue}{\left(\frac{1}{8} \cdot pi\right)} \cdot \frac{4}{3} \]

    7. lower-*.f64N/A

      \[\leadsto \tan \color{blue}{\left(\frac{1}{8} \cdot pi\right)} \cdot \frac{4}{3} \]

    8. metadata-eval99.5%

      \[\leadsto \tan \left(\color{blue}{0.125} \cdot pi\right) \cdot \frac{4}{3} \]

    9. lift-/.f64N/A

      \[\leadsto \tan \left(\frac{1}{8} \cdot pi\right) \cdot \color{blue}{\frac{4}{3}} \]

    10. metadata-eval99.5%

      \[\leadsto \tan \left(0.125 \cdot pi\right) \cdot \color{blue}{1.3333333333333333} \]

  3. Applied rewrites99.5%

    \[\leadsto \color{blue}{\tan \left(0.125 \cdot pi\right) \cdot 1.3333333333333333} \]
  4. Add Preprocessing

Alternative 3: 34.1% accurate, 1.6× speedup?

\[\frac{1}{\frac{6 + -0.03125 \cdot {pi}^{2}}{pi}} \]
(FPCore (pi)
  :precision binary64
  (/ 1.0 (/ (+ 6.0 (* -0.03125 (pow pi 2.0))) pi)))
double code(double pi) {
	return 1.0 / ((6.0 + (-0.03125 * pow(pi, 2.0))) / pi);
}

real(8) function code(pi)
    real(8), intent (in) :: pi
    code = 1.0d0 / ((6.0d0 + ((-0.03125d0) * (pi ** 2.0d0))) / pi)
end function

public static double code(double pi) {
	return 1.0 / ((6.0 + (-0.03125 * Math.pow(pi, 2.0))) / pi);
}

def code(pi):
	return 1.0 / ((6.0 + (-0.03125 * math.pow(pi, 2.0))) / pi)

function code(pi)
	return Float64(1.0 / Float64(Float64(6.0 + Float64(-0.03125 * (pi ^ 2.0))) / pi))
end

function tmp = code(pi)
	tmp = 1.0 / ((6.0 + (-0.03125 * (pi ^ 2.0))) / pi);
end

code[pi_] := N[(1.0 / N[(N[(6.0 + N[(-0.03125 * N[Power[pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / pi), $MachinePrecision]), $MachinePrecision]

\frac{1}{\frac{6 + -0.03125 \cdot {pi}^{2}}{pi}}
Derivation

  1. Initial program 99.5%

    \[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]
  2. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right)} \]

    2. lift-tan.f64N/A

      \[\leadsto \frac{4}{3} \cdot \color{blue}{\tan \left(\frac{pi}{8}\right)} \]

    3. tan-quotN/A

      \[\leadsto \frac{4}{3} \cdot \color{blue}{\frac{\sin \left(\frac{pi}{8}\right)}{\cos \left(\frac{pi}{8}\right)}} \]

    4. associate-*r/N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}{\cos \left(\frac{pi}{8}\right)}} \]

    5. div-flipN/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(\frac{pi}{8}\right)}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}}} \]

    6. lower-unsound-/.f64N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(\frac{pi}{8}\right)}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}}} \]

    7. lower-unsound-/.f64N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\cos \left(\frac{pi}{8}\right)}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}}} \]

    8. lift-/.f64N/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(\frac{pi}{8}\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    9. frac-2negN/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(pi\right)}{\mathsf{neg}\left(8\right)}\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    10. distribute-frac-neg2N/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    11. cos-negN/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\cos \left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    12. lower-cos.f64N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\cos \left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    13. frac-2negN/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(pi\right)\right)\right)}{\mathsf{neg}\left(8\right)}\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    14. remove-double-negN/A

      \[\leadsto \frac{1}{\frac{\cos \left(\frac{\color{blue}{pi}}{\mathsf{neg}\left(8\right)}\right)}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    15. mult-flipN/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(pi \cdot \frac{1}{\mathsf{neg}\left(8\right)}\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    16. *-commutativeN/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(\frac{1}{\mathsf{neg}\left(8\right)} \cdot pi\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    17. lower-*.f64N/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(\frac{1}{\mathsf{neg}\left(8\right)} \cdot pi\right)}}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    18. metadata-evalN/A

      \[\leadsto \frac{1}{\frac{\cos \left(\frac{1}{\color{blue}{-8}} \cdot pi\right)}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    19. metadata-evalN/A

      \[\leadsto \frac{1}{\frac{\cos \left(\color{blue}{\frac{-1}{8}} \cdot pi\right)}{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}} \]

