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

Percentage Accurate: 56.9% → 98.6%

Time: 34.0s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 56.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-lft-in N/A

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

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

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

   7. *-commutative N/A

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

   8. cos-diff N/A

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

   9. lower-+.f64 N/A

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

3. Applied rewrites 56.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. sin-neg-rev N/A

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

   7. sin-+PI-rev N/A

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

   8. lower-sin.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-+.f64 98.5%

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

5. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

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

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

   6. lift-*.f64 N/A

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

   7. cos-neg-rev N/A

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

   8. lift-cos.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 98.5%

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

7. Applied rewrites 98.5%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|z0\right|\\ t_1 := \left(\pi + \pi\right) \cdot \left|z0\right|\\ \frac{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos t\_1\right) \cdot 2 + \left(\cos \left(\left(t\_0 - \pi\right) - t\_0\right) - \cos \left(t\_1 + \pi\right)\right)}{2} \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* PI (fabs z0))) (t_1 (* (+ PI PI) (fabs z0))))
  (/
   (+
    (* (+ 1/2 (* 1/2 (cos t_1))) 2)
    (- (cos (- (- t_0 PI) t_0)) (cos (+ t_1 PI))))
   2)))

C

double code(double z0) {
	double t_0 = ((double) M_PI) * fabs(z0);
	double t_1 = (((double) M_PI) + ((double) M_PI)) * fabs(z0);
	return (((0.5 + (0.5 * cos(t_1))) * 2.0) + (cos(((t_0 - ((double) M_PI)) - t_0)) - cos((t_1 + ((double) M_PI))))) / 2.0;
}

Java

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

Python

def code(z0):
	t_0 = math.pi * math.fabs(z0)
	t_1 = (math.pi + math.pi) * math.fabs(z0)
	return (((0.5 + (0.5 * math.cos(t_1))) * 2.0) + (math.cos(((t_0 - math.pi) - t_0)) - math.cos((t_1 + math.pi)))) / 2.0

Julia

function code(z0)
	t_0 = Float64(pi * abs(z0))
	t_1 = Float64(Float64(pi + pi) * abs(z0))
	return Float64(Float64(Float64(Float64(0.5 + Float64(0.5 * cos(t_1))) * 2.0) + Float64(cos(Float64(Float64(t_0 - pi) - t_0)) - cos(Float64(t_1 + pi)))) / 2.0)
end

MATLAB

function tmp = code(z0)
	t_0 = pi * abs(z0);
	t_1 = (pi + pi) * abs(z0);
	tmp = (((0.5 + (0.5 * cos(t_1))) * 2.0) + (cos(((t_0 - pi) - t_0)) - cos((t_1 + pi)))) / 2.0;
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-lft-in N/A

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

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

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

   7. *-commutative N/A

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

   8. cos-diff N/A

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

   9. lower-+.f64 N/A

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

3. Applied rewrites 56.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. sin-neg-rev N/A

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

   7. sin-+PI-rev N/A

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

   8. lower-sin.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-+.f64 98.5%

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

5. Applied rewrites 98.5%

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

   1. lift-*.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-neg.f64 N/A

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

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

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

   6. lift-*.f64 N/A

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

   7. cos-neg-rev N/A

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

   8. lift-cos.f64 N/A

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

   9. pow2 N/A

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

   10. lower-pow.f64 98.5%

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

7. Applied rewrites 98.5%

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

9. Applied rewrites 98.6%

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

Alternative 3: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left|z0\right|\\ t_1 := t\_0 + \pi\\ \frac{\cos \left(t\_1 - t\_0\right) - \cos \left(t\_1 + t\_0\right)}{2} + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \left(\pi \cdot \left(\left|z0\right| + \left|z0\right|\right)\right)\right) \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* PI (fabs z0))) (t_1 (+ t_0 PI)))
  (+
   (/ (- (cos (- t_1 t_0)) (cos (+ t_1 t_0))) 2)
   (+ 1/2 (* 1/2 (cos (* PI (+ (fabs z0) (fabs z0)))))))))

