(sin (* -2 (* PI z0)))

Percentage Accurate: 53.4% → 53.4%

Time: 3.7s

Alternatives: 5

Speedup: 0.3×

Specification

Math

\[\sin \left(-2 \cdot \left(\pi \cdot z0\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* -2.0 (* PI z0))))

C

double code(double z0) {
	return sin((-2.0 * (((double) M_PI) * z0)));
}

Java

public static double code(double z0) {
	return Math.sin((-2.0 * (Math.PI * z0)));
}

Python

def code(z0):
	return math.sin((-2.0 * (math.pi * z0)))

Julia

function code(z0)
	return sin(Float64(-2.0 * Float64(pi * z0)))
end

MATLAB

function tmp = code(z0)
	tmp = sin((-2.0 * (pi * z0)));
end

Wolfram

code[z0_] := N[Sin[N[(-2.0 * N[(Pi * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(-2 \cdot \left(\pi \cdot z0\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* -2.0 (* PI z0))))

C

double code(double z0) {
	return sin((-2.0 * (((double) M_PI) * z0)));
}

Java

public static double code(double z0) {
	return Math.sin((-2.0 * (Math.PI * z0)));
}

Python

def code(z0):
	return math.sin((-2.0 * (math.pi * z0)))

Julia

function code(z0)
	return sin(Float64(-2.0 * Float64(pi * z0)))
end

MATLAB

function tmp = code(z0)
	tmp = sin((-2.0 * (pi * z0)));
end

Wolfram

code[z0_] := N[Sin[N[(-2.0 * N[(Pi * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 53.4% accurate, 0.3× speedup

Math

\[\sin \left({\pi}^{0.5555555555555556} \cdot \left(\left({\pi}^{0.1111111111111111} \cdot -2.9291837751230467\right) \cdot z0\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin
 (*
  (pow PI 0.5555555555555556)
  (* (* (pow PI 0.1111111111111111) -2.9291837751230467) z0))))

C

double code(double z0) {
	return sin((pow(((double) M_PI), 0.5555555555555556) * ((pow(((double) M_PI), 0.1111111111111111) * -2.9291837751230467) * z0)));
}

Java

public static double code(double z0) {
	return Math.sin((Math.pow(Math.PI, 0.5555555555555556) * ((Math.pow(Math.PI, 0.1111111111111111) * -2.9291837751230467) * z0)));
}

Python

def code(z0):
	return math.sin((math.pow(math.pi, 0.5555555555555556) * ((math.pow(math.pi, 0.1111111111111111) * -2.9291837751230467) * z0)))

Julia

function code(z0)
	return sin(Float64((pi ^ 0.5555555555555556) * Float64(Float64((pi ^ 0.1111111111111111) * -2.9291837751230467) * z0)))
end

MATLAB

function tmp = code(z0)
	tmp = sin(((pi ^ 0.5555555555555556) * (((pi ^ 0.1111111111111111) * -2.9291837751230467) * z0)));
end

Wolfram

code[z0_] := N[Sin[N[(N[Power[Pi, 0.5555555555555556], $MachinePrecision] * N[(N[(N[Power[Pi, 0.1111111111111111], $MachinePrecision] * -2.9291837751230467), $MachinePrecision] * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\sin \left(-2 \cdot \left(\pi \cdot z0\right)\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

      \[\leadsto \sin \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right) \]

   5. add-cube-cbrt N/A

      \[\leadsto \sin \left(\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot z0\right) \cdot -2\right) \]

   6. associate-*l* N/A

      \[\leadsto \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right)\right)} \cdot -2\right) \]

   7. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right)} \]

   9. lift-PI.f64 N/A

      \[\leadsto \sin \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   10. pow1/3 N/A

       \[\leadsto \sin \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   11. lift-PI.f64 N/A

       \[\leadsto \sin \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   12. pow1/3 N/A

       \[\leadsto \sin \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   13. pow-prod-up N/A

       \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   14. lower-pow.f64 N/A

       \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right)} \cdot -2\right)\right) \]

   18. lift-PI.f64 N/A

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

   19. lower-cbrt.f64 52.9%

       \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\left(\color{blue}{\sqrt[3]{\pi}} \cdot z0\right) \cdot -2\right)\right) \]

3. Applied rewrites 52.9%

   \[\leadsto \sin \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\left(\sqrt[3]{\pi} \cdot z0\right) \cdot -2\right)\right)} \]

