(sin (* PI (* z1 (- z0))))

Percentage Accurate: 52.1% → 96.1%

Time: 1.7s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\sin \left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (sin (* PI (* z1 (- z0)))))

C

double code(double z1, double z0) {
	return sin((((double) M_PI) * (z1 * -z0)));
}

Java

public static double code(double z1, double z0) {
	return Math.sin((Math.PI * (z1 * -z0)));
}

Python

def code(z1, z0):
	return math.sin((math.pi * (z1 * -z0)))

Julia

function code(z1, z0)
	return sin(Float64(pi * Float64(z1 * Float64(-z0))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = sin((pi * (z1 * -z0)));
end

Wolfram

code[z1_, z0_] := N[Sin[N[(Pi * N[(z1 * (-z0)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\sin \left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (sin (* PI (* z1 (- z0)))))

C

double code(double z1, double z0) {
	return sin((((double) M_PI) * (z1 * -z0)));
}

Java

public static double code(double z1, double z0) {
	return Math.sin((Math.PI * (z1 * -z0)));
}

Python

def code(z1, z0):
	return math.sin((math.pi * (z1 * -z0)))

Julia

function code(z1, z0)
	return sin(Float64(pi * Float64(z1 * Float64(-z0))))
end

MATLAB

function tmp = code(z1, z0)
	tmp = sin((pi * (z1 * -z0)));
end

Wolfram

code[z1_, z0_] := N[Sin[N[(Pi * N[(z1 * (-z0)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 96.1% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z1\right) \cdot \left(\mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\pi \cdot \left(\left|z1\right| \cdot \left(-\left|z0\right|\right)\right) \leq -2 \cdot 10^{+32}:\\ \;\;\;\;\sin \left(\left(-1.5707963267948966\right) - -0.5 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(-3.141592653589793 \cdot \left(\left|z0\right| \cdot \left|z1\right|\right)\right)\\ \end{array}\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (*
 (copysign 1.0 z1)
 (*
  (copysign 1.0 z0)
  (if (<= (* PI (* (fabs z1) (- (fabs z0)))) -2e+32)
    (sin (- (- 1.5707963267948966) (* -0.5 PI)))
    (sin (* -3.141592653589793 (* (fabs z0) (fabs z1))))))))

C

double code(double z1, double z0) {
	double tmp;
	if ((((double) M_PI) * (fabs(z1) * -fabs(z0))) <= -2e+32) {
		tmp = sin((-1.5707963267948966 - (-0.5 * ((double) M_PI))));
	} else {
		tmp = sin((-3.141592653589793 * (fabs(z0) * fabs(z1))));
	}
	return copysign(1.0, z1) * (copysign(1.0, z0) * tmp);
}

Java

public static double code(double z1, double z0) {
	double tmp;
	if ((Math.PI * (Math.abs(z1) * -Math.abs(z0))) <= -2e+32) {
		tmp = Math.sin((-1.5707963267948966 - (-0.5 * Math.PI)));
	} else {
		tmp = Math.sin((-3.141592653589793 * (Math.abs(z0) * Math.abs(z1))));
	}
	return Math.copySign(1.0, z1) * (Math.copySign(1.0, z0) * tmp);
}

Python

def code(z1, z0):
	tmp = 0
	if (math.pi * (math.fabs(z1) * -math.fabs(z0))) <= -2e+32:
		tmp = math.sin((-1.5707963267948966 - (-0.5 * math.pi)))
	else:
		tmp = math.sin((-3.141592653589793 * (math.fabs(z0) * math.fabs(z1))))
	return math.copysign(1.0, z1) * (math.copysign(1.0, z0) * tmp)

Julia

function code(z1, z0)
	tmp = 0.0
	if (Float64(pi * Float64(abs(z1) * Float64(-abs(z0)))) <= -2e+32)
		tmp = sin(Float64(Float64(-1.5707963267948966) - Float64(-0.5 * pi)));
	else
		tmp = sin(Float64(-3.141592653589793 * Float64(abs(z0) * abs(z1))));
	end
	return Float64(copysign(1.0, z1) * Float64(copysign(1.0, z0) * tmp))
end

