(tan (+ (- z0) (* 1/2 PI)))

Percentage Accurate: 5.5% → 99.8%

Time: 9.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (tan (+ (- z0) (* 1/2 PI))))

C

double code(double z0) {
	return tan((-z0 + (0.5 * ((double) M_PI))));
}

Java

public static double code(double z0) {
	return Math.tan((-z0 + (0.5 * Math.PI)));
}

Python

def code(z0):
	return math.tan((-z0 + (0.5 * math.pi)))

Julia

function code(z0)
	return tan(Float64(Float64(-z0) + Float64(0.5 * pi)))
end

MATLAB

function tmp = code(z0)
	tmp = tan((-z0 + (0.5 * pi)));
end

Wolfram

code[z0_] := N[Tan[N[((-z0) + N[(1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 5.5% accurate, 1.0× speedup

Math

\[\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (tan (+ (- z0) (* 1/2 PI))))

C

double code(double z0) {
	return tan((-z0 + (0.5 * ((double) M_PI))));
}

Java

public static double code(double z0) {
	return Math.tan((-z0 + (0.5 * Math.PI)));
}

Python

def code(z0):
	return math.tan((-z0 + (0.5 * math.pi)))

Julia

function code(z0)
	return tan(Float64(Float64(-z0) + Float64(0.5 * pi)))
end

MATLAB

function tmp = code(z0)
	tmp = tan((-z0 + (0.5 * pi)));
end

Wolfram

code[z0_] := N[Tan[N[((-z0) + N[(1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\frac{1}{\tan z0} \]

FPCore

(FPCore (z0)
  :precision binary64
  (/ 1 (tan z0)))

C

double code(double z0) {
	return 1.0 / tan(z0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = 1.0d0 / tan(z0)
end function

Java

public static double code(double z0) {
	return 1.0 / Math.tan(z0);
}

Python

def code(z0):
	return 1.0 / math.tan(z0)

Julia

function code(z0)
	return Float64(1.0 / tan(z0))
end

MATLAB

function tmp = code(z0)
	tmp = 1.0 / tan(z0);
end

Wolfram

code[z0_] := N[(1 / N[Tan[z0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 5.5%

   \[\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)} \]

   2. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}{\cos \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   3. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}} \]

   4. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}} \]

   5. lower-unsound-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\cos \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   10. mult-flip N/A

       \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \color{blue}{\frac{\pi}{2}}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   11. lift-PI.f64 N/A

       \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   12. cos-+PI/2-rev N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{neg}\left(\sin \left(-z0\right)\right)}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   13. sin-neg-rev N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\sin \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   14. lower-sin.f64 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\sin \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{1}{\frac{\sin \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z0\right)\right)}\right)\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   16. remove-double-neg N/A

       \[\leadsto \frac{1}{\frac{\sin \color{blue}{z0}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{\sin z0}{\sin \color{blue}{\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\sin z0}{\sin \left(\left(-z0\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}} \]

   19. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\sin z0}{\sin \left(\left(-z0\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}} \]

   20. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\sin z0}{\sin \left(\left(-z0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin z0}{\cos z0}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin z0}{\cos z0}}} \]

   2. lift-sin.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\sin z0}}{\cos z0}} \]

   3. lift-cos.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin z0}{\color{blue}{\cos z0}}} \]

   4. quot-tan N/A

      \[\leadsto \frac{1}{\color{blue}{\tan z0}} \]

   5. lower-tan.f64 99.8%

      \[\leadsto \frac{1}{\color{blue}{\tan z0}} \]

5. Applied rewrites 99.8%

   \[\leadsto \frac{1}{\color{blue}{\tan z0}} \]
6. Add Preprocessing

Alternative 2: 7.5% accurate, 1.0× speedup

Math

\[\left|\tan \left(\pi \cdot \frac{1}{2} - z0\right)\right| \]

FPCore

(FPCore (z0)
  :precision binary64
  (fabs (tan (- (* PI 1/2) z0))))