4. Evaluated real constant 53.4%

   \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\left(\color{blue}{1.4645918875615234} \cdot z0\right) \cdot -2\right)\right) \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \sin \left(\color{blue}{{\pi}^{\frac{2}{3}}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \sin \left({\pi}^{\color{blue}{\left(\frac{2}{3}\right)}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   3. pow-cbrt N/A

      \[\leadsto \sin \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   4. lift-cbrt.f64 N/A

      \[\leadsto \sin \left({\color{blue}{\left(\sqrt[3]{\pi}\right)}}^{2} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   5. lift-pow.f64 52.7%

      \[\leadsto \sin \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \left(\left(1.4645918875615234 \cdot z0\right) \cdot -2\right)\right) \]

6. Applied rewrites 52.7%

   \[\leadsto \sin \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \left(\left(1.4645918875615234 \cdot z0\right) \cdot -2\right)\right) \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left({\left(\sqrt[3]{\pi}\right)}^{2} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)} \]

   2. lift-pow.f64 N/A

      \[\leadsto \sin \left(\color{blue}{{\left(\sqrt[3]{\pi}\right)}^{2}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   3. lift-cbrt.f64 N/A

      \[\leadsto \sin \left({\color{blue}{\left(\sqrt[3]{\pi}\right)}}^{2} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   4. pow-cbrt N/A

      \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{2}{3}\right)}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   6. metadata-eval N/A

      \[\leadsto \sin \left({\pi}^{\color{blue}{\left(\frac{5}{9} + \frac{1}{9}\right)}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \sin \left({\pi}^{\left(\color{blue}{\frac{\frac{5}{3}}{3}} + \frac{1}{9}\right)} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   8. pow-prod-up N/A

      \[\leadsto \sin \left(\color{blue}{\left({\pi}^{\left(\frac{\frac{5}{3}}{3}\right)} \cdot {\pi}^{\frac{1}{9}}\right)} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   9. cbrt-pow N/A

      \[\leadsto \sin \left(\left(\color{blue}{\sqrt[3]{{\pi}^{\frac{5}{3}}}} \cdot {\pi}^{\frac{1}{9}}\right) \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   10. lift-pow.f64 N/A

       \[\leadsto \sin \left(\left(\sqrt[3]{\color{blue}{{\pi}^{\frac{5}{3}}}} \cdot {\pi}^{\frac{1}{9}}\right) \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   11. lift-cbrt.f64 N/A

       \[\leadsto \sin \left(\left(\color{blue}{\sqrt[3]{{\pi}^{\frac{5}{3}}}} \cdot {\pi}^{\frac{1}{9}}\right) \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   12. lift-pow.f64 N/A

       \[\leadsto \sin \left(\left(\sqrt[3]{{\pi}^{\frac{5}{3}}} \cdot \color{blue}{{\pi}^{\frac{1}{9}}}\right) \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right) \]

   13. associate-*l* N/A

       \[\leadsto \sin \color{blue}{\left(\sqrt[3]{{\pi}^{\frac{5}{3}}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)\right)} \]

   14. lower-*.f64 N/A

       \[\leadsto \sin \color{blue}{\left(\sqrt[3]{{\pi}^{\frac{5}{3}}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)\right)} \]

   15. lift-cbrt.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\sqrt[3]{{\pi}^{\frac{5}{3}}}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)\right) \]

   16. lift-pow.f64 N/A

       \[\leadsto \sin \left(\sqrt[3]{\color{blue}{{\pi}^{\frac{5}{3}}}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)\right) \]

   17. cbrt-pow N/A

       \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{\frac{5}{3}}{3}\right)}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)\right) \]

   18. lower-pow.f64 N/A

       \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{\frac{5}{3}}{3}\right)}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)\right) \]

   19. metadata-eval N/A

       \[\leadsto \sin \left({\pi}^{\color{blue}{\frac{5}{9}}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)\right) \]

   20. lift-*.f64 N/A

       \[\leadsto \sin \left({\pi}^{\frac{5}{9}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)}\right)\right) \]

   21. *-commutative N/A

       \[\leadsto \sin \left({\pi}^{\frac{5}{9}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \color{blue}{\left(-2 \cdot \left(\frac{824491934883991}{562949953421312} \cdot z0\right)\right)}\right)\right) \]

   22. lift-*.f64 N/A

       \[\leadsto \sin \left({\pi}^{\frac{5}{9}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \left(-2 \cdot \color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot z0\right)}\right)\right)\right) \]

