(sin (* (+ PI PI) z0))

Percentage Accurate: 53.4% → 98.7%

Time: 3.8s

Alternatives: 4

Speedup: 1.0×

Specification

Math

\[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (+ PI PI) z0)))

C

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

Java

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

Python

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

Julia

function code(z0)
	return sin(Float64(Float64(pi + pi) * z0))
end

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (+ PI PI) z0)))

C

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

Java

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

Python

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

Julia

function code(z0)
	return sin(Float64(Float64(pi + pi) * z0))
end

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 28000000000000:\\ \;\;\;\;\sin \left(2.1450293971110255 \cdot \left(2.9291837751230467 \cdot \left|z0\right|\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\cos \left(\pi \cdot \left(\left|z0\right| - 0.5\right)\right) - \cos \left(\left(0.5 + \left|z0\right|\right) \cdot \pi\right)\right) \cdot 1\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (copysign 1.0 z0)
 (if (<= (fabs z0) 28000000000000.0)
   (sin (* 2.1450293971110255 (* 2.9291837751230467 (fabs z0))))
   (*
    (- (cos (* PI (- (fabs z0) 0.5))) (cos (* (+ 0.5 (fabs z0)) PI)))
    1.0))))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 28000000000000.0) {
		tmp = sin((2.1450293971110255 * (2.9291837751230467 * fabs(z0))));
	} else {
		tmp = (cos((((double) M_PI) * (fabs(z0) - 0.5))) - cos(((0.5 + fabs(z0)) * ((double) M_PI)))) * 1.0;
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 28000000000000.0) {
		tmp = Math.sin((2.1450293971110255 * (2.9291837751230467 * Math.abs(z0))));
	} else {
		tmp = (Math.cos((Math.PI * (Math.abs(z0) - 0.5))) - Math.cos(((0.5 + Math.abs(z0)) * Math.PI))) * 1.0;
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 28000000000000.0:
		tmp = math.sin((2.1450293971110255 * (2.9291837751230467 * math.fabs(z0))))
	else:
		tmp = (math.cos((math.pi * (math.fabs(z0) - 0.5))) - math.cos(((0.5 + math.fabs(z0)) * math.pi))) * 1.0
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 28000000000000.0)
		tmp = sin(Float64(2.1450293971110255 * Float64(2.9291837751230467 * abs(z0))));
	else
		tmp = Float64(Float64(cos(Float64(pi * Float64(abs(z0) - 0.5))) - cos(Float64(Float64(0.5 + abs(z0)) * pi))) * 1.0);
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 28000000000000.0)
		tmp = sin((2.1450293971110255 * (2.9291837751230467 * abs(z0))));
	else
		tmp = (cos((pi * (abs(z0) - 0.5))) - cos(((0.5 + abs(z0)) * pi))) * 1.0;
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 28000000000000.0], N[Sin[N[(2.1450293971110255 * N[(2.9291837751230467 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[(N[Cos[N[(Pi * N[(N[Abs[z0], $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Cos[N[(N[(0.5 + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z0 < 2.8e13

   1. Initial program 53.4%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. add-cube-cbrt N/A

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

      5. lift-PI.f64 N/A

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

      6. add-cube-cbrt N/A

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

      7. distribute-lft-out N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow1/3 N/A

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

      12. lift-PI.f64 N/A

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

      13. pow1/3 N/A

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

      14. pow-prod-up N/A

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

      15. lower-pow.f64 N/A

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

      16. metadata-eval N/A

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

      17. lower-*.f64 N/A

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

   3. Applied rewrites 52.9%

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

   4. Evaluated real constant 53.4%

      \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{2.9291837751230467} \cdot z0\right)\right) \]

   5. Evaluated real constant 53.4%

      \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(2.9291837751230467 \cdot z0\right)\right) \]

   if 2.8e13 < z0

   1. Initial program 53.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l* N/A

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

      6. sin-2 N/A

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

      7. lower-*.f64 N/A

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

      8. *-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-*.f64 53.4%

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

   3. Applied rewrites 53.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. cos-neg-rev N/A

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

      5. sin-+PI/2-rev N/A

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

      6. lift-sin.f64 N/A

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

      7. sin-mult N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 7.1%

      \[\leadsto 2 \cdot \color{blue}{\frac{\cos \left(\pi \cdot \left(\left(-z0\right) + 0.5\right) - \pi \cdot z0\right) - \cos \left(\pi \cdot \left(\left(-z0\right) + 0.5\right) + \pi \cdot z0\right)}{2}} \]

   6. Taylor expanded in z0 around 0

      \[\leadsto 2 \cdot \frac{\cos \left(\pi \cdot \color{blue}{\frac{1}{2}} - \pi \cdot z0\right) - \cos \left(\pi \cdot \left(\left(-z0\right) + 0.5\right) + \pi \cdot z0\right)}{2} \]
   7. Step-by-step derivation