C

double code(double z0) {
	return fabs(tan(((((double) M_PI) * 0.5) - z0)));
}

Java

public static double code(double z0) {
	return Math.abs(Math.tan(((Math.PI * 0.5) - z0)));
}

Python

def code(z0):
	return math.fabs(math.tan(((math.pi * 0.5) - z0)))

Julia

function code(z0)
	return abs(tan(Float64(Float64(pi * 0.5) - z0)))
end

MATLAB

function tmp = code(z0)
	tmp = abs(tan(((pi * 0.5) - z0)));
end

Wolfram

code[z0_] := N[Abs[N[Tan[N[(N[(Pi * 1/2), $MachinePrecision] - z0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 5.5%

   \[\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)} \]

   2. tan-quot N/A

      \[\leadsto \color{blue}{\frac{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}{\cos \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   3. div-flip N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}} \]

   4. lower-unsound-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}} \]

   5. lower-unsound-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\cos \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}} \]

   6. lift-+.f64 N/A

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   8. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   10. mult-flip N/A

       \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \color{blue}{\frac{\pi}{2}}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   11. lift-PI.f64 N/A

       \[\leadsto \frac{1}{\frac{\cos \left(\left(-z0\right) + \frac{\color{blue}{\mathsf{PI}\left(\right)}}{2}\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   12. cos-+PI/2-rev N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{neg}\left(\sin \left(-z0\right)\right)}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   13. sin-neg-rev N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\sin \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   14. lower-sin.f64 N/A

       \[\leadsto \frac{1}{\frac{\color{blue}{\sin \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   15. lift-neg.f64 N/A

       \[\leadsto \frac{1}{\frac{\sin \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z0\right)\right)}\right)\right)}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   16. remove-double-neg N/A

       \[\leadsto \frac{1}{\frac{\sin \color{blue}{z0}}{\sin \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{1}{\frac{\sin z0}{\sin \color{blue}{\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{\sin z0}{\sin \left(\left(-z0\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right)}} \]

   19. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\sin z0}{\sin \left(\left(-z0\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)}} \]

   20. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\sin z0}{\sin \left(\left(-z0\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)}} \]

3. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{\sin z0}{\cos z0}}} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin z0}{\cos z0}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\sin z0}{\cos z0}}} \]

   3. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\sin z0} \cdot \cos z0} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sin z0} \cdot \cos z0} \]

   5. lower-/.f64 99.7%

      \[\leadsto \color{blue}{\frac{1}{\sin z0}} \cdot \cos z0 \]

5. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{1}{\sin z0} \cdot \cos z0} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sin z0} \cdot \cos z0} \]

   2. *-rgt-identity N/A

      \[\leadsto \frac{1}{\sin z0} \cdot \color{blue}{\left(\cos z0 \cdot 1\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{1}{\sin z0} \cdot \color{blue}{\left(\cos z0 \cdot 1\right)} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\sin z0}} \cdot \left(\cos z0 \cdot 1\right) \]

   5. associate-/r/ N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\sin z0}{\cos z0 \cdot 1}}} \]

   6. lift-sin.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\sin z0}}{\cos z0 \cdot 1}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin z0}{\color{blue}{\cos z0 \cdot 1}}} \]

   8. *-rgt-identity N/A

      \[\leadsto \frac{1}{\frac{\sin z0}{\color{blue}{\cos z0}}} \]

   9. lift-cos.f64 N/A

      \[\leadsto \frac{1}{\frac{\sin z0}{\color{blue}{\cos z0}}} \]

   10. tan-quot N/A

       \[\leadsto \frac{1}{\color{blue}{\tan z0}} \]

   11. lift-tan.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\tan z0}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{\left|1\right|}}{\tan z0} \]

   13. rem-exp-log N/A

       \[\leadsto \frac{\left|1\right|}{\color{blue}{e^{\log \tan z0}}} \]

   14. lift-log.f64 N/A

       \[\leadsto \frac{\left|1\right|}{e^{\color{blue}{\log \tan z0}}} \]

