(sin (* (- 1/2 (+ z0 z0)) PI))

Percentage Accurate: 57.5% → 98.5%

Time: 3.0s

Alternatives: 8

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (- 0.5 (+ z0 z0)) PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z0_] := N[Sin[N[(N[(0.5 - N[(z0 + z0), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]

Initial Program: 57.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* (- 0.5 (+ z0 z0)) PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z0_] := N[Sin[N[(N[(0.5 - N[(z0 + z0), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]

Alternative 1: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - \left|z0\right|\\ t_1 := \left(\sin \left(\pi \cdot \left(\left(1 - \frac{\left|z0\right|}{t\_0}\right) \cdot t\_0\right)\right) + \cos \left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\right)\right) \cdot 0.5\\ t_2 := \cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot \left|z0\right|\right) \cdot \pi\right)\\ \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\frac{{t\_1}^{3} + {t\_2}^{3}}{t\_1 \cdot t\_1 + \left(t\_2 \cdot t\_2 - t\_1 \cdot t\_2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (- 0.5 (fabs z0)))
       (t_1
        (*
         (+
          (sin (* PI (* (- 1.0 (/ (fabs z0) t_0)) t_0)))
          (cos (* (+ (fabs z0) (fabs z0)) PI)))
         0.5))
       (t_2 (* (cos (* PI 0.5)) (sin (* (* -2.0 (fabs z0)) PI)))))
  (if (<= (fabs z0) 85000.0)
    (/
     (+ (pow t_1 3.0) (pow t_2 3.0))
     (+ (* t_1 t_1) (- (* t_2 t_2) (* t_1 t_2))))
    (sin (* 0.5 PI)))))

C

double code(double z0) {
	double t_0 = 0.5 - fabs(z0);
	double t_1 = (sin((((double) M_PI) * ((1.0 - (fabs(z0) / t_0)) * t_0))) + cos(((fabs(z0) + fabs(z0)) * ((double) M_PI)))) * 0.5;
	double t_2 = cos((((double) M_PI) * 0.5)) * sin(((-2.0 * fabs(z0)) * ((double) M_PI)));
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = (pow(t_1, 3.0) + pow(t_2, 3.0)) / ((t_1 * t_1) + ((t_2 * t_2) - (t_1 * t_2)));
	} else {
		tmp = sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = 0.5 - Math.abs(z0);
	double t_1 = (Math.sin((Math.PI * ((1.0 - (Math.abs(z0) / t_0)) * t_0))) + Math.cos(((Math.abs(z0) + Math.abs(z0)) * Math.PI))) * 0.5;
	double t_2 = Math.cos((Math.PI * 0.5)) * Math.sin(((-2.0 * Math.abs(z0)) * Math.PI));
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = (Math.pow(t_1, 3.0) + Math.pow(t_2, 3.0)) / ((t_1 * t_1) + ((t_2 * t_2) - (t_1 * t_2)));
	} else {
		tmp = Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	t_0 = 0.5 - math.fabs(z0)
	t_1 = (math.sin((math.pi * ((1.0 - (math.fabs(z0) / t_0)) * t_0))) + math.cos(((math.fabs(z0) + math.fabs(z0)) * math.pi))) * 0.5
	t_2 = math.cos((math.pi * 0.5)) * math.sin(((-2.0 * math.fabs(z0)) * math.pi))
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = (math.pow(t_1, 3.0) + math.pow(t_2, 3.0)) / ((t_1 * t_1) + ((t_2 * t_2) - (t_1 * t_2)))
	else:
		tmp = math.sin((0.5 * math.pi))
	return tmp