MATLAB

function tmp_2 = code(z1, z0)
	tmp = 0.0;
	if ((pi * (abs(z1) * -abs(z0))) <= -2e+32)
		tmp = sin((-1.5707963267948966 - (-0.5 * pi)));
	else
		tmp = sin((-3.141592653589793 * (abs(z0) * abs(z1))));
	end
	tmp_2 = (sign(z1) * abs(1.0)) * ((sign(z0) * abs(1.0)) * tmp);
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(Pi * N[(N[Abs[z1], $MachinePrecision] * (-N[Abs[z0], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], -2e+32], N[Sin[N[((-1.5707963267948966) - N[(-0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sin[N[(-3.141592653589793 * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (PI.f64) (*.f64 z1 (neg.f64 z0))) < -2.0000000000000001e32

   1. Initial program 52.1%

      \[\sin \left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sin \left(\pi \cdot \color{blue}{\left(z1 \cdot \left(-z0\right)\right)}\right) \]

      4. associate-*r* N/A

         \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot z1\right) \cdot \left(-z0\right)\right)} \]

      5. lift-neg.f64 N/A

         \[\leadsto \sin \left(\left(\pi \cdot z1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z0\right)\right)}\right) \]

      6. distribute-rgt-neg-out N/A

         \[\leadsto \sin \color{blue}{\left(\mathsf{neg}\left(\left(\pi \cdot z1\right) \cdot z0\right)\right)} \]

      7. sin-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\sin \left(\left(\pi \cdot z1\right) \cdot z0\right)\right)} \]

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

         \[\leadsto \color{blue}{\cos \left(\left(\pi \cdot z1\right) \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      9. lower-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \left(\left(\pi \cdot z1\right) \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      10. lower-+.f64 N/A

          \[\leadsto \cos \color{blue}{\left(\left(\pi \cdot z1\right) \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \cos \left(\color{blue}{\left(\pi \cdot z1\right) \cdot z0} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      12. *-commutative N/A

          \[\leadsto \cos \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \cos \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \frac{\color{blue}{\pi}}{2}\right) \]

      15. metadata-eval N/A

          \[\leadsto \cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \frac{\pi}{\color{blue}{1 + 1}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \color{blue}{\frac{\pi}{1 + 1}}\right) \]

      17. metadata-eval 6.1%

          \[\leadsto \cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \frac{\pi}{\color{blue}{2}}\right) \]

   3. Applied rewrites 6.1%

      \[\leadsto \color{blue}{\cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \frac{\pi}{2}\right)} \]

   4. Taylor expanded in z1 around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \cos \left(\frac{1}{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

      2. lower-PI.f64 3.8%

         \[\leadsto \cos \left(0.5 \cdot \pi\right) \]

   6. Applied rewrites 3.8%

      \[\leadsto \cos \color{blue}{\left(0.5 \cdot \pi\right)} \]

   7. Evaluated real constant 3.8%

      \[\leadsto \cos 1.5707963267948966 \]
   8. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto \color{blue}{\cos \frac{884279719003555}{562949953421312}} \]

      2. cos-neg-rev N/A

         \[\leadsto \color{blue}{\cos \left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right)} \]

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

         \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      4. lower-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

      5. lift-PI.f64 N/A

         \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) + \frac{\color{blue}{\pi}}{2}\right) \]

      6. mult-flip N/A

         \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right) \]

      7. metadata-eval N/A

         \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right) \]

      8. *-commutative N/A

         \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right) \]

      9. fp-cancel-sign-sub-inv N/A

         \[\leadsto \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) - \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \pi\right)} \]

      10. metadata-eval N/A

          \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) - \color{blue}{\frac{-1}{2}} \cdot \pi\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \sin \left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) - \color{blue}{\frac{-1}{2} \cdot \pi}\right) \]

      12. lower--.f64 N/A

          \[\leadsto \sin \color{blue}{\left(\left(\mathsf{neg}\left(\frac{884279719003555}{562949953421312}\right)\right) - \frac{-1}{2} \cdot \pi\right)} \]