C

double code(double z0) {
	double t_0 = ((double) M_PI) * fabs(z0);
	double t_1 = t_0 + ((double) M_PI);
	return ((cos((t_1 - t_0)) - cos((t_1 + t_0))) / 2.0) + (0.5 + (0.5 * cos((((double) M_PI) * (fabs(z0) + fabs(z0))))));
}

Java

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

Python

def code(z0):
	t_0 = math.pi * math.fabs(z0)
	t_1 = t_0 + math.pi
	return ((math.cos((t_1 - t_0)) - math.cos((t_1 + t_0))) / 2.0) + (0.5 + (0.5 * math.cos((math.pi * (math.fabs(z0) + math.fabs(z0))))))

Julia

function code(z0)
	t_0 = Float64(pi * abs(z0))
	t_1 = Float64(t_0 + pi)
	return Float64(Float64(Float64(cos(Float64(t_1 - t_0)) - cos(Float64(t_1 + t_0))) / 2.0) + Float64(0.5 + Float64(0.5 * cos(Float64(pi * Float64(abs(z0) + abs(z0)))))))
end

MATLAB

function tmp = code(z0)
	t_0 = pi * abs(z0);
	t_1 = t_0 + pi;
	tmp = ((cos((t_1 - t_0)) - cos((t_1 + t_0))) / 2.0) + (0.5 + (0.5 * cos((pi * (abs(z0) + abs(z0))))));
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-+.f64 N/A

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

   6. distribute-rgt-in N/A

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

   7. associate-+l+ N/A

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

   8. sin-sum N/A

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

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

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

   10. lower-+.f64 N/A

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

3. Applied rewrites 59.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

5. Applied rewrites 98.6%

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

Alternative 4: 98.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. distribute-lft-in N/A

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

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

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

   7. *-commutative N/A

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

   8. cos-diff N/A

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

   9. lower-+.f64 N/A

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

3. Applied rewrites 56.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-neg.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. sin-neg-rev N/A

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

   7. sin-+PI-rev N/A

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

   8. lower-sin.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lower-+.f64 98.5%

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

5. Applied rewrites 98.5%

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

7. Applied rewrites 98.6%

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

Alternative 5: 62.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(\left|z0\right| + \left|z0\right|\right)\\ t_1 := \pi \cdot \left|z0\right|\\ t_2 := \left(\pi \cdot \pi\right) \cdot \left|z0\right|\\ t_3 := \sin t\_1 \cdot \cos \left(\pi \cdot \left(\left|z0\right| - \frac{-1}{2}\right)\right) + \left(\frac{1}{2} + \frac{1}{2} \cdot \cos t\_0\right)\\ t_4 := \left(t\_1 \cdot \left|z0\right|\right) \cdot \pi\\ \mathbf{if}\;\left|z0\right| \leq 53999999999999998142254366619909478518300619818547181722574336008055160832:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\left|z0\right| \leq 1400000000000000092200127673790131818671141844599020033696032063961720193169100099114440065407460048896:\\ \;\;\;\;\cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(t\_0, \left(\left(\left|z0\right| \cdot \left|z0\right|\right) \cdot \pi\right), \left(\pi + \pi\right), t\_2, \left(\left|z0\right|\right)\right)}{\frac{0 \cdot 0 - t\_4 \cdot t\_4}{\left(-\left|z0\right|\right) \cdot t\_2} + 0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (* PI (+ (fabs z0) (fabs z0))))
       (t_1 (* PI (fabs z0)))
       (t_2 (* (* PI PI) (fabs z0)))
       (t_3
        (+
         (* (sin t_1) (cos (* PI (- (fabs z0) -1/2))))
         (+ 1/2 (* 1/2 (cos t_0)))))
       (t_4 (* (* t_1 (fabs z0)) PI)))
  (if (<=
       (fabs z0)
       53999999999999998142254366619909478518300619818547181722574336008055160832)
    t_3
    (if (<=
         (fabs z0)
         1400000000000000092200127673790131818671141844599020033696032063961720193169100099114440065407460048896)
      (cos
       (/
        (304-z0z1z2z3z4
         t_0
         (* (* (fabs z0) (fabs z0)) PI)
         (+ PI PI)
         t_2
         (fabs z0))
        (+ (/ (- (* 0 0) (* t_4 t_4)) (* (- (fabs z0)) t_2)) 0)))
      t_3))))