Julia

function code(z0)
	t_0 = Float64(0.5 - abs(z0))
	t_1 = Float64(Float64(sin(Float64(pi * Float64(Float64(1.0 - Float64(abs(z0) / t_0)) * t_0))) + cos(Float64(Float64(abs(z0) + abs(z0)) * pi))) * 0.5)
	t_2 = Float64(cos(Float64(pi * 0.5)) * sin(Float64(Float64(-2.0 * abs(z0)) * pi)))
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = Float64(Float64((t_1 ^ 3.0) + (t_2 ^ 3.0)) / Float64(Float64(t_1 * t_1) + Float64(Float64(t_2 * t_2) - Float64(t_1 * t_2))));
	else
		tmp = sin(Float64(0.5 * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = 0.5 - abs(z0);
	t_1 = (sin((pi * ((1.0 - (abs(z0) / t_0)) * t_0))) + cos(((abs(z0) + abs(z0)) * pi))) * 0.5;
	t_2 = cos((pi * 0.5)) * sin(((-2.0 * abs(z0)) * pi));
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = ((t_1 ^ 3.0) + (t_2 ^ 3.0)) / ((t_1 * t_1) + ((t_2 * t_2) - (t_1 * t_2)));
	else
		tmp = sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(0.5 - N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[N[(Pi * N[(N[(1.0 - N[(N[Abs[z0], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(-2.0 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + N[Power[t$95$2, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[(t$95$2 * t$95$2), $MachinePrecision] - N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\sin \left(\left(0.5 - \left(z0 + z0\right)\right) \cdot \pi\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. sin-sum N/A

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

      8. flip3-+ N/A

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

   3. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.5%

      \[\leadsto \frac{{\color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 56.8%

      \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 56.8%

      \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. sin-cos-mult N/A

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

       5. mult-flip N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 56.8%

       \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   12. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. sub-to-mult N/A

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

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

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

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

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

       5. lower-unsound-/.f64 56.8%

          \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \color{blue}{\frac{z0}{0.5 - z0}}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

   13. Applied rewrites 56.8%

       \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \color{blue}{\left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)}\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. sub-to-mult N/A

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

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

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

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

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

       5. lower-unsound-/.f64 56.8%

          \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \color{blue}{\frac{z0}{0.5 - z0}}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

   15. Applied rewrites 56.8%

       \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \color{blue}{\left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)}\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   16. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. sub-to-mult N/A

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

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

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

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

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

       5. lower-unsound-/.f64 56.8%

          \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(1 - \color{blue}{\frac{z0}{0.5 - z0}}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

   17. Applied rewrites 56.8%

       \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \color{blue}{\left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)}\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   18. Step-by-step derivation

       1. lift--.f64 N/A

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

       2. sub-to-mult N/A

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

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

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

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

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

       5. lower-unsound-/.f64 56.8%

          \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\left(\sin \left(\pi \cdot \left(\left(1 - \color{blue}{\frac{z0}{0.5 - z0}}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

   19. Applied rewrites 56.8%

       \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\left(\sin \left(\pi \cdot \color{blue}{\left(\left(1 - \frac{z0}{0.5 - z0}\right) \cdot \left(0.5 - z0\right)\right)}\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

   if 85000 < z0

   1. Initial program 57.5%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 2: 98.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot \left|z0\right|\right) \cdot \pi\right)\\ t_1 := \left(\sin \left(\pi \cdot \left(\left(0.5 - \left|z0\right|\right) - \left|z0\right|\right)\right) + \cos \left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\right)\right) \cdot 0.5\\ \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\frac{{t\_1}^{3} + {t\_0}^{3}}{t\_1 \cdot t\_1 + \left(t\_0 \cdot t\_0 - t\_1 \cdot t\_0\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* (cos (* PI 0.5)) (sin (* (* -2.0 (fabs z0)) PI))))
       (t_1
        (*
         (+
          (sin (* PI (- (- 0.5 (fabs z0)) (fabs z0))))
          (cos (* (+ (fabs z0) (fabs z0)) PI)))
         0.5)))
  (if (<= (fabs z0) 85000.0)
    (/
     (+ (pow t_1 3.0) (pow t_0 3.0))
     (+ (* t_1 t_1) (- (* t_0 t_0) (* t_1 t_0))))
    (sin (* 0.5 PI)))))

