(cos (* (+ PI PI) z0))

Percentage Accurate: 56.9% → 98.7%

Time: 8.8s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 56.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \pi\\ {\cos t\_0}^{2} + \sin t\_0 \cdot \sin \left(t\_0 + \pi\right) \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) PI)))
  (+ (pow (cos t_0) 2) (* (sin t_0) (sin (+ t_0 PI))))))

C

double code(double z0) {
	double t_0 = fabs(z0) * ((double) M_PI);
	return pow(cos(t_0), 2.0) + (sin(t_0) * sin((t_0 + ((double) M_PI))));
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) * Math.PI;
	return Math.pow(Math.cos(t_0), 2.0) + (Math.sin(t_0) * Math.sin((t_0 + Math.PI)));
}

Python

def code(z0):
	t_0 = math.fabs(z0) * math.pi
	return math.pow(math.cos(t_0), 2.0) + (math.sin(t_0) * math.sin((t_0 + math.pi)))

Julia

function code(z0)
	t_0 = Float64(abs(z0) * pi)
	return Float64((cos(t_0) ^ 2.0) + Float64(sin(t_0) * sin(Float64(t_0 + pi))))
end

MATLAB

function tmp = code(z0)
	t_0 = abs(z0) * pi;
	tmp = (cos(t_0) ^ 2.0) + (sin(t_0) * sin((t_0 + pi)));
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Power[N[Cos[t$95$0], $MachinePrecision], 2], $MachinePrecision] + N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[N[(t$95$0 + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. remove-double-neg N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-rgt-in N/A

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

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

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

   10. cos-diff N/A

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

   11. lower-+.f64 N/A

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

3. Applied rewrites 56.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. sin-neg-rev N/A

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

   8. sin-+PI-rev N/A

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

   9. lower-sin.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-+.f64 98.5%

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

5. Applied rewrites 98.5%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. cos-neg-rev N/A

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

   8. lift-cos.f64 98.5%

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 98.5%

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

7. Applied rewrites 98.5%

   \[\leadsto \color{blue}{{\cos \left(z0 \cdot \pi\right)}^{2}} + \sin \left(z0 \cdot \pi\right) \cdot \sin \left(z0 \cdot \pi + \pi\right) \]
8. Add Preprocessing

Alternative 2: 98.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double z0) {
	return ((cos(((((z0 - -0.5) * ((double) M_PI)) - (-0.5 * ((double) M_PI))) - (z0 * ((double) M_PI)))) - cos(((((z0 - -0.5) + z0) * ((double) M_PI)) - (-0.5 * ((double) M_PI))))) / 2.0) + (0.5 + (0.5 * cos((z0 * (((double) M_PI) + ((double) M_PI))))));
}

Java

public static double code(double z0) {
	return ((Math.cos(((((z0 - -0.5) * Math.PI) - (-0.5 * Math.PI)) - (z0 * Math.PI))) - Math.cos(((((z0 - -0.5) + z0) * Math.PI) - (-0.5 * Math.PI)))) / 2.0) + (0.5 + (0.5 * Math.cos((z0 * (Math.PI + Math.PI)))));
}

Python

def code(z0):
	return ((math.cos(((((z0 - -0.5) * math.pi) - (-0.5 * math.pi)) - (z0 * math.pi))) - math.cos(((((z0 - -0.5) + z0) * math.pi) - (-0.5 * math.pi)))) / 2.0) + (0.5 + (0.5 * math.cos((z0 * (math.pi + math.pi)))))

Julia

function code(z0)
	return Float64(Float64(Float64(cos(Float64(Float64(Float64(Float64(z0 - -0.5) * pi) - Float64(-0.5 * pi)) - Float64(z0 * pi))) - cos(Float64(Float64(Float64(Float64(z0 - -0.5) + z0) * pi) - Float64(-0.5 * pi)))) / 2.0) + Float64(0.5 + Float64(0.5 * cos(Float64(z0 * Float64(pi + pi))))))
end