Derivation

1. Split input into 2 regimes

2. if z0 < 5.3999999999999998e73 or 1.4000000000000001e102 < z0

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. sin-sum N/A

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

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

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

      10. lower-+.f64 N/A

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

   3. Applied rewrites 59.0%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. metadata-eval 59.0%

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

   5. Applied rewrites 59.0%

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

   if 5.3999999999999998e73 < z0 < 1.4000000000000001e102

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. flip3-+ N/A

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

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

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

      7. lower-unsound-+.f64 N/A

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

      8. lower-unsound-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 29.7%

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

      1. lift-+.f64 N/A

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

      2. count-2 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. count-2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-rgt-identity N/A

          \[\leadsto \cos \left(\frac{\left(\pi \cdot \left(z0 + z0\right)\right) \cdot \color{blue}{\left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + 0\right)}}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}\right) \]

      14. +-inverses N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 N/A

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

   5. Applied rewrites 29.6%

      \[\leadsto \cos \left(\frac{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{\left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}}\right) \]

      2. +-inverses 29.6%

         \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{0}}\right) \]

   7. Applied rewrites 29.6%

      \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{0}}\right) \]

   9. Applied rewrites 20.5%

      \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\color{blue}{\frac{0 \cdot 0 - \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi\right) \cdot \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi\right)}{\left(-z0\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot z0\right)}} + 0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 60.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| + \left|z0\right|\\ t_1 := \left(\pi \cdot \pi\right) \cdot \left|z0\right|\\ t_2 := \left(\left(\pi \cdot \left|z0\right|\right) \cdot \left|z0\right|\right) \cdot \pi\\ \mathbf{if}\;\left|z0\right| \leq 120000000000000003914723800577605532769170469312866435853625461833728:\\ \;\;\;\;\cos \left(\frac{1}{\frac{1}{\left(\pi + \pi\right) \cdot \left|z0\right|}}\right)\\ \mathbf{elif}\;\left|z0\right| \leq 1400000000000000092200127673790131818671141844599020033696032063961720193169100099114440065407460048896:\\ \;\;\;\;\cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot t\_0\right), \left(\left(\left|z0\right| \cdot \left|z0\right|\right) \cdot \pi\right), \left(\pi + \pi\right), t\_1, \left(\left|z0\right|\right)\right)}{\frac{0 \cdot 0 - t\_2 \cdot t\_2}{\left(-\left|z0\right|\right) \cdot t\_1} + 0}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\left(t\_0 - \frac{-1}{2}\right) \cdot {\pi}^{\frac{2}{3}}\right) \cdot \sqrt[3]{\pi}\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z0) (fabs z0)))
       (t_1 (* (* PI PI) (fabs z0)))
       (t_2 (* (* (* PI (fabs z0)) (fabs z0)) PI)))
  (if (<=
       (fabs z0)
       120000000000000003914723800577605532769170469312866435853625461833728)
    (cos (/ 1 (/ 1 (* (+ PI PI) (fabs z0)))))
    (if (<=
         (fabs z0)
         1400000000000000092200127673790131818671141844599020033696032063961720193169100099114440065407460048896)
      (cos
       (/
        (304-z0z1z2z3z4
         (* PI t_0)
         (* (* (fabs z0) (fabs z0)) PI)
         (+ PI PI)
         t_1
         (fabs z0))
        (+ (/ (- (* 0 0) (* t_2 t_2)) (* (- (fabs z0)) t_1)) 0)))
      (sin (* (* (- t_0 -1/2) (pow PI 2/3)) (cbrt PI)))))))