C

double code(double z0) {
	double t_0 = cos((((double) M_PI) * 0.5)) * sin(((-2.0 * fabs(z0)) * ((double) M_PI)));
	double t_1 = (sin((((double) M_PI) * ((0.5 - fabs(z0)) - fabs(z0)))) + cos(((fabs(z0) + fabs(z0)) * ((double) M_PI)))) * 0.5;
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = (pow(t_1, 3.0) + pow(t_0, 3.0)) / ((t_1 * t_1) + ((t_0 * t_0) - (t_1 * t_0)));
	} else {
		tmp = sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.cos((Math.PI * 0.5)) * Math.sin(((-2.0 * Math.abs(z0)) * Math.PI));
	double t_1 = (Math.sin((Math.PI * ((0.5 - Math.abs(z0)) - Math.abs(z0)))) + Math.cos(((Math.abs(z0) + Math.abs(z0)) * Math.PI))) * 0.5;
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = (Math.pow(t_1, 3.0) + Math.pow(t_0, 3.0)) / ((t_1 * t_1) + ((t_0 * t_0) - (t_1 * t_0)));
	} else {
		tmp = Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.cos((math.pi * 0.5)) * math.sin(((-2.0 * math.fabs(z0)) * math.pi))
	t_1 = (math.sin((math.pi * ((0.5 - math.fabs(z0)) - math.fabs(z0)))) + math.cos(((math.fabs(z0) + math.fabs(z0)) * math.pi))) * 0.5
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = (math.pow(t_1, 3.0) + math.pow(t_0, 3.0)) / ((t_1 * t_1) + ((t_0 * t_0) - (t_1 * t_0)))
	else:
		tmp = math.sin((0.5 * math.pi))
	return tmp

Julia

function code(z0)
	t_0 = Float64(cos(Float64(pi * 0.5)) * sin(Float64(Float64(-2.0 * abs(z0)) * pi)))
	t_1 = Float64(Float64(sin(Float64(pi * Float64(Float64(0.5 - abs(z0)) - abs(z0)))) + cos(Float64(Float64(abs(z0) + abs(z0)) * pi))) * 0.5)
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = Float64(Float64((t_1 ^ 3.0) + (t_0 ^ 3.0)) / Float64(Float64(t_1 * t_1) + Float64(Float64(t_0 * t_0) - Float64(t_1 * t_0))));
	else
		tmp = sin(Float64(0.5 * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = cos((pi * 0.5)) * sin(((-2.0 * abs(z0)) * pi));
	t_1 = (sin((pi * ((0.5 - abs(z0)) - abs(z0)))) + cos(((abs(z0) + abs(z0)) * pi))) * 0.5;
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = ((t_1 ^ 3.0) + (t_0 ^ 3.0)) / ((t_1 * t_1) + ((t_0 * t_0) - (t_1 * t_0)));
	else
		tmp = sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Cos[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(-2.0 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sin[N[(Pi * N[(N[(0.5 - N[Abs[z0], $MachinePrecision]), $MachinePrecision] - N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] + N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(N[(t$95$0 * t$95$0), $MachinePrecision] - N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\sin \left(\left(0.5 - \left(z0 + z0\right)\right) \cdot \pi\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. sin-sum N/A

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

      8. flip3-+ N/A

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

   3. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.5%

      \[\leadsto \frac{{\color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 56.8%

      \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 56.8%

      \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. sin-cos-mult N/A

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

       5. mult-flip N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 56.8%

       \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

   if 85000 < z0

   1. Initial program 57.5%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 3: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| + \left|z0\right|\\ t_1 := t\_0 \cdot \pi\\ \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\frac{\left(\sin \left(\left(\left(0.5 - \left|z0\right|\right) - \left|z0\right|\right) \cdot \pi\right) + \sin \left(\pi \cdot \left(0.5 + t\_0\right)\right)\right) + \left(\cos t\_1 + \sin \left(-\left(t\_1 + 0.5 \cdot \pi\right)\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z0) (fabs z0))) (t_1 (* t_0 PI)))
  (if (<= (fabs z0) 85000.0)
    (/
     (+
      (+
       (sin (* (- (- 0.5 (fabs z0)) (fabs z0)) PI))
       (sin (* PI (+ 0.5 t_0))))
      (+ (cos t_1) (sin (- (+ t_1 (* 0.5 PI))))))
     2.0)
    (sin (* 0.5 PI)))))