      13. lower-neg.f64 59.1%

          \[\leadsto \sin \left(\color{blue}{\left(-1.5707963267948966\right)} - -0.5 \cdot \pi\right) \]

   9. Applied rewrites 59.1%

      \[\leadsto \color{blue}{\sin \left(\left(-1.5707963267948966\right) - -0.5 \cdot \pi\right)} \]

   if -2.0000000000000001e32 < (*.f64 (PI.f64) (*.f64 z1 (neg.f64 z0)))

   1. Initial program 52.1%

      \[\sin \left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \color{blue}{\left(\left(z1 \cdot \left(-z0\right)\right) \cdot \pi\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(z1 \cdot \left(-z0\right)\right)} \cdot \pi\right) \]

      4. *-commutative N/A

         \[\leadsto \sin \left(\color{blue}{\left(\left(-z0\right) \cdot z1\right)} \cdot \pi\right) \]

      5. associate-*l* N/A

         \[\leadsto \sin \color{blue}{\left(\left(-z0\right) \cdot \left(z1 \cdot \pi\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto \sin \left(\left(-z0\right) \cdot \color{blue}{\left(\pi \cdot z1\right)}\right) \]

      7. lift-PI.f64 N/A

         \[\leadsto \sin \left(\left(-z0\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      8. add-cube-cbrt N/A

         \[\leadsto \sin \left(\left(-z0\right) \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot z1\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \sin \left(\left(-z0\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right)}\right) \]

      10. associate-*r* N/A

          \[\leadsto \sin \color{blue}{\left(\left(\left(-z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \sin \color{blue}{\left(\left(\left(-z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-z0\right)\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-z0\right)\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      14. lift-PI.f64 N/A

          \[\leadsto \sin \left(\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      15. pow1/3 N/A

          \[\leadsto \sin \left(\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      16. lift-PI.f64 N/A

          \[\leadsto \sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      17. pow1/3 N/A

          \[\leadsto \sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      18. pow-prod-up N/A

          \[\leadsto \sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      19. lower-pow.f64 N/A

          \[\leadsto \sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \sin \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   3. Applied rewrites 51.8%

      \[\leadsto \sin \color{blue}{\left(\left({\pi}^{0.6666666666666666} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot z1\right)\right)} \]

   4. Evaluated real constant 51.8%

      \[\leadsto \sin \left(\left(\color{blue}{2.1450293971110255} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot z1\right)\right) \]

   5. Evaluated real constant 52.2%

      \[\leadsto \sin \left(\left(2.1450293971110255 \cdot \left(-z0\right)\right) \cdot \left(\color{blue}{1.4645918875615234} \cdot z1\right)\right) \]

   6. Taylor expanded in z1 around 0

      \[\leadsto \sin \color{blue}{\left(\frac{-3982441812995697399929051632117}{1267650600228229401496703205376} \cdot \left(z0 \cdot z1\right)\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sin \left(\frac{-3982441812995697399929051632117}{1267650600228229401496703205376} \cdot \color{blue}{\left(z0 \cdot z1\right)}\right) \]

      2. lower-*.f64 52.1%

         \[\leadsto \sin \left(-3.141592653589793 \cdot \left(z0 \cdot \color{blue}{z1}\right)\right) \]

   8. Applied rewrites 52.1%

      \[\leadsto \sin \color{blue}{\left(-3.141592653589793 \cdot \left(z0 \cdot z1\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 52.1% accurate, 1.0× speedup

Math

\[\sin \left(-3.141592653589793 \cdot \left(z0 \cdot z1\right)\right) \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (sin (* -3.141592653589793 (* z0 z1))))

C

double code(double z1, double z0) {
	return sin((-3.141592653589793 * (z0 * z1)));
}

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(z1, z0)
use fmin_fmax_functions
    real(8), intent (in) :: z1
    real(8), intent (in) :: z0
    code = sin(((-3.141592653589793d0) * (z0 * z1)))
end function

Java

public static double code(double z1, double z0) {
	return Math.sin((-3.141592653589793 * (z0 * z1)));
}