Derivation

1. Split input into 3 regimes

2. if z0 < 1.2e68

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow1/3 N/A

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

      12. pow-prod-up N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 56.8%

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

   3. Applied rewrites 56.8%

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

   5. Applied rewrites 29.7%

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

   7. Applied rewrites 56.9%

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

   if 1.2e68 < z0 < 1.4000000000000001e102

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. flip3-+ N/A

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

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

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

      7. lower-unsound-+.f64 N/A

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

      8. lower-unsound-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 29.7%

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

      1. lift-+.f64 N/A

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

      2. count-2 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. count-2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-rgt-identity N/A

          \[\leadsto \cos \left(\frac{\left(\pi \cdot \left(z0 + z0\right)\right) \cdot \color{blue}{\left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + 0\right)}}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}\right) \]

      14. +-inverses N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 N/A

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

   5. Applied rewrites 29.6%

      \[\leadsto \cos \left(\frac{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{\left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}}\right) \]

      2. +-inverses 29.6%

         \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{0}}\right) \]

   7. Applied rewrites 29.6%

      \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{0}}\right) \]

   9. Applied rewrites 20.5%

      \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\color{blue}{\frac{0 \cdot 0 - \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi\right) \cdot \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi\right)}{\left(-z0\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot z0\right)}} + 0}\right) \]

   if 1.4000000000000001e102 < z0

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-PI.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. metadata-eval 56.9%

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

   3. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. rem-3cbrt-lft N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-PI.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lift-cbrt.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-cbrt.f64 N/A

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

      10. lift-cbrt.f64 N/A

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

   5. Applied rewrites 56.8%

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

Alternative 7: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{1}{\frac{1}{\left(\pi + \pi\right) \cdot \left|z0\right|}}\right)\\ t_1 := \left(\pi \cdot \pi\right) \cdot \left|z0\right|\\ t_2 := \left(\left(\pi \cdot \left|z0\right|\right) \cdot \left|z0\right|\right) \cdot \pi\\ \mathbf{if}\;\left|z0\right| \leq 120000000000000003914723800577605532769170469312866435853625461833728:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\left|z0\right| \leq 1400000000000000092200127673790131818671141844599020033696032063961720193169100099114440065407460048896:\\ \;\;\;\;\cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(\left|z0\right| + \left|z0\right|\right)\right), \left(\left(\left|z0\right| \cdot \left|z0\right|\right) \cdot \pi\right), \left(\pi + \pi\right), t\_1, \left(\left|z0\right|\right)\right)}{\frac{0 \cdot 0 - t\_2 \cdot t\_2}{\left(-\left|z0\right|\right) \cdot t\_1} + 0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (cos (/ 1 (/ 1 (* (+ PI PI) (fabs z0))))))
       (t_1 (* (* PI PI) (fabs z0)))
       (t_2 (* (* (* PI (fabs z0)) (fabs z0)) PI)))
  (if (<=
       (fabs z0)
       120000000000000003914723800577605532769170469312866435853625461833728)
    t_0
    (if (<=
         (fabs z0)
         1400000000000000092200127673790131818671141844599020033696032063961720193169100099114440065407460048896)
      (cos
       (/
        (304-z0z1z2z3z4
         (* PI (+ (fabs z0) (fabs z0)))
         (* (* (fabs z0) (fabs z0)) PI)
         (+ PI PI)
         t_1
         (fabs z0))
        (+ (/ (- (* 0 0) (* t_2 t_2)) (* (- (fabs z0)) t_1)) 0)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if z0 < 1.2e68 or 1.4000000000000001e102 < z0