   15. exp-fabs N/A

       \[\leadsto \frac{\left|1\right|}{\color{blue}{\left|e^{\log \tan z0}\right|}} \]

   16. lift-log.f64 N/A

       \[\leadsto \frac{\left|1\right|}{\left|e^{\color{blue}{\log \tan z0}}\right|} \]

   17. rem-exp-log N/A

       \[\leadsto \frac{\left|1\right|}{\left|\color{blue}{\tan z0}\right|} \]

   18. fabs-div N/A

       \[\leadsto \color{blue}{\left|\frac{1}{\tan z0}\right|} \]

   19. lift-/.f64 N/A

       \[\leadsto \left|\color{blue}{\frac{1}{\tan z0}}\right| \]

   20. lower-fabs.f64 50.2%

       \[\leadsto \color{blue}{\left|\frac{1}{\tan z0}\right|} \]

   21. lift-/.f64 N/A

       \[\leadsto \left|\color{blue}{\frac{1}{\tan z0}}\right| \]

   22. lift-tan.f64 N/A

       \[\leadsto \left|\frac{1}{\color{blue}{\tan z0}}\right| \]

   23. tan-+PI/2-rev N/A

       \[\leadsto \left|\color{blue}{\tan \left(\left(\mathsf{neg}\left(z0\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)}\right| \]

   24. lift-PI.f64 N/A

       \[\leadsto \left|\tan \left(\left(\mathsf{neg}\left(z0\right)\right) + \frac{\color{blue}{\pi}}{2}\right)\right| \]

   25. mult-flip N/A

       \[\leadsto \left|\tan \left(\left(\mathsf{neg}\left(z0\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)\right| \]

   26. metadata-eval N/A

       \[\leadsto \left|\tan \left(\left(\mathsf{neg}\left(z0\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right)\right| \]

   27. lift-*.f64 N/A

       \[\leadsto \left|\tan \left(\left(\mathsf{neg}\left(z0\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right)\right| \]

7. Applied rewrites 7.5%

   \[\leadsto \color{blue}{\left|\tan \left(\pi \cdot \frac{1}{2} - z0\right)\right|} \]
8. Add Preprocessing

Alternative 3: 5.5% accurate, 1.0× speedup

Math

\[\tan \left(\pi \cdot \frac{1}{2} - z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (tan (- (* PI 1/2) z0)))

C

double code(double z0) {
	return tan(((((double) M_PI) * 0.5) - z0));
}

Java

public static double code(double z0) {
	return Math.tan(((Math.PI * 0.5) - z0));
}

Python

def code(z0):
	return math.tan(((math.pi * 0.5) - z0))

Julia

function code(z0)
	return tan(Float64(Float64(pi * 0.5) - z0))
end

MATLAB

function tmp = code(z0)
	tmp = tan(((pi * 0.5) - z0));
end

Wolfram

code[z0_] := N[Tan[N[(N[(Pi * 1/2), $MachinePrecision] - z0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 5.5%

   \[\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \tan \color{blue}{\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)} \]

   2. +-commutative N/A

      \[\leadsto \tan \color{blue}{\left(\frac{1}{2} \cdot \pi + \left(-z0\right)\right)} \]

   3. add-flip N/A

      \[\leadsto \tan \color{blue}{\left(\frac{1}{2} \cdot \pi - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \tan \color{blue}{\left(\frac{1}{2} \cdot \pi - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \tan \left(\color{blue}{\frac{1}{2} \cdot \pi} - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right) \]

   6. *-commutative N/A

      \[\leadsto \tan \left(\color{blue}{\pi \cdot \frac{1}{2}} - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \tan \left(\color{blue}{\pi \cdot \frac{1}{2}} - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right) \]

   8. lift-neg.f64 N/A

      \[\leadsto \tan \left(\pi \cdot \frac{1}{2} - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z0\right)\right)}\right)\right)\right) \]

   9. remove-double-neg 5.5%

      \[\leadsto \tan \left(\pi \cdot \frac{1}{2} - \color{blue}{z0}\right) \]