   23. associate-*r* N/A

       \[\leadsto \sin \left({\pi}^{\frac{5}{9}} \cdot \left({\pi}^{\frac{1}{9}} \cdot \color{blue}{\left(\left(-2 \cdot \frac{824491934883991}{562949953421312}\right) \cdot z0\right)}\right)\right) \]

8. Applied rewrites 53.4%

   \[\leadsto \sin \color{blue}{\left({\pi}^{0.5555555555555556} \cdot \left(\left({\pi}^{0.1111111111111111} \cdot -2.9291837751230467\right) \cdot z0\right)\right)} \]
9. Add Preprocessing

Alternative 2: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(\left(z0 \cdot 1.4645918875615234\right) \cdot -4.290058794222051\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (* z0 1.4645918875615234) -4.290058794222051)))

C

double code(double z0) {
	return sin(((z0 * 1.4645918875615234) * -4.290058794222051));
}

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 = sin(((z0 * 1.4645918875615234d0) * (-4.290058794222051d0)))
end function

Java

public static double code(double z0) {
	return Math.sin(((z0 * 1.4645918875615234) * -4.290058794222051));
}

Python

def code(z0):
	return math.sin(((z0 * 1.4645918875615234) * -4.290058794222051))

Julia

function code(z0)
	return sin(Float64(Float64(z0 * 1.4645918875615234) * -4.290058794222051))
end

MATLAB

function tmp = code(z0)
	tmp = sin(((z0 * 1.4645918875615234) * -4.290058794222051));
end

Wolfram

code[z0_] := N[Sin[N[(N[(z0 * 1.4645918875615234), $MachinePrecision] * -4.290058794222051), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\sin \left(-2 \cdot \left(\pi \cdot z0\right)\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

      \[\leadsto \sin \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right) \]

   5. add-cube-cbrt N/A

      \[\leadsto \sin \left(\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot z0\right) \cdot -2\right) \]

   6. associate-*l* N/A

      \[\leadsto \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right)\right)} \cdot -2\right) \]

   7. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right)} \]

   9. lift-PI.f64 N/A

      \[\leadsto \sin \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   10. pow1/3 N/A

       \[\leadsto \sin \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   11. lift-PI.f64 N/A

       \[\leadsto \sin \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   12. pow1/3 N/A

       \[\leadsto \sin \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   13. pow-prod-up N/A

       \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   14. lower-pow.f64 N/A

       \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right)} \cdot -2\right)\right) \]

   18. lift-PI.f64 N/A

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

   19. lower-cbrt.f64 52.9%

       \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\left(\color{blue}{\sqrt[3]{\pi}} \cdot z0\right) \cdot -2\right)\right) \]

3. Applied rewrites 52.9%

   \[\leadsto \sin \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\left(\sqrt[3]{\pi} \cdot z0\right) \cdot -2\right)\right)} \]

4. Evaluated real constant 53.4%

   \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\left(\color{blue}{1.4645918875615234} \cdot z0\right) \cdot -2\right)\right) \]

5. Evaluated real constant 53.4%

   \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(\left(1.4645918875615234 \cdot z0\right) \cdot -2\right)\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(\left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right) \cdot \frac{4830176796763987}{2251799813685248}\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)} \cdot \frac{4830176796763987}{2251799813685248}\right) \]

   4. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot \left(-2 \cdot \frac{4830176796763987}{2251799813685248}\right)\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot \left(-2 \cdot \frac{4830176796763987}{2251799813685248}\right)\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot z0\right)} \cdot \left(-2 \cdot \frac{4830176796763987}{2251799813685248}\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \sin \left(\color{blue}{\left(z0 \cdot \frac{824491934883991}{562949953421312}\right)} \cdot \left(-2 \cdot \frac{4830176796763987}{2251799813685248}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(z0 \cdot \frac{824491934883991}{562949953421312}\right)} \cdot \left(-2 \cdot \frac{4830176796763987}{2251799813685248}\right)\right) \]

   9. metadata-eval 53.4%

      \[\leadsto \sin \left(\left(z0 \cdot 1.4645918875615234\right) \cdot \color{blue}{-4.290058794222051}\right) \]

7. Applied rewrites 53.4%

   \[\leadsto \sin \color{blue}{\left(\left(z0 \cdot 1.4645918875615234\right) \cdot -4.290058794222051\right)} \]
8. Add Preprocessing

Alternative 3: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(z0 \cdot -6.283185307179586\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* z0 -6.283185307179586)))

C

double code(double z0) {
	return sin((z0 * -6.283185307179586));
}

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 = sin((z0 * (-6.283185307179586d0)))
end function