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow1/3 N/A

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

      12. pow-prod-up N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 56.8%

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

   3. Applied rewrites 56.8%

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

   5. Applied rewrites 29.7%

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

   7. Applied rewrites 56.9%

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

   if 1.2e68 < z0 < 1.4000000000000001e102

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. flip3-+ N/A

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

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

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

      7. lower-unsound-+.f64 N/A

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

      8. lower-unsound-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 29.7%

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

      1. lift-+.f64 N/A

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

      2. count-2 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. count-2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-rgt-identity N/A

          \[\leadsto \cos \left(\frac{\left(\pi \cdot \left(z0 + z0\right)\right) \cdot \color{blue}{\left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + 0\right)}}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}\right) \]

      14. +-inverses N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 N/A

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

   5. Applied rewrites 29.6%

      \[\leadsto \cos \left(\frac{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{\left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}}\right) \]

      2. +-inverses 29.6%

         \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{0}}\right) \]

   7. Applied rewrites 29.6%

      \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{0}}\right) \]

   9. Applied rewrites 20.5%

      \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\color{blue}{\frac{0 \cdot 0 - \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi\right) \cdot \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi\right)}{\left(-z0\right) \cdot \left(\left(\pi \cdot \pi\right) \cdot z0\right)}} + 0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 60.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{1}{\frac{1}{\left(\pi + \pi\right) \cdot \left|z0\right|}}\right)\\ t_1 := \left(\left(\pi \cdot \left|z0\right|\right) \cdot \left|z0\right|\right) \cdot \pi\\ \mathbf{if}\;\left|z0\right| \leq 999999999999999939709166371603178586112:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\left|z0\right| \leq 1400000000000000092200127673790131818671141844599020033696032063961720193169100099114440065407460048896:\\ \;\;\;\;\cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(\left|z0\right| + \left|z0\right|\right)\right), \left(\left(\left|z0\right| \cdot \left|z0\right|\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot \left|z0\right|\right), \left(\left|z0\right|\right)\right)}{\frac{t\_1 \cdot t\_1 - 0 \cdot 0}{t\_1} + 0}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (cos (/ 1 (/ 1 (* (+ PI PI) (fabs z0))))))
       (t_1 (* (* (* PI (fabs z0)) (fabs z0)) PI)))
  (if (<= (fabs z0) 999999999999999939709166371603178586112)
    t_0
    (if (<=
         (fabs z0)
         1400000000000000092200127673790131818671141844599020033696032063961720193169100099114440065407460048896)
      (cos
       (/
        (304-z0z1z2z3z4
         (* PI (+ (fabs z0) (fabs z0)))
         (* (* (fabs z0) (fabs z0)) PI)
         (+ PI PI)
         (* (* PI PI) (fabs z0))
         (fabs z0))
        (+ (/ (- (* t_1 t_1) (* 0 0)) t_1) 0)))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if z0 < 9.9999999999999994e38 or 1.4000000000000001e102 < z0

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-PI.f64 N/A

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

      3. add-cube-cbrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. pow1/3 N/A

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

      10. lift-PI.f64 N/A

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

      11. pow1/3 N/A

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

      12. pow-prod-up N/A

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

      13. lower-pow.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-PI.f64 N/A

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

      16. lower-cbrt.f64 56.8%

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

   3. Applied rewrites 56.8%

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

   5. Applied rewrites 29.7%

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

   7. Applied rewrites 56.9%

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

   if 9.9999999999999994e38 < z0 < 1.4000000000000001e102

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. flip3-+ N/A

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

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

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

      7. lower-unsound-+.f64 N/A

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

      8. lower-unsound-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-unsound-pow.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lower-unsound-+.f64 N/A

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

   3. Applied rewrites 29.7%

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

      1. lift-+.f64 N/A

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

      2. count-2 N/A

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

      3. lift-pow.f64 N/A

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

      4. cube-mult N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. count-2 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-lft-in N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-rgt-identity N/A

          \[\leadsto \cos \left(\frac{\left(\pi \cdot \left(z0 + z0\right)\right) \cdot \color{blue}{\left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + 0\right)}}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}\right) \]

      14. +-inverses N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 N/A

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

   5. Applied rewrites 29.6%

      \[\leadsto \cos \left(\frac{\color{blue}{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}\right) \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{\left(\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) - \left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right)\right)}}\right) \]