    20. *-commutativeN/A

      \[\leadsto \frac{1}{\frac{\cos \left(\frac{-1}{8} \cdot pi\right)}{\color{blue}{\sin \left(\frac{pi}{8}\right) \cdot \frac{4}{3}}}} \]

    21. lower-*.f64N/A

      \[\leadsto \frac{1}{\frac{\cos \left(\frac{-1}{8} \cdot pi\right)}{\color{blue}{\sin \left(\frac{pi}{8}\right) \cdot \frac{4}{3}}}} \]

  3. Applied rewrites99.3%

    \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(-0.125 \cdot pi\right)}{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}}} \]

  4. Taylor expanded in pi around 0

    \[\leadsto \frac{1}{\color{blue}{\frac{6 + \frac{-1}{32} \cdot {pi}^{2}}{pi}}} \]
  5. Step-by-step derivation

    1. lower-/.f64N/A

      \[\leadsto \frac{1}{\frac{6 + \frac{-1}{32} \cdot {pi}^{2}}{\color{blue}{pi}}} \]

    2. lower-+.f64N/A

      \[\leadsto \frac{1}{\frac{6 + \frac{-1}{32} \cdot {pi}^{2}}{pi}} \]

    3. lower-*.f64N/A

      \[\leadsto \frac{1}{\frac{6 + \frac{-1}{32} \cdot {pi}^{2}}{pi}} \]

    4. lower-pow.f6434.1%

      \[\leadsto \frac{1}{\frac{6 + -0.03125 \cdot {pi}^{2}}{pi}} \]

  6. Applied rewrites34.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{6 + -0.03125 \cdot {pi}^{2}}{pi}}} \]
  7. Add Preprocessing

Alternative 4: 34.1% accurate, 1.4× speedup?

\[\frac{1}{\frac{8 + -0.041666666666666664 \cdot {pi}^{2}}{pi} \cdot 0.75} \]
(FPCore (pi)
  :precision binary64
  (/ 1.0 (* (/ (+ 8.0 (* -0.041666666666666664 (pow pi 2.0))) pi) 0.75)))
double code(double pi) {
	return 1.0 / (((8.0 + (-0.041666666666666664 * pow(pi, 2.0))) / pi) * 0.75);
}

real(8) function code(pi)
    real(8), intent (in) :: pi
    code = 1.0d0 / (((8.0d0 + ((-0.041666666666666664d0) * (pi ** 2.0d0))) / pi) * 0.75d0)
end function

public static double code(double pi) {
	return 1.0 / (((8.0 + (-0.041666666666666664 * Math.pow(pi, 2.0))) / pi) * 0.75);
}

def code(pi):
	return 1.0 / (((8.0 + (-0.041666666666666664 * math.pow(pi, 2.0))) / pi) * 0.75)

function code(pi)
	return Float64(1.0 / Float64(Float64(Float64(8.0 + Float64(-0.041666666666666664 * (pi ^ 2.0))) / pi) * 0.75))
end

function tmp = code(pi)
	tmp = 1.0 / (((8.0 + (-0.041666666666666664 * (pi ^ 2.0))) / pi) * 0.75);
end

code[pi_] := N[(1.0 / N[(N[(N[(8.0 + N[(-0.041666666666666664 * N[Power[pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / pi), $MachinePrecision] * 0.75), $MachinePrecision]), $MachinePrecision]

\frac{1}{\frac{8 + -0.041666666666666664 \cdot {pi}^{2}}{pi} \cdot 0.75}
Derivation

  1. Initial program 99.5%

    \[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]
  2. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto \color{blue}{\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right)} \]