Java

public static double code(double z0) {
	return Math.sin((z0 * -6.283185307179586));
}

Python

def code(z0):
	return math.sin((z0 * -6.283185307179586))

Julia

function code(z0)
	return sin(Float64(z0 * -6.283185307179586))
end

MATLAB

function tmp = code(z0)
	tmp = sin((z0 * -6.283185307179586));
end

Wolfram

code[z0_] := N[Sin[N[(z0 * -6.283185307179586), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\sin \left(-2 \cdot \left(\pi \cdot z0\right)\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

      \[\leadsto \sin \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right) \]

   5. add-cube-cbrt N/A

      \[\leadsto \sin \left(\left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot z0\right) \cdot -2\right) \]

   6. associate-*l* N/A

      \[\leadsto \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right)\right)} \cdot -2\right) \]

   7. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right)} \]

   9. lift-PI.f64 N/A

      \[\leadsto \sin \left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   10. pow1/3 N/A

       \[\leadsto \sin \left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   11. lift-PI.f64 N/A

       \[\leadsto \sin \left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   12. pow1/3 N/A

       \[\leadsto \sin \left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   13. pow-prod-up N/A

       \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   14. lower-pow.f64 N/A

       \[\leadsto \sin \left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \sin \left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \sin \left({\pi}^{\frac{2}{3}} \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right) \cdot -2\right)}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \sin \left({\pi}^{\frac{2}{3}} \cdot \left(\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z0\right)} \cdot -2\right)\right) \]

   18. lift-PI.f64 N/A

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

   19. lower-cbrt.f64 52.9%

       \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\left(\color{blue}{\sqrt[3]{\pi}} \cdot z0\right) \cdot -2\right)\right) \]

3. Applied rewrites 52.9%

   \[\leadsto \sin \color{blue}{\left({\pi}^{0.6666666666666666} \cdot \left(\left(\sqrt[3]{\pi} \cdot z0\right) \cdot -2\right)\right)} \]

4. Evaluated real constant 53.4%

   \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\left(\color{blue}{1.4645918875615234} \cdot z0\right) \cdot -2\right)\right) \]

5. Evaluated real constant 53.4%

   \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(\left(1.4645918875615234 \cdot z0\right) \cdot -2\right)\right) \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{4830176796763987}{2251799813685248} \cdot \left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{4830176796763987}{2251799813685248} \cdot \color{blue}{\left(\left(\frac{824491934883991}{562949953421312} \cdot z0\right) \cdot -2\right)}\right) \]

   3. *-commutative N/A

      \[\leadsto \sin \left(\frac{4830176796763987}{2251799813685248} \cdot \color{blue}{\left(-2 \cdot \left(\frac{824491934883991}{562949953421312} \cdot z0\right)\right)}\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{4830176796763987}{2251799813685248} \cdot \left(-2 \cdot \color{blue}{\left(\frac{824491934883991}{562949953421312} \cdot z0\right)}\right)\right) \]

   5. associate-*r* N/A

      \[\leadsto \sin \left(\frac{4830176796763987}{2251799813685248} \cdot \color{blue}{\left(\left(-2 \cdot \frac{824491934883991}{562949953421312}\right) \cdot z0\right)}\right) \]

   6. metadata-eval N/A

      \[\leadsto \sin \left(\frac{4830176796763987}{2251799813685248} \cdot \left(\color{blue}{\frac{-824491934883991}{281474976710656}} \cdot z0\right)\right) \]

   7. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\frac{4830176796763987}{2251799813685248} \cdot \frac{-824491934883991}{281474976710656}\right) \cdot z0\right)} \]

   8. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(z0 \cdot \left(\frac{4830176796763987}{2251799813685248} \cdot \frac{-824491934883991}{281474976710656}\right)\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(z0 \cdot \left(\frac{4830176796763987}{2251799813685248} \cdot \frac{-824491934883991}{281474976710656}\right)\right)} \]

   10. metadata-eval 53.4%

       \[\leadsto \sin \left(z0 \cdot \color{blue}{-6.283185307179586}\right) \]

7. Applied rewrites 53.4%

   \[\leadsto \sin \color{blue}{\left(z0 \cdot -6.283185307179586\right)} \]
8. Add Preprocessing

Alternative 4: 51.1% accurate, 3.3× speedup

Math

\[\left(\pi \cdot \left(\left(\left(\left(z0 \cdot z0\right) \cdot 1.3333333333333333\right) \cdot \pi\right) \cdot \pi + -2\right)\right) \cdot z0 \]