Python

def code(z1, z0):
	return math.sin((-3.141592653589793 * (z0 * z1)))

Julia

function code(z1, z0)
	return sin(Float64(-3.141592653589793 * Float64(z0 * z1)))
end

MATLAB

function tmp = code(z1, z0)
	tmp = sin((-3.141592653589793 * (z0 * z1)));
end

Wolfram

code[z1_, z0_] := N[Sin[N[(-3.141592653589793 * N[(z0 * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\sin \left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \sin \color{blue}{\left(\left(z1 \cdot \left(-z0\right)\right) \cdot \pi\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\left(z1 \cdot \left(-z0\right)\right)} \cdot \pi\right) \]

   4. *-commutative N/A

      \[\leadsto \sin \left(\color{blue}{\left(\left(-z0\right) \cdot z1\right)} \cdot \pi\right) \]

   5. associate-*l* N/A

      \[\leadsto \sin \color{blue}{\left(\left(-z0\right) \cdot \left(z1 \cdot \pi\right)\right)} \]

   6. *-commutative N/A

      \[\leadsto \sin \left(\left(-z0\right) \cdot \color{blue}{\left(\pi \cdot z1\right)}\right) \]

   7. lift-PI.f64 N/A

      \[\leadsto \sin \left(\left(-z0\right) \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   8. add-cube-cbrt N/A

      \[\leadsto \sin \left(\left(-z0\right) \cdot \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot z1\right)\right) \]

   9. associate-*l* N/A

      \[\leadsto \sin \left(\left(-z0\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right)}\right) \]

   10. associate-*r* N/A

       \[\leadsto \sin \color{blue}{\left(\left(\left(-z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \sin \color{blue}{\left(\left(\left(-z0\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right)} \]

   12. *-commutative N/A

       \[\leadsto \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-z0\right)\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-z0\right)\right)} \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   14. lift-PI.f64 N/A

       \[\leadsto \sin \left(\left(\left(\sqrt[3]{\color{blue}{\pi}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   15. pow1/3 N/A

       \[\leadsto \sin \left(\left(\left(\color{blue}{{\pi}^{\frac{1}{3}}} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   16. lift-PI.f64 N/A

       \[\leadsto \sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \sqrt[3]{\color{blue}{\pi}}\right) \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   17. pow1/3 N/A

       \[\leadsto \sin \left(\left(\left({\pi}^{\frac{1}{3}} \cdot \color{blue}{{\pi}^{\frac{1}{3}}}\right) \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   18. pow-prod-up N/A

       \[\leadsto \sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   19. lower-pow.f64 N/A

       \[\leadsto \sin \left(\left(\color{blue}{{\pi}^{\left(\frac{1}{3} + \frac{1}{3}\right)}} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

   20. metadata-eval N/A

       \[\leadsto \sin \left(\left({\pi}^{\color{blue}{\frac{2}{3}}} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot z1\right)\right) \]

3. Applied rewrites 51.8%

   \[\leadsto \sin \color{blue}{\left(\left({\pi}^{0.6666666666666666} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot z1\right)\right)} \]

4. Evaluated real constant 51.8%

   \[\leadsto \sin \left(\left(\color{blue}{2.1450293971110255} \cdot \left(-z0\right)\right) \cdot \left(\sqrt[3]{\pi} \cdot z1\right)\right) \]

5. Evaluated real constant 52.2%

   \[\leadsto \sin \left(\left(2.1450293971110255 \cdot \left(-z0\right)\right) \cdot \left(\color{blue}{1.4645918875615234} \cdot z1\right)\right) \]

6. Taylor expanded in z1 around 0

   \[\leadsto \sin \color{blue}{\left(\frac{-3982441812995697399929051632117}{1267650600228229401496703205376} \cdot \left(z0 \cdot z1\right)\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \sin \left(\frac{-3982441812995697399929051632117}{1267650600228229401496703205376} \cdot \color{blue}{\left(z0 \cdot z1\right)}\right) \]

   2. lower-*.f64 52.1%

      \[\leadsto \sin \left(-3.141592653589793 \cdot \left(z0 \cdot \color{blue}{z1}\right)\right) \]