    2. lift-tan.f64N/A

      \[\leadsto \frac{4}{3} \cdot \color{blue}{\tan \left(\frac{pi}{8}\right)} \]

    3. tan-quotN/A

      \[\leadsto \frac{4}{3} \cdot \color{blue}{\frac{\sin \left(\frac{pi}{8}\right)}{\cos \left(\frac{pi}{8}\right)}} \]

    4. associate-*r/N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}{\cos \left(\frac{pi}{8}\right)}} \]

    5. lower-/.f64N/A

      \[\leadsto \color{blue}{\frac{\frac{4}{3} \cdot \sin \left(\frac{pi}{8}\right)}{\cos \left(\frac{pi}{8}\right)}} \]

    6. *-commutativeN/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{pi}{8}\right) \cdot \frac{4}{3}}}{\cos \left(\frac{pi}{8}\right)} \]

    7. lower-*.f64N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{pi}{8}\right) \cdot \frac{4}{3}}}{\cos \left(\frac{pi}{8}\right)} \]

    8. lower-sin.f64N/A

      \[\leadsto \frac{\color{blue}{\sin \left(\frac{pi}{8}\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    9. lift-/.f64N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{pi}{8}\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    10. mult-flipN/A

      \[\leadsto \frac{\sin \color{blue}{\left(pi \cdot \frac{1}{8}\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    11. *-commutativeN/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{1}{8} \cdot pi\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    12. lower-*.f64N/A

      \[\leadsto \frac{\sin \color{blue}{\left(\frac{1}{8} \cdot pi\right)} \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    13. metadata-evalN/A

      \[\leadsto \frac{\sin \left(\color{blue}{\frac{1}{8}} \cdot pi\right) \cdot \frac{4}{3}}{\cos \left(\frac{pi}{8}\right)} \]

    14. lift-/.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \color{blue}{\frac{4}{3}}}{\cos \left(\frac{pi}{8}\right)} \]

    15. metadata-evalN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \color{blue}{\frac{4}{3}}}{\cos \left(\frac{pi}{8}\right)} \]

    16. lift-/.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{pi}{8}\right)}} \]

    17. frac-2negN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(pi\right)}{\mathsf{neg}\left(8\right)}\right)}} \]

    18. distribute-frac-neg2N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)\right)}} \]

    19. cos-negN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\cos \left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)}} \]

    20. lower-cos.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\cos \left(\frac{\mathsf{neg}\left(pi\right)}{8}\right)}} \]

    21. frac-2negN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{\mathsf{neg}\left(\left(\mathsf{neg}\left(pi\right)\right)\right)}{\mathsf{neg}\left(8\right)}\right)}} \]

    22. remove-double-negN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \left(\frac{\color{blue}{pi}}{\mathsf{neg}\left(8\right)}\right)} \]

    23. mult-flipN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(pi \cdot \frac{1}{\mathsf{neg}\left(8\right)}\right)}} \]

    24. *-commutativeN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{1}{\mathsf{neg}\left(8\right)} \cdot pi\right)}} \]

    25. lower-*.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \color{blue}{\left(\frac{1}{\mathsf{neg}\left(8\right)} \cdot pi\right)}} \]

    26. metadata-evalN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\cos \left(\frac{1}{\color{blue}{-8}} \cdot pi\right)} \]

    27. metadata-eval99.3%

      \[\leadsto \frac{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}{\cos \left(\color{blue}{-0.125} \cdot pi\right)} \]

  3. Applied rewrites99.3%

    \[\leadsto \color{blue}{\frac{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}{\cos \left(-0.125 \cdot pi\right)}} \]
  4. Step-by-step derivation

    1. lift-cos.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\cos \left(\frac{-1}{8} \cdot pi\right)}} \]

    2. sin-+PI/2-revN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\sin \left(\frac{-1}{8} \cdot pi + \frac{\mathsf{PI}\left(\right)}{2}\right)}} \]

    3. lower-sin.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\color{blue}{\sin \left(\frac{-1}{8} \cdot pi + \frac{\mathsf{PI}\left(\right)}{2}\right)}} \]

    4. lift-*.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \left(\color{blue}{\frac{-1}{8} \cdot pi} + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