C

double code(double z0) {
	double t_0 = fabs(z0) + fabs(z0);
	double t_1 = t_0 * ((double) M_PI);
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = ((sin((((0.5 - fabs(z0)) - fabs(z0)) * ((double) M_PI))) + sin((((double) M_PI) * (0.5 + t_0)))) + (cos(t_1) + sin(-(t_1 + (0.5 * ((double) M_PI)))))) / 2.0;
	} else {
		tmp = sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) + Math.abs(z0);
	double t_1 = t_0 * Math.PI;
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = ((Math.sin((((0.5 - Math.abs(z0)) - Math.abs(z0)) * Math.PI)) + Math.sin((Math.PI * (0.5 + t_0)))) + (Math.cos(t_1) + Math.sin(-(t_1 + (0.5 * Math.PI))))) / 2.0;
	} else {
		tmp = Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) + math.fabs(z0)
	t_1 = t_0 * math.pi
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = ((math.sin((((0.5 - math.fabs(z0)) - math.fabs(z0)) * math.pi)) + math.sin((math.pi * (0.5 + t_0)))) + (math.cos(t_1) + math.sin(-(t_1 + (0.5 * math.pi))))) / 2.0
	else:
		tmp = math.sin((0.5 * math.pi))
	return tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) + abs(z0))
	t_1 = Float64(t_0 * pi)
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(Float64(0.5 - abs(z0)) - abs(z0)) * pi)) + sin(Float64(pi * Float64(0.5 + t_0)))) + Float64(cos(t_1) + sin(Float64(-Float64(t_1 + Float64(0.5 * pi)))))) / 2.0);
	else
		tmp = sin(Float64(0.5 * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) + abs(z0);
	t_1 = t_0 * pi;
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = ((sin((((0.5 - abs(z0)) - abs(z0)) * pi)) + sin((pi * (0.5 + t_0)))) + (cos(t_1) + sin(-(t_1 + (0.5 * pi))))) / 2.0;
	else
		tmp = sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * Pi), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[(N[(N[(N[Sin[N[(N[(N[(0.5 - N[Abs[z0], $MachinePrecision]), $MachinePrecision] - N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(Pi * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[t$95$1], $MachinePrecision] + N[Sin[(-N[(t$95$1 + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\sin \left(\left(0.5 - \left(z0 + z0\right)\right) \cdot \pi\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. sin-sum N/A

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

      8. flip3-+ N/A

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

   3. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. count-2 N/A

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

      9. lift-+.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. lift-PI.f64 N/A

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

      18. cos-PI/2 N/A

          \[\leadsto \frac{{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\sin \left(\mathsf{neg}\left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{0}\right)}^{3}}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

      19. sin-PI N/A

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

      20. lift-PI.f64 N/A

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

   5. Applied rewrites 57.5%

      \[\leadsto \frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\color{blue}{\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. count-2 N/A

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

      9. lift-+.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. lift-PI.f64 N/A

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

      18. cos-PI/2 N/A

          \[\leadsto \frac{{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\sin \left(\mathsf{neg}\left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{0}\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

      19. sin-PI N/A

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

      20. lift-PI.f64 N/A

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

   7. Applied rewrites 57.5%

      \[\leadsto \frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\color{blue}{\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}} \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. count-2 N/A

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

      9. lift-+.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. lift-PI.f64 N/A

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

      18. cos-PI/2 N/A

          \[\leadsto \frac{{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} \cdot \left(\sin \left(\mathsf{neg}\left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{0}\right) - \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

      19. sin-PI N/A

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

      20. lift-PI.f64 N/A

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

   9. Applied rewrites 57.5%

      \[\leadsto \frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} \cdot \color{blue}{\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}} - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. metadata-eval N/A

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

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

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

       8. count-2 N/A

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

       9. lift-+.f64 N/A

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

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

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. lift-cos.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. metadata-eval N/A

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

       16. mult-flip N/A

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

       17. lift-PI.f64 N/A

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

       18. cos-PI/2 N/A

           \[\leadsto \frac{{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} \cdot \frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} - \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\mathsf{neg}\left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{0}\right)\right)} \]

       19. sin-PI N/A

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

       20. lift-PI.f64 N/A

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

   11. Applied rewrites 57.5%

       \[\leadsto \frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} \cdot \frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}}\right)} \]

   13. Applied rewrites 53.6%

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

       1. lift-neg.f64 N/A

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

       2. lift-cos.f64 N/A

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

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

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

       4. sin-neg-rev N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-neg.f64 N/A

