(sin (* z0 (- PI)))

Percentage Accurate: 53.5% → 98.6%

Time: 3.6s

Alternatives: 4

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \pi\\ \mathsf{copysign}\left(1, z0\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 4 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(\left|z0\right| \cdot -3.141592653589793\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(\sin \left(t\_0 - \pi\right) - \sin t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) PI)))
  (*
   (copysign 1.0 z0)
   (if (<= (fabs z0) 4e-5)
     (sin (* (fabs z0) -3.141592653589793))
     (* 0.5 (- (sin (- t_0 PI)) (sin t_0)))))))

C

double code(double z0) {
	double t_0 = fabs(z0) * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 4e-5) {
		tmp = sin((fabs(z0) * -3.141592653589793));
	} else {
		tmp = 0.5 * (sin((t_0 - ((double) M_PI))) - sin(t_0));
	}
	return copysign(1.0, z0) * tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 4e-5) {
		tmp = Math.sin((Math.abs(z0) * -3.141592653589793));
	} else {
		tmp = 0.5 * (Math.sin((t_0 - Math.PI)) - Math.sin(t_0));
	}
	return Math.copySign(1.0, z0) * tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) * math.pi
	tmp = 0
	if math.fabs(z0) <= 4e-5:
		tmp = math.sin((math.fabs(z0) * -3.141592653589793))
	else:
		tmp = 0.5 * (math.sin((t_0 - math.pi)) - math.sin(t_0))
	return math.copysign(1.0, z0) * tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) * pi)
	tmp = 0.0
	if (abs(z0) <= 4e-5)
		tmp = sin(Float64(abs(z0) * -3.141592653589793));
	else
		tmp = Float64(0.5 * Float64(sin(Float64(t_0 - pi)) - sin(t_0)));
	end
	return Float64(copysign(1.0, z0) * tmp)
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) * pi;
	tmp = 0.0;
	if (abs(z0) <= 4e-5)
		tmp = sin((abs(z0) * -3.141592653589793));
	else
		tmp = 0.5 * (sin((t_0 - pi)) - sin(t_0));
	end
	tmp_2 = (sign(z0) * abs(1.0)) * tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z0], $MachinePrecision], 4e-5], N[Sin[N[(N[Abs[z0], $MachinePrecision] * -3.141592653589793), $MachinePrecision]], $MachinePrecision], N[(0.5 * N[(N[Sin[N[(t$95$0 - Pi), $MachinePrecision]], $MachinePrecision] - N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 4.0000000000000003e-5

   1. Initial program 53.5%

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

   2. Evaluated real constant 53.5%

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

   if 4.0000000000000003e-5 < z0

   1. Initial program 53.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-neg.f64 N/A

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

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

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

      5. sin-neg N/A

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

      6. sin-+PI-rev N/A

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

      7. lower-sin.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 6.4%

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

   3. Applied rewrites 6.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. sin-+PI N/A

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

      5. mul-1-neg N/A

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

      6. cos-PI N/A

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

      7. lift-PI.f64 N/A

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

      8. cos-neg-rev N/A

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

      9. lift-neg.f64 N/A

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

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

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

      11. sin-mult N/A

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

      12. lower-/.f64 N/A

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

   5. Applied rewrites 52.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. sin-mult-rev N/A

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

      8. *-commutative N/A

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

   7. Applied rewrites 51.6%

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

   8. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \left(\sin \left(z0 \cdot \mathsf{PI}\left(\right) - \mathsf{PI}\left(\right)\right) - \sin \left(z0 \cdot \mathsf{PI}\left(\right)\right)\right) \]

      6. lower-PI.f64 N/A

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

      7. lower-PI.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-PI.f64 51.6%

          \[\leadsto 0.5 \cdot \left(\sin \left(z0 \cdot \pi - \pi\right) - \sin \left(z0 \cdot \pi\right)\right) \]

   10. Applied rewrites 51.6%

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

Alternative 2: 53.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (-3.141592653589793d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.5%

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

2. Evaluated real constant 53.5%

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

Alternative 3: 3.7% accurate, 1.0× speedup

Math

\[\cos \left(1.5707963267948966 - \pi\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (cos (- 1.5707963267948966 PI)))

C

double code(double z0) {
	return cos((1.5707963267948966 - ((double) M_PI)));
}

Java

public static double code(double z0) {
	return Math.cos((1.5707963267948966 - Math.PI));
}

Python

def code(z0):
	return math.cos((1.5707963267948966 - math.pi))

Julia

function code(z0)
	return cos(Float64(1.5707963267948966 - pi))
end

MATLAB

function tmp = code(z0)
	tmp = cos((1.5707963267948966 - pi));
end

Wolfram

code[z0_] := N[Cos[N[(1.5707963267948966 - Pi), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 53.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. sin-neg N/A

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

   6. sin-+PI-rev N/A

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

   7. lower-sin.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 6.4%

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

3. Applied rewrites 6.4%

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

   1. lift-sin.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. add-flip N/A

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

   4. lift-neg.f64 N/A

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

   5. sub-negate-rev N/A

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

   6. sin-neg N/A

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

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

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

   8. lower-cos.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lift-neg.f64 N/A

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

   12. add-flip N/A

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

   13. lift-+.f64 N/A

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

   14. lower-neg.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. mult-flip N/A

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

   17. lower-*.f64 N/A

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

   18. metadata-eval 6.4%

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

5. Applied rewrites 6.4%

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

6. Evaluated real constant 6.4%

   \[\leadsto \cos \left(\left(-\left(\pi \cdot z0 + \pi\right)\right) + \color{blue}{1.5707963267948966}\right) \]

7. Taylor expanded in z0 around 0

   \[\leadsto \cos \color{blue}{\left(\frac{884279719003555}{562949953421312} - \pi\right)} \]
8. Step-by-step derivation

   1. lower--.f64 N/A

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

   2. lower-PI.f64 3.7%

      \[\leadsto \cos \left(1.5707963267948966 - \pi\right) \]

9. Applied rewrites 3.7%

   \[\leadsto \cos \color{blue}{\left(1.5707963267948966 - \pi\right)} \]
10. Add Preprocessing

Alternative 4: 3.7% accurate, 1.1× speedup

Math

\[\sin \pi \]

FPCore

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

C

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

Java

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

Python

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

Julia

function code(z0)
	return sin(pi)
end

MATLAB

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

Wolfram

code[z0_] := N[Sin[Pi], $MachinePrecision]

Derivation

1. Initial program 53.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. sin-neg N/A

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

   6. sin-+PI-rev N/A

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

   7. lower-sin.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lower-+.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 6.4%

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

3. Applied rewrites 6.4%

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

4. Taylor expanded in z0 around 0

   \[\leadsto \sin \color{blue}{\pi} \]
5. Step-by-step derivation

   1. lower-PI.f64 3.7%

      \[\leadsto \sin \pi \]

6. Applied rewrites 3.7%

   \[\leadsto \sin \color{blue}{\pi} \]
7. Add Preprocessing

Reproduce

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