    5. lower-fma.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \color{blue}{\left(\mathsf{fma}\left(\frac{-1}{8}, pi, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)}} \]

    6. lift-PI.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \left(\mathsf{fma}\left(\frac{-1}{8}, pi, \frac{\color{blue}{\pi}}{2}\right)\right)} \]

    7. mult-flipN/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \left(\mathsf{fma}\left(\frac{-1}{8}, pi, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right)} \]

    8. lower-*.f64N/A

      \[\leadsto \frac{\sin \left(\frac{1}{8} \cdot pi\right) \cdot \frac{4}{3}}{\sin \left(\mathsf{fma}\left(\frac{-1}{8}, pi, \color{blue}{\pi \cdot \frac{1}{2}}\right)\right)} \]

    9. metadata-eval99.4%

      \[\leadsto \frac{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}{\sin \left(\mathsf{fma}\left(-0.125, pi, \pi \cdot \color{blue}{0.5}\right)\right)} \]

  5. Applied rewrites99.4%

    \[\leadsto \frac{\sin \left(0.125 \cdot pi\right) \cdot 1.3333333333333333}{\color{blue}{\sin \left(\mathsf{fma}\left(-0.125, pi, \pi \cdot 0.5\right)\right)}} \]

  7. Applied rewrites99.4%

    \[\leadsto \color{blue}{\frac{1}{\frac{1}{\tan \left(pi \cdot 0.125\right)} \cdot 0.75}} \]

  8. Taylor expanded in pi around 0

    \[\leadsto \frac{1}{\color{blue}{\frac{8 + \frac{-1}{24} \cdot {pi}^{2}}{pi}} \cdot 0.75} \]
  9. Step-by-step derivation

    1. lower-/.f64N/A

      \[\leadsto \frac{1}{\frac{8 + \frac{-1}{24} \cdot {pi}^{2}}{\color{blue}{pi}} \cdot \frac{3}{4}} \]

    2. lower-+.f64N/A

      \[\leadsto \frac{1}{\frac{8 + \frac{-1}{24} \cdot {pi}^{2}}{pi} \cdot \frac{3}{4}} \]

    3. lower-*.f64N/A

      \[\leadsto \frac{1}{\frac{8 + \frac{-1}{24} \cdot {pi}^{2}}{pi} \cdot \frac{3}{4}} \]

    4. lower-pow.f6434.1%

      \[\leadsto \frac{1}{\frac{8 + -0.041666666666666664 \cdot {pi}^{2}}{pi} \cdot 0.75} \]

  10. Applied rewrites34.1%

    \[\leadsto \frac{1}{\color{blue}{\frac{8 + -0.041666666666666664 \cdot {pi}^{2}}{pi}} \cdot 0.75} \]
  11. Add Preprocessing

Alternative 5: 30.3% accurate, 2.5× speedup?

\[pi \cdot \left(\left(1 - \frac{-0.0008680555555555555 \cdot \left(pi \cdot pi\right)}{0.16666666666666666}\right) \cdot 0.16666666666666666\right) \]
(FPCore (pi)
  :precision binary64
  (*
 pi
 (*
  (- 1.0 (/ (* -0.0008680555555555555 (* pi pi)) 0.16666666666666666))
  0.16666666666666666)))
double code(double pi) {
	return pi * ((1.0 - ((-0.0008680555555555555 * (pi * pi)) / 0.16666666666666666)) * 0.16666666666666666);
}

real(8) function code(pi)
    real(8), intent (in) :: pi
    code = pi * ((1.0d0 - (((-0.0008680555555555555d0) * (pi * pi)) / 0.16666666666666666d0)) * 0.16666666666666666d0)
end function

public static double code(double pi) {
	return pi * ((1.0 - ((-0.0008680555555555555 * (pi * pi)) / 0.16666666666666666)) * 0.16666666666666666);
}

def code(pi):
	return pi * ((1.0 - ((-0.0008680555555555555 * (pi * pi)) / 0.16666666666666666)) * 0.16666666666666666)