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

       7. lift-PI.f64 N/A

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

       8. mult-flip-rev N/A

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

       9. metadata-eval N/A

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

       10. lift-*.f64 N/A

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

       11. lower-+.f64 56.8%

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

       12. lift-*.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 56.8%

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

   15. Applied rewrites 56.8%

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

   if 85000 < z0

   1. Initial program 57.5%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 4: 98.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| + \left|z0\right|\\ \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\left(\sin \left(\left(\left(0.5 - \left|z0\right|\right) - \left|z0\right|\right) \cdot \pi\right) + \cos \left(t\_0 \cdot \pi\right)\right) \cdot 0.5 - \sin \left(\left(-\pi\right) \cdot t\_0\right) \cdot \cos \left(-0.5 \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z0) (fabs z0))))
  (if (<= (fabs z0) 85000.0)
    (-
     (*
      (+
       (sin (* (- (- 0.5 (fabs z0)) (fabs z0)) PI))
       (cos (* t_0 PI)))
      0.5)
     (* (sin (* (- PI) t_0)) (cos (* -0.5 PI))))
    (sin (* 0.5 PI)))))

C

double code(double z0) {
	double t_0 = fabs(z0) + fabs(z0);
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = ((sin((((0.5 - fabs(z0)) - fabs(z0)) * ((double) M_PI))) + cos((t_0 * ((double) M_PI)))) * 0.5) - (sin((-((double) M_PI) * t_0)) * cos((-0.5 * ((double) M_PI))));
	} else {
		tmp = sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) + Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = ((Math.sin((((0.5 - Math.abs(z0)) - Math.abs(z0)) * Math.PI)) + Math.cos((t_0 * Math.PI))) * 0.5) - (Math.sin((-Math.PI * t_0)) * Math.cos((-0.5 * Math.PI)));
	} else {
		tmp = Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) + math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = ((math.sin((((0.5 - math.fabs(z0)) - math.fabs(z0)) * math.pi)) + math.cos((t_0 * math.pi))) * 0.5) - (math.sin((-math.pi * t_0)) * math.cos((-0.5 * math.pi)))
	else:
		tmp = math.sin((0.5 * math.pi))
	return tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) + abs(z0))
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = Float64(Float64(Float64(sin(Float64(Float64(Float64(0.5 - abs(z0)) - abs(z0)) * pi)) + cos(Float64(t_0 * pi))) * 0.5) - Float64(sin(Float64(Float64(-pi) * t_0)) * cos(Float64(-0.5 * pi))));
	else
		tmp = sin(Float64(0.5 * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) + abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = ((sin((((0.5 - abs(z0)) - abs(z0)) * pi)) + cos((t_0 * pi))) * 0.5) - (sin((-pi * t_0)) * cos((-0.5 * pi)));
	else
		tmp = sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[(N[(N[(N[Sin[N[(N[(N[(0.5 - N[Abs[z0], $MachinePrecision]), $MachinePrecision] - N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] + N[Cos[N[(t$95$0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[Sin[N[((-Pi) * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\sin \left(\left(0.5 - \left(z0 + z0\right)\right) \cdot \pi\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. sin-sum N/A

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

      8. flip3-+ N/A

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

   3. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 56.5%

      \[\leadsto \frac{{\color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   7. Applied rewrites 56.8%

      \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. sin-cos-mult N/A

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

      5. mult-flip N/A

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

      6. metadata-eval N/A

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

      7. lower-*.f64 N/A

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

   9. Applied rewrites 56.8%

      \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-cos.f64 N/A

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

       4. sin-cos-mult N/A

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

       5. mult-flip N/A

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

       6. metadata-eval N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 56.8%

       \[\leadsto \frac{{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) \cdot \left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \color{blue}{\left(\left(\sin \left(\pi \cdot \left(\left(0.5 - z0\right) - z0\right)\right) + \cos \left(\left(z0 + z0\right) \cdot \pi\right)\right) \cdot 0.5\right)} \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

   13. Applied rewrites 56.8%

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

   if 85000 < z0

   1. Initial program 57.5%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 5: 98.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := 2 \cdot \left|z0\right|\\ \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;0.5 \cdot \left(\sin \left(\pi \cdot \left(0.5 + t\_0\right)\right) + \sin \left(\pi \cdot \left(0.5 - t\_0\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* 2.0 (fabs z0))))
  (if (<= (fabs z0) 85000.0)
    (* 0.5 (+ (sin (* PI (+ 0.5 t_0))) (sin (* PI (- 0.5 t_0)))))
    (sin (* 0.5 PI)))))