   8. Applied rewrites 4.8%

      \[\leadsto 2 \cdot \frac{\cos \left(\pi \cdot \color{blue}{0.5} - \pi \cdot z0\right) - \cos \left(\pi \cdot \left(\left(-z0\right) + 0.5\right) + \pi \cdot z0\right)}{2} \]

   9. Taylor expanded in z0 around 0

      \[\leadsto 2 \cdot \frac{\cos \left(\pi \cdot 0.5 - \pi \cdot z0\right) - \cos \left(\pi \cdot \color{blue}{\frac{1}{2}} + \pi \cdot z0\right)}{2} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.7%

       \[\leadsto 2 \cdot \frac{\cos \left(\pi \cdot 0.5 - \pi \cdot z0\right) - \cos \left(\pi \cdot \color{blue}{0.5} + \pi \cdot z0\right)}{2} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-/.f64 N/A

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

       4. mult-flip N/A

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

       5. metadata-eval N/A

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

       6. associate-*l* N/A

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

   13. Applied rewrites 50.7%

       \[\leadsto \color{blue}{\left(\cos \left(\pi \cdot \left(z0 - 0.5\right)\right) - \cos \left(\left(0.5 + z0\right) \cdot \pi\right)\right) \cdot 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 53.4% accurate, 1.0× speedup

Math

\[\sin \left(2.1450293971110255 \cdot \left(2.9291837751230467 \cdot z0\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* 2.1450293971110255 (* 2.9291837751230467 z0))))

C

double code(double z0) {
	return sin((2.1450293971110255 * (2.9291837751230467 * 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 = sin((2.1450293971110255d0 * (2.9291837751230467d0 * z0)))
end function

Java

public static double code(double z0) {
	return Math.sin((2.1450293971110255 * (2.9291837751230467 * z0)));
}

Python

def code(z0):
	return math.sin((2.1450293971110255 * (2.9291837751230467 * z0)))

Julia

function code(z0)
	return sin(Float64(2.1450293971110255 * Float64(2.9291837751230467 * z0)))
end

MATLAB

function tmp = code(z0)
	tmp = sin((2.1450293971110255 * (2.9291837751230467 * z0)));
end

Wolfram

code[z0_] := N[Sin[N[(2.1450293971110255 * N[(2.9291837751230467 * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.4%

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

   1. lift-*.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. add-cube-cbrt N/A

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

   5. lift-PI.f64 N/A

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

   6. add-cube-cbrt N/A

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

   7. distribute-lft-out N/A

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

   8. associate-*l* N/A

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

   9. lower-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. pow1/3 N/A

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

   12. lift-PI.f64 N/A

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

   13. pow1/3 N/A

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

   14. pow-prod-up N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval N/A

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

   17. lower-*.f64 N/A

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

3. Applied rewrites 52.9%

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

4. Evaluated real constant 53.4%

   \[\leadsto \sin \left({\pi}^{0.6666666666666666} \cdot \left(\color{blue}{2.9291837751230467} \cdot z0\right)\right) \]

5. Evaluated real constant 53.4%

   \[\leadsto \sin \left(\color{blue}{2.1450293971110255} \cdot \left(2.9291837751230467 \cdot z0\right)\right) \]
6. Add Preprocessing

Alternative 3: 53.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double z0) {
	return sin((6.283185307179586 * 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 = sin((6.283185307179586d0 * z0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\sin \left(\left(\pi + \pi\right) \cdot z0\right) \]

2. Evaluated real constant 53.4%

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

Alternative 4: 51.1% accurate, 12.1× speedup

Math

\[\left(z0 + z0\right) \cdot \pi \]

FPCore

(FPCore (z0)
  :precision binary64
  (* (+ z0 z0) PI))

C

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

Java

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

Python

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

Julia

function code(z0)
	return Float64(Float64(z0 + z0) * pi)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2 N/A

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

   5. associate-*l* N/A

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

   6. sin-2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 53.4%

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

3. Applied rewrites 53.4%

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

4. Taylor expanded in z0 around 0

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-PI.f64 51.1%

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

6. Applied rewrites 51.1%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lift-*.f64 N/A

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

   5. lower-*.f64 51.1%

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

   6. lift-*.f64 N/A

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

   7. count-2-rev N/A

      \[\leadsto \left(z0 + z0\right) \cdot \pi \]

   8. lower-+.f64 51.1%

      \[\leadsto \left(z0 + z0\right) \cdot \pi \]

8. Applied rewrites 51.1%

   \[\leadsto \left(z0 + z0\right) \cdot \color{blue}{\pi} \]
9. Add Preprocessing

Reproduce

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