3. Applied rewrites 5.5%

   \[\leadsto \tan \color{blue}{\left(\pi \cdot \frac{1}{2} - z0\right)} \]
4. Add Preprocessing

Alternative 4: 5.4% accurate, 1.0× speedup

Math

\[\tan \left(\frac{3}{2} \cdot \pi - z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (tan (- (* 3/2 PI) z0)))

C

double code(double z0) {
	return tan(((1.5 * ((double) M_PI)) - z0));
}

Java

public static double code(double z0) {
	return Math.tan(((1.5 * Math.PI) - z0));
}

Python

def code(z0):
	return math.tan(((1.5 * math.pi) - z0))

Julia

function code(z0)
	return tan(Float64(Float64(1.5 * pi) - z0))
end

MATLAB

function tmp = code(z0)
	tmp = tan(((1.5 * pi) - z0));
end

Wolfram

code[z0_] := N[Tan[N[(N[(3/2 * Pi), $MachinePrecision] - z0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 5.5%

   \[\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) \]
2. Step-by-step derivation

   1. lift-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)} \]

   2. tan-+PI-rev N/A

      \[\leadsto \color{blue}{\tan \left(\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) + \mathsf{PI}\left(\right)\right)} \]

   3. lower-tan.f64 N/A

      \[\leadsto \color{blue}{\tan \left(\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) + \mathsf{PI}\left(\right)\right)} \]

   4. lift-PI.f64 N/A

      \[\leadsto \tan \left(\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right) + \color{blue}{\pi}\right) \]

   5. +-commutative N/A

      \[\leadsto \tan \color{blue}{\left(\pi + \left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)\right)} \]

   6. lift-+.f64 N/A

      \[\leadsto \tan \left(\pi + \color{blue}{\left(\left(-z0\right) + \frac{1}{2} \cdot \pi\right)}\right) \]

   7. +-commutative N/A

      \[\leadsto \tan \left(\pi + \color{blue}{\left(\frac{1}{2} \cdot \pi + \left(-z0\right)\right)}\right) \]

   8. add-flip N/A

      \[\leadsto \tan \left(\pi + \color{blue}{\left(\frac{1}{2} \cdot \pi - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right)}\right) \]

   9. associate-+r- N/A

      \[\leadsto \tan \color{blue}{\left(\left(\pi + \frac{1}{2} \cdot \pi\right) - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right)} \]

   10. +-commutative N/A

       \[\leadsto \tan \left(\color{blue}{\left(\frac{1}{2} \cdot \pi + \pi\right)} - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right) \]

   11. lower--.f64 N/A

       \[\leadsto \tan \color{blue}{\left(\left(\frac{1}{2} \cdot \pi + \pi\right) - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \tan \left(\left(\color{blue}{\frac{1}{2} \cdot \pi} + \pi\right) - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right) \]

   13. distribute-lft1-in N/A

       \[\leadsto \tan \left(\color{blue}{\left(\frac{1}{2} + 1\right) \cdot \pi} - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \tan \left(\color{blue}{\left(\frac{1}{2} + 1\right) \cdot \pi} - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \tan \left(\color{blue}{\frac{3}{2}} \cdot \pi - \left(\mathsf{neg}\left(\left(-z0\right)\right)\right)\right) \]

   16. lift-neg.f64 N/A

       \[\leadsto \tan \left(\frac{3}{2} \cdot \pi - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(z0\right)\right)}\right)\right)\right) \]

   17. remove-double-neg 5.4%

       \[\leadsto \tan \left(\frac{3}{2} \cdot \pi - \color{blue}{z0}\right) \]

3. Applied rewrites 5.4%

   \[\leadsto \color{blue}{\tan \left(\frac{3}{2} \cdot \pi - z0\right)} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0)
  :name "(tan (+ (- z0) (* 1/2 PI)))"
  :precision binary64
  (tan (+ (- z0) (* 1/2 PI))))