MATLAB

function tmp = code(z0)
	tmp = ((cos(((((z0 - -0.5) * pi) - (-0.5 * pi)) - (z0 * pi))) - cos(((((z0 - -0.5) + z0) * pi) - (-0.5 * pi)))) / 2.0) + (0.5 + (0.5 * cos((z0 * (pi + pi)))));
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

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

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-+l+ N/A

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

   11. sin-sum N/A

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

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

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

   13. lower-+.f64 N/A

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

3. Applied rewrites 59.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

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

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

   5. lift-sin.f64 N/A

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

   6. sin-mult N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 98.7%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift--.f64 N/A

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

   4. associate-+r- N/A

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

7. Applied rewrites 98.7%

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

Alternative 3: 98.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double z0) {
	return ((cos((((double) M_PI) * ((fabs(z0) - -1.0) - fabs(z0)))) - cos(((((fabs(z0) - -0.5) * ((double) M_PI)) - (-0.5 * ((double) M_PI))) + (fabs(z0) * ((double) M_PI))))) / 2.0) + (0.5 + (0.5 * cos((fabs(z0) * (((double) M_PI) + ((double) M_PI))))));
}

Java

public static double code(double z0) {
	return ((Math.cos((Math.PI * ((Math.abs(z0) - -1.0) - Math.abs(z0)))) - Math.cos(((((Math.abs(z0) - -0.5) * Math.PI) - (-0.5 * Math.PI)) + (Math.abs(z0) * Math.PI)))) / 2.0) + (0.5 + (0.5 * Math.cos((Math.abs(z0) * (Math.PI + Math.PI)))));
}

Python

def code(z0):
	return ((math.cos((math.pi * ((math.fabs(z0) - -1.0) - math.fabs(z0)))) - math.cos(((((math.fabs(z0) - -0.5) * math.pi) - (-0.5 * math.pi)) + (math.fabs(z0) * math.pi)))) / 2.0) + (0.5 + (0.5 * math.cos((math.fabs(z0) * (math.pi + math.pi)))))

Julia

function code(z0)
	return Float64(Float64(Float64(cos(Float64(pi * Float64(Float64(abs(z0) - -1.0) - abs(z0)))) - cos(Float64(Float64(Float64(Float64(abs(z0) - -0.5) * pi) - Float64(-0.5 * pi)) + Float64(abs(z0) * pi)))) / 2.0) + Float64(0.5 + Float64(0.5 * cos(Float64(abs(z0) * Float64(pi + pi))))))
end

MATLAB

function tmp = code(z0)
	tmp = ((cos((pi * ((abs(z0) - -1.0) - abs(z0)))) - cos(((((abs(z0) - -0.5) * pi) - (-0.5 * pi)) + (abs(z0) * pi)))) / 2.0) + (0.5 + (0.5 * cos((abs(z0) * (pi + pi)))));
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

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

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-+l+ N/A

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

   11. sin-sum N/A

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

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

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

   13. lower-+.f64 N/A

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

3. Applied rewrites 59.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

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

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

   5. lift-sin.f64 N/A

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

   6. sin-mult N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 98.7%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. distribute-rgt-out-- N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. distribute-lft-out-- N/A

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

   9. lower-*.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lift--.f64 N/A

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

   12. associate--l- N/A

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

   13. metadata-eval N/A

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

   14. lower--.f64 98.7%

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

7. Applied rewrites 98.7%

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

Alternative 4: 98.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (z0)
  :precision binary64
  (-
 (-
  (*
   (sin (* (* (- 1 (/ -1 (fabs z0))) (fabs z0)) PI))
   (sin (* (fabs z0) PI)))
  -1/2)
 (* -1/2 (cos (* (fabs z0) (+ PI PI))))))