function code(pi)
	return Float64(pi * Float64(Float64(1.0 - Float64(Float64(-0.0008680555555555555 * Float64(pi * pi)) / 0.16666666666666666)) * 0.16666666666666666))
end

function tmp = code(pi)
	tmp = pi * ((1.0 - ((-0.0008680555555555555 * (pi * pi)) / 0.16666666666666666)) * 0.16666666666666666);
end

code[pi_] := N[(pi * N[(N[(1.0 - N[(N[(-0.0008680555555555555 * N[(pi * pi), $MachinePrecision]), $MachinePrecision] / 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

pi \cdot \left(\left(1 - \frac{-0.0008680555555555555 \cdot \left(pi \cdot pi\right)}{0.16666666666666666}\right) \cdot 0.16666666666666666\right)
Derivation

  1. Initial program 99.5%

    \[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]

  2. Taylor expanded in pi around 0

    \[\leadsto \color{blue}{pi \cdot \left(\frac{1}{6} + \frac{1}{1152} \cdot {pi}^{2}\right)} \]
  3. Step-by-step derivation

    1. lower-*.f64N/A

      \[\leadsto pi \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{1152} \cdot {pi}^{2}\right)} \]

    2. lower-+.f64N/A

      \[\leadsto pi \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{1152} \cdot {pi}^{2}}\right) \]

    3. lower-*.f64N/A

      \[\leadsto pi \cdot \left(\frac{1}{6} + \frac{1}{1152} \cdot \color{blue}{{pi}^{2}}\right) \]

    4. lower-pow.f6430.3%

      \[\leadsto pi \cdot \left(0.16666666666666666 + 0.0008680555555555555 \cdot {pi}^{\color{blue}{2}}\right) \]

  4. Applied rewrites30.3%

    \[\leadsto \color{blue}{pi \cdot \left(0.16666666666666666 + 0.0008680555555555555 \cdot {pi}^{2}\right)} \]
  5. Step-by-step derivation

    1. lift-+.f64N/A

      \[\leadsto pi \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{1152} \cdot {pi}^{2}}\right) \]

    2. add-flipN/A

      \[\leadsto pi \cdot \left(\frac{1}{6} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{1152} \cdot {pi}^{2}\right)\right)}\right) \]

    3. sub-to-multN/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{1152} \cdot {pi}^{2}\right)}{\frac{1}{6}}\right) \cdot \color{blue}{\frac{1}{6}}\right) \]

    4. lower-unsound-*.f64N/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{1152} \cdot {pi}^{2}\right)}{\frac{1}{6}}\right) \cdot \color{blue}{\frac{1}{6}}\right) \]

    5. lower-unsound--.f64N/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{1152} \cdot {pi}^{2}\right)}{\frac{1}{6}}\right) \cdot \frac{1}{6}\right) \]

    6. lower-unsound-/.f64N/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{1152} \cdot {pi}^{2}\right)}{\frac{1}{6}}\right) \cdot \frac{1}{6}\right) \]

    7. lift-*.f64N/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\mathsf{neg}\left(\frac{1}{1152} \cdot {pi}^{2}\right)}{\frac{1}{6}}\right) \cdot \frac{1}{6}\right) \]

    8. distribute-lft-neg-outN/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\left(\mathsf{neg}\left(\frac{1}{1152}\right)\right) \cdot {pi}^{2}}{\frac{1}{6}}\right) \cdot \frac{1}{6}\right) \]

    9. lower-*.f64N/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\left(\mathsf{neg}\left(\frac{1}{1152}\right)\right) \cdot {pi}^{2}}{\frac{1}{6}}\right) \cdot \frac{1}{6}\right) \]

    10. metadata-eval30.3%

      \[\leadsto pi \cdot \left(\left(1 - \frac{-0.0008680555555555555 \cdot {pi}^{2}}{0.16666666666666666}\right) \cdot 0.16666666666666666\right) \]

    11. lift-pow.f64N/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\frac{-1}{1152} \cdot {pi}^{2}}{\frac{1}{6}}\right) \cdot \frac{1}{6}\right) \]

    12. unpow2N/A

      \[\leadsto pi \cdot \left(\left(1 - \frac{\frac{-1}{1152} \cdot \left(pi \cdot pi\right)}{\frac{1}{6}}\right) \cdot \frac{1}{6}\right) \]