8. Applied rewrites 52.1%

   \[\leadsto \sin \color{blue}{\left(-3.141592653589793 \cdot \left(z0 \cdot z1\right)\right)} \]
9. Add Preprocessing

Alternative 3: 3.7% accurate, 0.5× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \cos 1.5707963267948966 \]

FPCore

(FPCore (z1 z0)
  :precision binary64
  (* (copysign 1.0 z0) (cos 1.5707963267948966)))

C

double code(double z1, double z0) {
	return copysign(1.0, z0) * cos(1.5707963267948966);
}

Java

public static double code(double z1, double z0) {
	return Math.copySign(1.0, z0) * Math.cos(1.5707963267948966);
}

Python

def code(z1, z0):
	return math.copysign(1.0, z0) * math.cos(1.5707963267948966)

Julia

function code(z1, z0)
	return Float64(copysign(1.0, z0) * cos(1.5707963267948966))
end

MATLAB

function tmp = code(z1, z0)
	tmp = (sign(z0) * abs(1.0)) * cos(1.5707963267948966);
end

Wolfram

code[z1_, z0_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[Cos[1.5707963267948966], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.1%

   \[\sin \left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right)} \]

   2. lift-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\pi \cdot \left(z1 \cdot \left(-z0\right)\right)\right)} \]

   3. lift-*.f64 N/A

      \[\leadsto \sin \left(\pi \cdot \color{blue}{\left(z1 \cdot \left(-z0\right)\right)}\right) \]

   4. associate-*r* N/A

      \[\leadsto \sin \color{blue}{\left(\left(\pi \cdot z1\right) \cdot \left(-z0\right)\right)} \]

   5. lift-neg.f64 N/A

      \[\leadsto \sin \left(\left(\pi \cdot z1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(z0\right)\right)}\right) \]

   6. distribute-rgt-neg-out N/A

      \[\leadsto \sin \color{blue}{\left(\mathsf{neg}\left(\left(\pi \cdot z1\right) \cdot z0\right)\right)} \]

   7. sin-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sin \left(\left(\pi \cdot z1\right) \cdot z0\right)\right)} \]

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

      \[\leadsto \color{blue}{\cos \left(\left(\pi \cdot z1\right) \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   9. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(\left(\pi \cdot z1\right) \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   10. lower-+.f64 N/A

       \[\leadsto \cos \color{blue}{\left(\left(\pi \cdot z1\right) \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto \cos \left(\color{blue}{\left(\pi \cdot z1\right) \cdot z0} + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   12. *-commutative N/A

       \[\leadsto \cos \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \cos \left(\color{blue}{\left(z1 \cdot \pi\right)} \cdot z0 + \frac{\mathsf{PI}\left(\right)}{2}\right) \]

   14. lift-PI.f64 N/A

       \[\leadsto \cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \frac{\color{blue}{\pi}}{2}\right) \]

   15. metadata-eval N/A

       \[\leadsto \cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \frac{\pi}{\color{blue}{1 + 1}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto \cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \color{blue}{\frac{\pi}{1 + 1}}\right) \]

   17. metadata-eval 6.1%

       \[\leadsto \cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \frac{\pi}{\color{blue}{2}}\right) \]

3. Applied rewrites 6.1%

   \[\leadsto \color{blue}{\cos \left(\left(z1 \cdot \pi\right) \cdot z0 + \frac{\pi}{2}\right)} \]

4. Taylor expanded in z1 around 0

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

   1. lower-*.f64 N/A

      \[\leadsto \cos \left(\frac{1}{2} \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \]

   2. lower-PI.f64 3.8%

      \[\leadsto \cos \left(0.5 \cdot \pi\right) \]

6. Applied rewrites 3.8%

   \[\leadsto \cos \color{blue}{\left(0.5 \cdot \pi\right)} \]

7. Evaluated real constant 3.8%

   \[\leadsto \cos 1.5707963267948966 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z1 z0)
  :name "(sin (* PI (* z1 (- z0))))"
  :precision binary64
  (sin (* PI (* z1 (- z0)))))