C

double code(double z0) {
	return ((sin((((1.0 - (-1.0 / fabs(z0))) * fabs(z0)) * ((double) M_PI))) * sin((fabs(z0) * ((double) M_PI)))) - -0.5) - (-0.5 * cos((fabs(z0) * (((double) M_PI) + ((double) M_PI)))));
}

Java

public static double code(double z0) {
	return ((Math.sin((((1.0 - (-1.0 / Math.abs(z0))) * Math.abs(z0)) * Math.PI)) * Math.sin((Math.abs(z0) * Math.PI))) - -0.5) - (-0.5 * Math.cos((Math.abs(z0) * (Math.PI + Math.PI))));
}

Python

def code(z0):
	return ((math.sin((((1.0 - (-1.0 / math.fabs(z0))) * math.fabs(z0)) * math.pi)) * math.sin((math.fabs(z0) * math.pi))) - -0.5) - (-0.5 * math.cos((math.fabs(z0) * (math.pi + math.pi))))

Julia

function code(z0)
	return Float64(Float64(Float64(sin(Float64(Float64(Float64(1.0 - Float64(-1.0 / abs(z0))) * abs(z0)) * pi)) * sin(Float64(abs(z0) * pi))) - -0.5) - Float64(-0.5 * cos(Float64(abs(z0) * Float64(pi + pi)))))
end

MATLAB

function tmp = code(z0)
	tmp = ((sin((((1.0 - (-1.0 / abs(z0))) * abs(z0)) * pi)) * sin((abs(z0) * pi))) - -0.5) - (-0.5 * cos((abs(z0) * (pi + pi))));
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

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

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-+l+ N/A

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

   11. sin-sum N/A

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

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

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

   13. lower-+.f64 N/A

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

3. Applied rewrites 59.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

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

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

   5. lift-sin.f64 N/A

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

   6. sin-mult N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 98.7%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate-+r+ N/A

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

   4. add-flip N/A

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

   5. lower--.f64 N/A

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

7. Applied rewrites 98.5%

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

   1. lift--.f64 N/A

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

   2. sub-to-mult N/A

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

   3. lower-unsound-*.f64 N/A

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

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

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

   5. lower-unsound-/.f64 98.5%

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

9. Applied rewrites 98.5%

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

Alternative 5: 98.5% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (z0)
  :precision binary64
  (+
 (*
  (sin (* (fabs z0) PI))
  (sin (- (* (- (fabs z0) -1/2) PI) (* -1/2 PI))))
 (+ 1/2 (* 1/2 (cos (* (fabs z0) (+ PI PI)))))))

C

double code(double z0) {
	return (sin((fabs(z0) * ((double) M_PI))) * sin((((fabs(z0) - -0.5) * ((double) M_PI)) - (-0.5 * ((double) M_PI))))) + (0.5 + (0.5 * cos((fabs(z0) * (((double) M_PI) + ((double) M_PI))))));
}

Java

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

Python

def code(z0):
	return (math.sin((math.fabs(z0) * math.pi)) * math.sin((((math.fabs(z0) - -0.5) * math.pi) - (-0.5 * math.pi)))) + (0.5 + (0.5 * math.cos((math.fabs(z0) * (math.pi + math.pi)))))

Julia

function code(z0)
	return Float64(Float64(sin(Float64(abs(z0) * pi)) * sin(Float64(Float64(Float64(abs(z0) - -0.5) * pi) - Float64(-0.5 * pi)))) + Float64(0.5 + Float64(0.5 * cos(Float64(abs(z0) * Float64(pi + pi))))))
end

MATLAB

function tmp = code(z0)
	tmp = (sin((abs(z0) * pi)) * sin((((abs(z0) - -0.5) * pi) - (-0.5 * pi)))) + (0.5 + (0.5 * cos((abs(z0) * (pi + pi)))));
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