C

double code(double z0) {
	double t_0 = 2.0 * fabs(z0);
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = 0.5 * (sin((((double) M_PI) * (0.5 + t_0))) + sin((((double) M_PI) * (0.5 - t_0))));
	} else {
		tmp = sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = 2.0 * Math.abs(z0);
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = 0.5 * (Math.sin((Math.PI * (0.5 + t_0))) + Math.sin((Math.PI * (0.5 - t_0))));
	} else {
		tmp = Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	t_0 = 2.0 * math.fabs(z0)
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = 0.5 * (math.sin((math.pi * (0.5 + t_0))) + math.sin((math.pi * (0.5 - t_0))))
	else:
		tmp = math.sin((0.5 * math.pi))
	return tmp

Julia

function code(z0)
	t_0 = Float64(2.0 * abs(z0))
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = Float64(0.5 * Float64(sin(Float64(pi * Float64(0.5 + t_0))) + sin(Float64(pi * Float64(0.5 - t_0)))));
	else
		tmp = sin(Float64(0.5 * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = 2.0 * abs(z0);
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = 0.5 * (sin((pi * (0.5 + t_0))) + sin((pi * (0.5 - t_0))));
	else
		tmp = sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(2.0 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[(0.5 * N[(N[Sin[N[(Pi * N[(0.5 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Sin[N[(Pi * N[(0.5 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\sin \left(\left(0.5 - \left(z0 + z0\right)\right) \cdot \pi\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. sub-flip N/A

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

      6. distribute-lft-in N/A

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

      7. sin-sum N/A

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

      8. flip3-+ N/A

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

   3. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)}} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. count-2 N/A

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

      9. lift-+.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. lift-PI.f64 N/A

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

      18. cos-PI/2 N/A

          \[\leadsto \frac{{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\sin \left(\mathsf{neg}\left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{0}\right)}^{3}}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

      19. sin-PI N/A

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

      20. lift-PI.f64 N/A

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

   5. Applied rewrites 57.5%

      \[\leadsto \frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\color{blue}{\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. count-2 N/A

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

      9. lift-+.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. lift-PI.f64 N/A

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

      18. cos-PI/2 N/A

          \[\leadsto \frac{{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\left(\sin \left(\mathsf{neg}\left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{0}\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

      19. sin-PI N/A

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

      20. lift-PI.f64 N/A

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

   7. Applied rewrites 57.5%

      \[\leadsto \frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\color{blue}{\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}} \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right) - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. metadata-eval N/A

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

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

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

      8. count-2 N/A

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

      9. lift-+.f64 N/A

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

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

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-cos.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. metadata-eval N/A

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

      16. mult-flip N/A

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

      17. lift-PI.f64 N/A

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

      18. cos-PI/2 N/A

          \[\leadsto \frac{{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} \cdot \left(\sin \left(\mathsf{neg}\left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{0}\right) - \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot \frac{1}{2}\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]

      19. sin-PI N/A

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

      20. lift-PI.f64 N/A

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

   9. Applied rewrites 57.5%

      \[\leadsto \frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} \cdot \color{blue}{\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}} - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\cos \left(\pi \cdot 0.5\right) \cdot \sin \left(\left(-2 \cdot z0\right) \cdot \pi\right)\right)\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. metadata-eval N/A

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

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

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

       8. count-2 N/A

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

       9. lift-+.f64 N/A

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

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

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

       11. *-commutative N/A

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

       12. lift-*.f64 N/A

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

       13. lift-cos.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. metadata-eval N/A

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

       16. mult-flip N/A

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

       17. lift-PI.f64 N/A

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

       18. cos-PI/2 N/A

           \[\leadsto \frac{{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} \cdot \frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} - \left(\sin \left(\pi \cdot \frac{1}{2}\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\mathsf{neg}\left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{0}\right)\right)} \]

       19. sin-PI N/A

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

       20. lift-PI.f64 N/A

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

   11. Applied rewrites 57.5%

       \[\leadsto \frac{{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right)}^{3} + {\left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}\right)}^{3}}{\left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) + \left(\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} \cdot \frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2} - \left(\sin \left(\pi \cdot 0.5\right) \cdot \cos \left(\pi \cdot \left(z0 + z0\right)\right)\right) \cdot \color{blue}{\frac{\cos \left(\left(\pi \cdot -2\right) \cdot z0 - \pi\right) - \cos \left(\left(\pi \cdot -2\right) \cdot z0 + \pi\right)}{2}}\right)} \]