    13. lower-*.f6430.3%

      \[\leadsto pi \cdot \left(\left(1 - \frac{-0.0008680555555555555 \cdot \left(pi \cdot pi\right)}{0.16666666666666666}\right) \cdot 0.16666666666666666\right) \]

  6. Applied rewrites30.3%

    \[\leadsto pi \cdot \left(\left(1 - \frac{-0.0008680555555555555 \cdot \left(pi \cdot pi\right)}{0.16666666666666666}\right) \cdot \color{blue}{0.16666666666666666}\right) \]
  7. Add Preprocessing

Alternative 6: 30.3% accurate, 4.1× speedup?

\[pi \cdot \mathsf{fma}\left(0.0008680555555555555 \cdot pi, pi, 0.16666666666666666\right) \]
(FPCore (pi)
  :precision binary64
  (* pi (fma (* 0.0008680555555555555 pi) pi 0.16666666666666666)))
double code(double pi) {
	return pi * fma((0.0008680555555555555 * pi), pi, 0.16666666666666666);
}

function code(pi)
	return Float64(pi * fma(Float64(0.0008680555555555555 * pi), pi, 0.16666666666666666))
end

code[pi_] := N[(pi * N[(N[(0.0008680555555555555 * pi), $MachinePrecision] * pi + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

pi \cdot \mathsf{fma}\left(0.0008680555555555555 \cdot pi, pi, 0.16666666666666666\right)
Derivation

  1. Initial program 99.5%

    \[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]

  2. Taylor expanded in pi around 0

    \[\leadsto \color{blue}{pi \cdot \left(\frac{1}{6} + \frac{1}{1152} \cdot {pi}^{2}\right)} \]
  3. Step-by-step derivation

    1. lower-*.f64N/A

      \[\leadsto pi \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{1152} \cdot {pi}^{2}\right)} \]

    2. lower-+.f64N/A

      \[\leadsto pi \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{1152} \cdot {pi}^{2}}\right) \]

    3. lower-*.f64N/A

      \[\leadsto pi \cdot \left(\frac{1}{6} + \frac{1}{1152} \cdot \color{blue}{{pi}^{2}}\right) \]

    4. lower-pow.f6430.3%

      \[\leadsto pi \cdot \left(0.16666666666666666 + 0.0008680555555555555 \cdot {pi}^{\color{blue}{2}}\right) \]

  4. Applied rewrites30.3%

    \[\leadsto \color{blue}{pi \cdot \left(0.16666666666666666 + 0.0008680555555555555 \cdot {pi}^{2}\right)} \]
  5. Step-by-step derivation

    1. lift-+.f64N/A

      \[\leadsto pi \cdot \left(\frac{1}{6} + \color{blue}{\frac{1}{1152} \cdot {pi}^{2}}\right) \]

    2. +-commutativeN/A

      \[\leadsto pi \cdot \left(\frac{1}{1152} \cdot {pi}^{2} + \color{blue}{\frac{1}{6}}\right) \]

    3. sum-to-multN/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\frac{1}{1152} \cdot {pi}^{2}}\right) \cdot \color{blue}{\left(\frac{1}{1152} \cdot {pi}^{2}\right)}\right) \]

    4. lower-unsound-*.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\frac{1}{1152} \cdot {pi}^{2}}\right) \cdot \color{blue}{\left(\frac{1}{1152} \cdot {pi}^{2}\right)}\right) \]

    5. lower-unsound-+.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\frac{1}{1152} \cdot {pi}^{2}}\right) \cdot \left(\color{blue}{\frac{1}{1152}} \cdot {pi}^{2}\right)\right) \]

    6. lower-unsound-/.f6430.3%

      \[\leadsto pi \cdot \left(\left(1 + \frac{0.16666666666666666}{0.0008680555555555555 \cdot {pi}^{2}}\right) \cdot \left(0.0008680555555555555 \cdot {pi}^{2}\right)\right) \]