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

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-+l+ N/A

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

   11. sin-sum N/A

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

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

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

   13. lower-+.f64 N/A

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

3. Applied rewrites 59.0%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. mult-flip N/A

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

   6. metadata-eval N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. add-flip N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

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

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

   13. metadata-eval N/A

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

   14. lift-*.f64 N/A

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

   15. lower--.f64 98.7%

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

5. Applied rewrites 98.6%

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

Alternative 6: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| \cdot \pi\\ \left(\frac{1}{2} + \cos \left(\left|z0\right| \cdot \left(\pi + \pi\right)\right) \cdot \frac{1}{2}\right) + \sin t\_0 \cdot \sin \left(t\_0 + \pi\right) \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (fabs z0) PI)))
  (+
   (+ 1/2 (* (cos (* (fabs z0) (+ PI PI))) 1/2))
   (* (sin t_0) (sin (+ t_0 PI))))))

C

double code(double z0) {
	double t_0 = fabs(z0) * ((double) M_PI);
	return (0.5 + (cos((fabs(z0) * (((double) M_PI) + ((double) M_PI)))) * 0.5)) + (sin(t_0) * sin((t_0 + ((double) M_PI))));
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) * Math.PI;
	return (0.5 + (Math.cos((Math.abs(z0) * (Math.PI + Math.PI))) * 0.5)) + (Math.sin(t_0) * Math.sin((t_0 + Math.PI)));
}

Python

def code(z0):
	t_0 = math.fabs(z0) * math.pi
	return (0.5 + (math.cos((math.fabs(z0) * (math.pi + math.pi))) * 0.5)) + (math.sin(t_0) * math.sin((t_0 + math.pi)))

Julia

function code(z0)
	t_0 = Float64(abs(z0) * pi)
	return Float64(Float64(0.5 + Float64(cos(Float64(abs(z0) * Float64(pi + pi))) * 0.5)) + Float64(sin(t_0) * sin(Float64(t_0 + pi))))
end

MATLAB

function tmp = code(z0)
	t_0 = abs(z0) * pi;
	tmp = (0.5 + (cos((abs(z0) * (pi + pi))) * 0.5)) + (sin(t_0) * sin((t_0 + pi)));
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[(1/2 + N[(N[Cos[N[(N[Abs[z0], $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1/2), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t$95$0], $MachinePrecision] * N[Sin[N[(t$95$0 + Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

   4. remove-double-neg N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-+.f64 N/A

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

   8. distribute-rgt-in N/A

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

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

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

   10. cos-diff N/A

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

   11. lower-+.f64 N/A

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

3. Applied rewrites 56.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. sin-neg-rev N/A

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

   8. sin-+PI-rev N/A

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

   9. lower-sin.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lower-+.f64 98.5%

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

5. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. remove-double-neg N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

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

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

   7. lift-neg.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-cos.f64 N/A

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

   10. cos-neg-rev N/A

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

   11. sqr-cos-a N/A

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

   12. metadata-eval N/A

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

   13. lift-*.f64 N/A

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

   14. lift-neg.f64 N/A

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

   15. distribute-lft-neg-out N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. remove-double-neg N/A

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

7. Applied rewrites 98.6%

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

Alternative 7: 59.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left(\pi + \pi\right) \cdot \left|z0\right|\right)\\ t_1 := \frac{1}{2} \cdot \sin \left(\left(-\left|z0\right|\right) \cdot \left(\pi + \pi\right) + \frac{1}{2} \cdot \pi\right)\\ \mathbf{if}\;t\_0 \leq \frac{3602879701896397}{36028797018963968}:\\ \;\;\;\;\left(\frac{1}{2} + t\_1\right) - \left(\frac{1}{2} - t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (cos (* (+ PI PI) (fabs z0))))
       (t_1 (* 1/2 (sin (+ (* (- (fabs z0)) (+ PI PI)) (* 1/2 PI))))))
  (if (<= t_0 3602879701896397/36028797018963968)
    (- (+ 1/2 t_1) (- 1/2 t_1))
    t_0)))