   13. Applied rewrites 53.6%

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

   14. Taylor expanded in z0 around inf

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

       1. lower-*.f64 N/A

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

       2. lower-+.f64 N/A

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

       3. lower-sin.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-PI.f64 N/A

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

       6. lower-+.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-sin.f64 N/A

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

       9. lower-*.f64 N/A

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

       10. lower-PI.f64 N/A

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

       11. lower--.f64 N/A

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

       12. lower-*.f64 53.6%

           \[\leadsto 0.5 \cdot \left(\sin \left(\pi \cdot \left(0.5 + 2 \cdot z0\right)\right) + \sin \left(\pi \cdot \left(0.5 - 2 \cdot z0\right)\right)\right) \]

   16. Applied rewrites 53.6%

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

   if 85000 < z0

   1. Initial program 57.5%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 6: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\sin \left(\left(0.5 - \left(\left|z0\right| + \left|z0\right|\right)\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= (fabs z0) 85000.0)
  (sin (* (- 0.5 (+ (fabs z0) (fabs z0))) PI))
  (sin (* 0.5 PI))))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = sin(((0.5 - (fabs(z0) + fabs(z0))) * ((double) M_PI)));
	} else {
		tmp = sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = Math.sin(((0.5 - (Math.abs(z0) + Math.abs(z0))) * Math.PI));
	} else {
		tmp = Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = math.sin(((0.5 - (math.fabs(z0) + math.fabs(z0))) * math.pi))
	else:
		tmp = math.sin((0.5 * math.pi))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = sin(Float64(Float64(0.5 - Float64(abs(z0) + abs(z0))) * pi));
	else
		tmp = sin(Float64(0.5 * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = sin(((0.5 - (abs(z0) + abs(z0))) * pi));
	else
		tmp = sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[Sin[N[(N[(0.5 - N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

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

   if 85000 < z0

   1. Initial program 57.5%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 7: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\cos \left(\left(\left|z0\right| + \left|z0\right|\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(0.5 \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= (fabs z0) 85000.0)
  (cos (* (+ (fabs z0) (fabs z0)) PI))
  (sin (* 0.5 PI))))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = cos(((fabs(z0) + fabs(z0)) * ((double) M_PI)));
	} else {
		tmp = sin((0.5 * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = Math.cos(((Math.abs(z0) + Math.abs(z0)) * Math.PI));
	} else {
		tmp = Math.sin((0.5 * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = math.cos(((math.fabs(z0) + math.fabs(z0)) * math.pi))
	else:
		tmp = math.sin((0.5 * math.pi))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = cos(Float64(Float64(abs(z0) + abs(z0)) * pi));
	else
		tmp = sin(Float64(0.5 * pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = cos(((abs(z0) + abs(z0)) * pi));
	else
		tmp = sin((0.5 * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[Cos[N[(N[(N[Abs[z0], $MachinePrecision] + N[Abs[z0], $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\sin \left(\left(0.5 - \left(z0 + z0\right)\right) \cdot \pi\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-negate-rev N/A

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

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

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 57.5%

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

   3. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\sin \left(\left(\left(z0 + z0\right) - 0.5\right) \cdot \pi + \pi\right)} \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. distribute-lft1-in N/A

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

      5. *-commutative N/A

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

      6. lift--.f64 N/A

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

      7. sub-flip N/A

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

      8. associate-+l+ N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. distribute-lft-out N/A

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

      12. lift-*.f64 N/A

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

      13. metadata-eval N/A

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

      14. mult-flip N/A

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

      15. lift-PI.f64 N/A

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

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

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

      17. lift-cos.f64 57.5%

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

   5. Applied rewrites 57.5%

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

   if 85000 < z0

   1. Initial program 57.5%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 8: 97.2% accurate, 1.1× speedup

Math

\[\sin \left(0.5 \cdot \pi\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (sin (* 0.5 PI)))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z0_] := N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 57.5%

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

2. Taylor expanded in z0 around 0

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

4. Applied rewrites 97.2%

   \[\leadsto \sin \left(\color{blue}{0.5} \cdot \pi\right) \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(sin (* (- 1/2 (+ z0 z0)) PI))"
  :precision binary64
  (sin (* (- 0.5 (+ z0 z0)) PI)))