FPCore

(FPCore (z0)
  :precision binary64
  (* (* PI (+ (* (* (* (* z0 z0) 1.3333333333333333) PI) PI) -2.0)) z0))

C

double code(double z0) {
	return (((double) M_PI) * (((((z0 * z0) * 1.3333333333333333) * ((double) M_PI)) * ((double) M_PI)) + -2.0)) * z0;
}

Java

public static double code(double z0) {
	return (Math.PI * (((((z0 * z0) * 1.3333333333333333) * Math.PI) * Math.PI) + -2.0)) * z0;
}

Python

def code(z0):
	return (math.pi * (((((z0 * z0) * 1.3333333333333333) * math.pi) * math.pi) + -2.0)) * z0

Julia

function code(z0)
	return Float64(Float64(pi * Float64(Float64(Float64(Float64(Float64(z0 * z0) * 1.3333333333333333) * pi) * pi) + -2.0)) * z0)
end

MATLAB

function tmp = code(z0)
	tmp = (pi * (((((z0 * z0) * 1.3333333333333333) * pi) * pi) + -2.0)) * z0;
end

Wolfram

code[z0_] := N[(N[(Pi * N[(N[(N[(N[(N[(z0 * z0), $MachinePrecision] * 1.3333333333333333), $MachinePrecision] * Pi), $MachinePrecision] * Pi), $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\sin \left(-2 \cdot \left(\pi \cdot z0\right)\right) \]

2. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto z0 \cdot \color{blue}{\left(-2 \cdot \mathsf{PI}\left(\right) + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{4}{3}} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{3}}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}\right)\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}\right)\right) \]

   9. lower-PI.f64 51.1%

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + 1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{z0 \cdot \left(-2 \cdot \pi + 1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto z0 \cdot \color{blue}{\left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \cdot \color{blue}{z0} \]

   3. lower-*.f64 51.1%

      \[\leadsto \left(-2 \cdot \pi + 1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \cdot \color{blue}{z0} \]

6. Applied rewrites 51.1%

   \[\leadsto \left(\pi \cdot \left(\left(\left(\left(z0 \cdot z0\right) \cdot 1.3333333333333333\right) \cdot \pi\right) \cdot \pi + -2\right)\right) \cdot \color{blue}{z0} \]
7. Add Preprocessing

Alternative 5: 51.1% accurate, 18.5× speedup

Math

\[z0 \cdot -6.283185307179586 \]

FPCore

(FPCore (z0)
  :precision binary64
  (* z0 -6.283185307179586))

C

double code(double z0) {
	return z0 * -6.283185307179586;
}

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 = z0 * (-6.283185307179586d0)
end function

Java

public static double code(double z0) {
	return z0 * -6.283185307179586;
}

Python

def code(z0):
	return z0 * -6.283185307179586

Julia

function code(z0)
	return Float64(z0 * -6.283185307179586)
end

MATLAB

function tmp = code(z0)
	tmp = z0 * -6.283185307179586;
end

Wolfram

code[z0_] := N[(z0 * -6.283185307179586), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\sin \left(-2 \cdot \left(\pi \cdot z0\right)\right) \]

2. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto z0 \cdot \color{blue}{\left(-2 \cdot \mathsf{PI}\left(\right) + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right) \]

   3. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \mathsf{PI}\left(\right) + \color{blue}{\frac{4}{3}} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \]

   4. lower-PI.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \color{blue}{\left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{3}\right)}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{3}}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{3}\right)\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + \frac{4}{3} \cdot \left({z0}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{3}}\right)\right) \]

   9. lower-PI.f64 51.1%

      \[\leadsto z0 \cdot \left(-2 \cdot \pi + 1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right) \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{z0 \cdot \left(-2 \cdot \pi + 1.3333333333333333 \cdot \left({z0}^{2} \cdot {\pi}^{3}\right)\right)} \]

5. Taylor expanded in z0 around 0

   \[\leadsto z0 \cdot \left(-2 \cdot \color{blue}{\pi}\right) \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto z0 \cdot \left(-2 \cdot \mathsf{PI}\left(\right)\right) \]

   2. lower-PI.f64 51.1%

      \[\leadsto z0 \cdot \left(-2 \cdot \pi\right) \]

7. Applied rewrites 51.1%

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

8. Evaluated real constant 51.1%

   \[\leadsto z0 \cdot -6.283185307179586 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(sin (* -2 (* PI z0)))"
  :precision binary64
  (sin (* -2.0 (* PI z0))))