C

double code(double z0) {
	double t_0 = cos(((((double) M_PI) + ((double) M_PI)) * fabs(z0)));
	double t_1 = 0.5 * sin(((-fabs(z0) * (((double) M_PI) + ((double) M_PI))) + (0.5 * ((double) M_PI))));
	double tmp;
	if (t_0 <= 0.1) {
		tmp = (0.5 + t_1) - (0.5 - t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.cos(((Math.PI + Math.PI) * Math.abs(z0)));
	double t_1 = 0.5 * Math.sin(((-Math.abs(z0) * (Math.PI + Math.PI)) + (0.5 * Math.PI)));
	double tmp;
	if (t_0 <= 0.1) {
		tmp = (0.5 + t_1) - (0.5 - t_1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.cos(((math.pi + math.pi) * math.fabs(z0)))
	t_1 = 0.5 * math.sin(((-math.fabs(z0) * (math.pi + math.pi)) + (0.5 * math.pi)))
	tmp = 0
	if t_0 <= 0.1:
		tmp = (0.5 + t_1) - (0.5 - t_1)
	else:
		tmp = t_0
	return tmp

Julia

function code(z0)
	t_0 = cos(Float64(Float64(pi + pi) * abs(z0)))
	t_1 = Float64(0.5 * sin(Float64(Float64(Float64(-abs(z0)) * Float64(pi + pi)) + Float64(0.5 * pi))))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = Float64(Float64(0.5 + t_1) - Float64(0.5 - t_1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = cos(((pi + pi) * abs(z0)));
	t_1 = 0.5 * sin(((-abs(z0) * (pi + pi)) + (0.5 * pi)));
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = (0.5 + t_1) - (0.5 - t_1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[Cos[N[(N[(Pi + Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1/2 * N[Sin[N[(N[((-N[Abs[z0], $MachinePrecision]) * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] + N[(1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 3602879701896397/36028797018963968], N[(N[(1/2 + t$95$1), $MachinePrecision] - N[(1/2 - t$95$1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) z0)) < 0.10000000000000001

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. count-2 N/A

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

      5. associate-*l* N/A

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

      6. cos-2 N/A

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

      7. lower--.f64 N/A

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

   3. Applied rewrites 56.9%

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

   5. Applied rewrites 56.2%

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

   7. Applied rewrites 56.9%

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

   if 0.10000000000000001 < (cos.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) z0))

   1. Initial program 56.9%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 59.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left(\pi + \pi\right) \cdot \left|z0\right|\right)\\ \mathbf{if}\;t\_0 \leq \frac{3602879701896397}{36028797018963968}:\\ \;\;\;\;\sin \left(\left(-2 \cdot \pi\right) \cdot \left|z0\right| - \pi \cdot \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (cos (* (+ PI PI) (fabs z0)))))
  (if (<= t_0 3602879701896397/36028797018963968)
    (sin (- (* (* -2 PI) (fabs z0)) (* PI -1/2)))
    t_0)))

C

double code(double z0) {
	double t_0 = cos(((((double) M_PI) + ((double) M_PI)) * fabs(z0)));
	double tmp;
	if (t_0 <= 0.1) {
		tmp = sin((((-2.0 * ((double) M_PI)) * fabs(z0)) - (((double) M_PI) * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.cos(((Math.PI + Math.PI) * Math.abs(z0)));
	double tmp;
	if (t_0 <= 0.1) {
		tmp = Math.sin((((-2.0 * Math.PI) * Math.abs(z0)) - (Math.PI * -0.5)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.cos(((math.pi + math.pi) * math.fabs(z0)))
	tmp = 0
	if t_0 <= 0.1:
		tmp = math.sin((((-2.0 * math.pi) * math.fabs(z0)) - (math.pi * -0.5)))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0)
	t_0 = cos(Float64(Float64(pi + pi) * abs(z0)))
	tmp = 0.0
	if (t_0 <= 0.1)
		tmp = sin(Float64(Float64(Float64(-2.0 * pi) * abs(z0)) - Float64(pi * -0.5)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = cos(((pi + pi) * abs(z0)));
	tmp = 0.0;
	if (t_0 <= 0.1)
		tmp = sin((((-2.0 * pi) * abs(z0)) - (pi * -0.5)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[Cos[N[(N[(Pi + Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, 3602879701896397/36028797018963968], N[Sin[N[(N[(N[(-2 * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] - N[(Pi * -1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) z0)) < 0.10000000000000001