    7. lift-*.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\frac{1}{1152} \cdot {pi}^{2}}\right) \cdot \left(\frac{1}{1152} \cdot {pi}^{2}\right)\right) \]

    8. *-commutativeN/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{{pi}^{2} \cdot \frac{1}{1152}}\right) \cdot \left(\frac{1}{1152} \cdot {pi}^{2}\right)\right) \]

    9. lower-*.f6430.3%

      \[\leadsto pi \cdot \left(\left(1 + \frac{0.16666666666666666}{{pi}^{2} \cdot 0.0008680555555555555}\right) \cdot \left(0.0008680555555555555 \cdot {pi}^{2}\right)\right) \]

    10. lift-pow.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{{pi}^{2} \cdot \frac{1}{1152}}\right) \cdot \left(\frac{1}{1152} \cdot {pi}^{2}\right)\right) \]

    11. unpow2N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\left(pi \cdot pi\right) \cdot \frac{1}{1152}}\right) \cdot \left(\frac{1}{1152} \cdot {pi}^{2}\right)\right) \]

    12. lower-*.f6430.3%

      \[\leadsto pi \cdot \left(\left(1 + \frac{0.16666666666666666}{\left(pi \cdot pi\right) \cdot 0.0008680555555555555}\right) \cdot \left(0.0008680555555555555 \cdot {pi}^{2}\right)\right) \]

    13. lift-*.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\left(pi \cdot pi\right) \cdot \frac{1}{1152}}\right) \cdot \left(\frac{1}{1152} \cdot \color{blue}{{pi}^{2}}\right)\right) \]

    14. *-commutativeN/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\left(pi \cdot pi\right) \cdot \frac{1}{1152}}\right) \cdot \left({pi}^{2} \cdot \color{blue}{\frac{1}{1152}}\right)\right) \]

    15. lower-*.f6430.3%

      \[\leadsto pi \cdot \left(\left(1 + \frac{0.16666666666666666}{\left(pi \cdot pi\right) \cdot 0.0008680555555555555}\right) \cdot \left({pi}^{2} \cdot \color{blue}{0.0008680555555555555}\right)\right) \]

    16. lift-pow.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\left(pi \cdot pi\right) \cdot \frac{1}{1152}}\right) \cdot \left({pi}^{2} \cdot \frac{1}{1152}\right)\right) \]

    17. unpow2N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\left(pi \cdot pi\right) \cdot \frac{1}{1152}}\right) \cdot \left(\left(pi \cdot pi\right) \cdot \frac{1}{1152}\right)\right) \]

    18. lower-*.f6430.3%

      \[\leadsto pi \cdot \left(\left(1 + \frac{0.16666666666666666}{\left(pi \cdot pi\right) \cdot 0.0008680555555555555}\right) \cdot \left(\left(pi \cdot pi\right) \cdot 0.0008680555555555555\right)\right) \]

  6. Applied rewrites30.3%

    \[\leadsto pi \cdot \left(\left(1 + \frac{0.16666666666666666}{\left(pi \cdot pi\right) \cdot 0.0008680555555555555}\right) \cdot \color{blue}{\left(\left(pi \cdot pi\right) \cdot 0.0008680555555555555\right)}\right) \]
  7. Step-by-step derivation

    1. lift-*.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\left(pi \cdot pi\right) \cdot \frac{1}{1152}}\right) \cdot \color{blue}{\left(\left(pi \cdot pi\right) \cdot \frac{1}{1152}\right)}\right) \]

    2. lift-+.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\left(pi \cdot pi\right) \cdot \frac{1}{1152}}\right) \cdot \left(\color{blue}{\left(pi \cdot pi\right)} \cdot \frac{1}{1152}\right)\right) \]

    3. lift-/.f64N/A

      \[\leadsto pi \cdot \left(\left(1 + \frac{\frac{1}{6}}{\left(pi \cdot pi\right) \cdot \frac{1}{1152}}\right) \cdot \left(\left(pi \cdot \color{blue}{pi}\right) \cdot \frac{1}{1152}\right)\right) \]

    4. sum-to-mult-revN/A

      \[\leadsto pi \cdot \left(\left(pi \cdot pi\right) \cdot \frac{1}{1152} + \color{blue}{\frac{1}{6}}\right) \]