      2. +-inverses 29.6%

         \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{0}}\right) \]

   7. Applied rewrites 29.6%

      \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\left(\pi \cdot z0\right) \cdot \left(\pi \cdot z0\right) + \color{blue}{0}}\right) \]

   9. Applied rewrites 20.5%

      \[\leadsto \cos \left(\frac{\mathsf{304\_z0z1z2z3z4}\left(\left(\pi \cdot \left(z0 + z0\right)\right), \left(\left(z0 \cdot z0\right) \cdot \pi\right), \left(\pi + \pi\right), \left(\left(\pi \cdot \pi\right) \cdot z0\right), z0\right)}{\color{blue}{\frac{\left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi\right) \cdot \left(\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi\right) - 0 \cdot 0}{\left(\left(\pi \cdot z0\right) \cdot z0\right) \cdot \pi}} + 0}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 59.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) + abs(z0);
	t_1 = cos((t_0 * pi));
	tmp = 0.0;
	if (t_1 <= 0.1)
		tmp = sin(((0.5 - t_0) * pi));
	else
		tmp = t_1;
	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[Cos[N[(t$95$0 * Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, 3602879701896397/36028797018963968], N[Sin[N[(N[(1/2 - t$95$0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lift-PI.f64 N/A

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

      9. mult-flip N/A

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

      10. distribute-lft-out N/A

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

      11. lower-*.f64 N/A

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

      12. lower-+.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. count-2 N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval 56.9%

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

   3. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 56.9%

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

      4. lift-+.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. metadata-eval N/A

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

      9. count-2 N/A

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

      10. lift-+.f64 N/A

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

      11. lower--.f64 56.9%

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

   5. Applied rewrites 56.9%

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

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

   1. Initial program 56.9%

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

Alternative 10: 59.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left|z0\right| + \left|z0\right|\\ t_1 := \cos \left(t\_0 \cdot \pi\right)\\ \mathbf{if}\;t\_1 \leq \frac{-5764607523034235}{288230376151711744}:\\ \;\;\;\;\sin \left(\pi \cdot \left(\frac{1}{2} + t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (+ (fabs z0) (fabs z0))) (t_1 (cos (* t_0 PI))))
  (if (<= t_1 -5764607523034235/288230376151711744)
    (sin (* PI (+ 1/2 t_0)))
    t_1)))

C

double code(double z0) {
	double t_0 = fabs(z0) + fabs(z0);
	double t_1 = cos((t_0 * ((double) M_PI)));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = sin((((double) M_PI) * (0.5 + t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.abs(z0) + Math.abs(z0);
	double t_1 = Math.cos((t_0 * Math.PI));
	double tmp;
	if (t_1 <= -0.02) {
		tmp = Math.sin((Math.PI * (0.5 + t_0)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.fabs(z0) + math.fabs(z0)
	t_1 = math.cos((t_0 * math.pi))
	tmp = 0
	if t_1 <= -0.02:
		tmp = math.sin((math.pi * (0.5 + t_0)))
	else:
		tmp = t_1
	return tmp

Julia

function code(z0)
	t_0 = Float64(abs(z0) + abs(z0))
	t_1 = cos(Float64(t_0 * pi))
	tmp = 0.0
	if (t_1 <= -0.02)
		tmp = sin(Float64(pi * Float64(0.5 + t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = abs(z0) + abs(z0);
	t_1 = cos((t_0 * pi));
	tmp = 0.0;
	if (t_1 <= -0.02)
		tmp = sin((pi * (0.5 + t_0)));
	else
		tmp = t_1;
	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[Cos[N[(t$95$0 * Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -5764607523034235/288230376151711744], N[Sin[N[(Pi * N[(1/2 + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-PI.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. distribute-lft-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. metadata-eval 56.9%

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

   3. Applied rewrites 56.9%

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

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

   1. Initial program 56.9%

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

Reproduce

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