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. add-flip N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. count-2 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-PI.f64 N/A

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

      16. mult-flip N/A

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

      17. distribute-rgt-neg-in N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

   3. Applied rewrites 56.9%

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

   if 0.10000000000000001 < (cos.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) z0))

   1. Initial program 56.9%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left(\pi + \pi\right) \cdot \left|z0\right|\right)\\ \mathbf{if}\;t\_0 \leq \frac{-5764607523034235}{288230376151711744}:\\ \;\;\;\;\sin \left(\pi \cdot \left(\left|z0\right| + \left(\left|z0\right| - \frac{-1}{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (cos (* (+ PI PI) (fabs z0)))))
  (if (<= t_0 -5764607523034235/288230376151711744)
    (sin (* PI (+ (fabs z0) (- (fabs z0) -1/2))))
    t_0)))

C

double code(double z0) {
	double t_0 = cos(((((double) M_PI) + ((double) M_PI)) * fabs(z0)));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = sin((((double) M_PI) * (fabs(z0) + (fabs(z0) - -0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.cos(((Math.PI + Math.PI) * Math.abs(z0)));
	double tmp;
	if (t_0 <= -0.02) {
		tmp = Math.sin((Math.PI * (Math.abs(z0) + (Math.abs(z0) - -0.5))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.cos(((math.pi + math.pi) * math.fabs(z0)))
	tmp = 0
	if t_0 <= -0.02:
		tmp = math.sin((math.pi * (math.fabs(z0) + (math.fabs(z0) - -0.5))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0)
	t_0 = cos(Float64(Float64(pi + pi) * abs(z0)))
	tmp = 0.0
	if (t_0 <= -0.02)
		tmp = sin(Float64(pi * Float64(abs(z0) + Float64(abs(z0) - -0.5))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = cos(((pi + pi) * abs(z0)));
	tmp = 0.0;
	if (t_0 <= -0.02)
		tmp = sin((pi * (abs(z0) + (abs(z0) - -0.5))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[Cos[N[(N[(Pi + Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5764607523034235/288230376151711744], N[Sin[N[(Pi * N[(N[Abs[z0], $MachinePrecision] + N[(N[Abs[z0], $MachinePrecision] - -1/2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) z0)) < -0.02

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. add-flip N/A

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

      5. lower--.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. mult-flip N/A

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

      11. distribute-rgt-neg-in N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 56.9%

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

   3. Applied rewrites 56.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate--l+ N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. metadata-eval N/A

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

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

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-+.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. distribute-rgt-out N/A

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

   5. Applied rewrites 56.9%

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

   if -0.02 < (cos.f64 (*.f64 (+.f64 (PI.f64) (PI.f64)) z0))

   1. Initial program 56.9%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

   3. cos-neg-rev N/A

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

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

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

   5. remove-double-neg N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-+.f64 N/A

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

   9. distribute-rgt-in N/A

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

   10. associate-+l+ N/A

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

   11. sin-sum N/A

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

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

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

   13. lower-+.f64 N/A

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

3. Applied rewrites 59.0%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. distribute-rgt-out N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. add-flip N/A

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

   8. metadata-eval N/A

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

   9. lower--.f64 59.0%

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

5. Applied rewrites 59.0%

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

Reproduce

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