    5. add-flipN/A

      \[\leadsto pi \cdot \left(\left(pi \cdot pi\right) \cdot \frac{1}{1152} - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right) \]

    6. *-lft-identityN/A

      \[\leadsto pi \cdot \left(1 \cdot \left(\left(pi \cdot pi\right) \cdot \frac{1}{1152}\right) - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{6}}\right)\right)\right) \]

    7. add-flipN/A

      \[\leadsto pi \cdot \left(1 \cdot \left(\left(pi \cdot pi\right) \cdot \frac{1}{1152}\right) + \color{blue}{\frac{1}{6}}\right) \]

    8. *-lft-identityN/A

      \[\leadsto pi \cdot \left(\left(pi \cdot pi\right) \cdot \frac{1}{1152} + \frac{1}{6}\right) \]

    9. lift-*.f64N/A

      \[\leadsto pi \cdot \left(\left(pi \cdot pi\right) \cdot \frac{1}{1152} + \frac{1}{6}\right) \]

    10. lift-*.f64N/A

      \[\leadsto pi \cdot \left(\left(pi \cdot pi\right) \cdot \frac{1}{1152} + \frac{1}{6}\right) \]

    11. associate-*l*N/A

      \[\leadsto pi \cdot \left(pi \cdot \left(pi \cdot \frac{1}{1152}\right) + \frac{1}{6}\right) \]

    12. *-commutativeN/A

      \[\leadsto pi \cdot \left(\left(pi \cdot \frac{1}{1152}\right) \cdot pi + \frac{1}{6}\right) \]

    13. lower-fma.f64N/A

      \[\leadsto pi \cdot \mathsf{fma}\left(pi \cdot \frac{1}{1152}, \color{blue}{pi}, \frac{1}{6}\right) \]

    14. *-commutativeN/A

      \[\leadsto pi \cdot \mathsf{fma}\left(\frac{1}{1152} \cdot pi, pi, \frac{1}{6}\right) \]

    15. lower-*.f6430.3%

      \[\leadsto pi \cdot \mathsf{fma}\left(0.0008680555555555555 \cdot pi, pi, 0.16666666666666666\right) \]

  8. Applied rewrites30.3%

    \[\leadsto pi \cdot \mathsf{fma}\left(0.0008680555555555555 \cdot pi, \color{blue}{pi}, 0.16666666666666666\right) \]
  9. Add Preprocessing

Alternative 7: 24.5% accurate, 12.2× speedup?

\[0.16666666666666666 \cdot pi \]
(FPCore (pi)
  :precision binary64
  (* 0.16666666666666666 pi))
double code(double pi) {
	return 0.16666666666666666 * pi;
}

real(8) function code(pi)
    real(8), intent (in) :: pi
    code = 0.16666666666666666d0 * pi
end function

public static double code(double pi) {
	return 0.16666666666666666 * pi;
}

def code(pi):
	return 0.16666666666666666 * pi

function code(pi)
	return Float64(0.16666666666666666 * pi)
end

function tmp = code(pi)
	tmp = 0.16666666666666666 * pi;
end

code[pi_] := N[(0.16666666666666666 * pi), $MachinePrecision]

0.16666666666666666 \cdot pi
Derivation

  1. Initial program 99.5%

    \[\frac{4}{3} \cdot \tan \left(\frac{pi}{8}\right) \]

  2. Taylor expanded in pi around 0

    \[\leadsto \color{blue}{\frac{1}{6} \cdot pi} \]
  3. Step-by-step derivation

    1. lower-*.f6424.5%

      \[\leadsto 0.16666666666666666 \cdot \color{blue}{pi} \]

  4. Applied rewrites24.5%

    \[\leadsto \color{blue}{0.16666666666666666 \cdot pi} \]
  5. Add Preprocessing

Reproduce

?
herbie shell --seed 1 
(FPCore (pi)
  :name "(4/3)*tan(pi/8)"
  :precision binary64
  :pre (and (<= 3.0 pi) (<= pi 4.0))
  (* (/ 4.0 3.0) (tan (/ pi 8.0))))