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

Percentage Accurate: 56.9% → 98.7%

Time: 7.2s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 56.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

double code(double z0) {
	double t_0 = -((double) M_PI) * z0;
	double t_1 = t_0 - (((double) M_PI) * -1.0);
	return ((cos((t_1 - t_0)) - cos((t_1 + t_0))) / 2.0) + (0.5 + (0.5 * cos(((((double) M_PI) + ((double) M_PI)) * z0))));
}

Java

public static double code(double z0) {
	double t_0 = -Math.PI * z0;
	double t_1 = t_0 - (Math.PI * -1.0);
	return ((Math.cos((t_1 - t_0)) - Math.cos((t_1 + t_0))) / 2.0) + (0.5 + (0.5 * Math.cos(((Math.PI + Math.PI) * z0))));
}

Python

def code(z0):
	t_0 = -math.pi * z0
	t_1 = t_0 - (math.pi * -1.0)
	return ((math.cos((t_1 - t_0)) - math.cos((t_1 + t_0))) / 2.0) + (0.5 + (0.5 * math.cos(((math.pi + math.pi) * z0))))

Julia

function code(z0)
	t_0 = Float64(Float64(-pi) * z0)
	t_1 = Float64(t_0 - Float64(pi * -1.0))
	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(Float64(pi + pi) * z0)))))
end

MATLAB

function tmp = code(z0)
	t_0 = -pi * z0;
	t_1 = t_0 - (pi * -1.0);
	tmp = ((cos((t_1 - t_0)) - cos((t_1 + t_0))) / 2.0) + (0.5 + (0.5 * cos(((pi + pi) * z0))));
end

Wolfram

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

Derivation

1. Initial program 56.9%

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

3. Applied rewrites 59.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-cos.f64 N/A

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

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

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

   5. lift-sin.f64 N/A

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

   6. sin-mult N/A

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

   7. lower-/.f64 N/A

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

5. Applied rewrites 98.7%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. metadata-eval N/A

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

   15. lift-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. mult-flip-rev N/A

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

7. Applied rewrites 98.6%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

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

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

   4. metadata-eval N/A

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

   5. *-commutative N/A

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

   6. lift--.f64 N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. lift-neg.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

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

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

   14. metadata-eval N/A

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

   15. lift-*.f64 N/A

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

   16. metadata-eval N/A

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

   17. mult-flip-rev N/A

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

9. Applied rewrites 98.6%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

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

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

    5. lower-*.f64 N/A

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

    6. metadata-eval 98.6%

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

11. Applied rewrites 98.6%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

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

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

    5. lower-*.f64 N/A

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

    6. metadata-eval 98.6%

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

13. Applied rewrites 98.6%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

3. Applied rewrites 59.1%

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

5. Applied rewrites 59.0%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lift-+.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. lift--.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. mult-flip-rev N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

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

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

   15. metadata-eval N/A

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

   16. lift-*.f64 N/A

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

   17. lift--.f64 N/A

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

7. Applied rewrites 60.5%

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

9. Applied rewrites 98.7%

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

Alternative 3: 98.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.9%

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

3. Applied rewrites 59.1%

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

   1. lift-cos.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. metadata-eval N/A

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

   6. mult-flip N/A

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

   7. lift-PI.f64 N/A

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

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

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

   9. sin-+PI-rev N/A

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

   10. lower-sin.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-+.f64 98.6%

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

5. Applied rewrites 98.6%

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

Alternative 4: 61.6% accurate, 0.2× speedup

Math

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

FPCore

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

C

double code(double z0) {
	double tmp;
	if (cos((fabs(z0) * (((double) M_PI) + ((double) M_PI)))) <= -0.02) {
		tmp = -cos((2.0 * ((((double) M_PI) * fabs(z0)) - (-0.5 * ((double) M_PI)))));
	} else {
		tmp = 0.5 - ((-0.5 * cos(((fabs(z0) + fabs(z0)) * ((double) M_PI)))) - (cos(((0.5 - fabs(z0)) * ((double) M_PI))) * sin((-((double) M_PI) * fabs(z0)))));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.cos((Math.abs(z0) * (Math.PI + Math.PI))) <= -0.02) {
		tmp = -Math.cos((2.0 * ((Math.PI * Math.abs(z0)) - (-0.5 * Math.PI))));
	} else {
		tmp = 0.5 - ((-0.5 * Math.cos(((Math.abs(z0) + Math.abs(z0)) * Math.PI))) - (Math.cos(((0.5 - Math.abs(z0)) * Math.PI)) * Math.sin((-Math.PI * Math.abs(z0)))));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if math.cos((math.fabs(z0) * (math.pi + math.pi))) <= -0.02:
		tmp = -math.cos((2.0 * ((math.pi * math.fabs(z0)) - (-0.5 * math.pi))))
	else:
		tmp = 0.5 - ((-0.5 * math.cos(((math.fabs(z0) + math.fabs(z0)) * math.pi))) - (math.cos(((0.5 - math.fabs(z0)) * math.pi)) * math.sin((-math.pi * math.fabs(z0)))))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (cos(Float64(abs(z0) * Float64(pi + pi))) <= -0.02)
		tmp = Float64(-cos(Float64(2.0 * Float64(Float64(pi * abs(z0)) - Float64(-0.5 * pi)))));
	else
		tmp = Float64(0.5 - Float64(Float64(-0.5 * cos(Float64(Float64(abs(z0) + abs(z0)) * pi))) - Float64(cos(Float64(Float64(0.5 - abs(z0)) * pi)) * sin(Float64(Float64(-pi) * abs(z0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (cos((abs(z0) * (pi + pi))) <= -0.02)
		tmp = -cos((2.0 * ((pi * abs(z0)) - (-0.5 * pi))));
	else
		tmp = 0.5 - ((-0.5 * cos(((abs(z0) + abs(z0)) * pi))) - (cos(((0.5 - abs(z0)) * pi)) * sin((-pi * abs(z0)))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. cos-sum N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 N/A

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

   3. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. difference-of-squares-rev N/A

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

      5. sub-negate-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lift-sin.f64 N/A

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

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

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

   5. Applied rewrites 57.0%

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

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

   1. Initial program 56.9%

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

   3. Applied rewrites 59.1%

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

   5. Applied rewrites 59.0%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lift-PI.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

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

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

      15. metadata-eval N/A

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

      16. lift-*.f64 N/A

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

      17. lift--.f64 N/A

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

   7. Applied rewrites 60.5%

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

   9. Applied rewrites 59.1%

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

Alternative 5: 61.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\left|z0\right| \cdot \left(\pi + \pi\right)\right)\\ \mathbf{if}\;t\_0 \leq \frac{-1152921504606847}{1152921504606846976}:\\ \;\;\;\;-\cos \left(2 \cdot \left(\pi \cdot \left|z0\right| - \frac{-1}{2} \cdot \pi\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (cos (* (fabs z0) (+ PI PI)))))
  (if (<= t_0 -1152921504606847/1152921504606846976)
    (- (cos (* 2 (- (* PI (fabs z0)) (* -1/2 PI)))))
    t_0)))

C

double code(double z0) {
	double t_0 = cos((fabs(z0) * (((double) M_PI) + ((double) M_PI))));
	double tmp;
	if (t_0 <= -0.001) {
		tmp = -cos((2.0 * ((((double) M_PI) * fabs(z0)) - (-0.5 * ((double) M_PI)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = Math.cos((Math.abs(z0) * (Math.PI + Math.PI)));
	double tmp;
	if (t_0 <= -0.001) {
		tmp = -Math.cos((2.0 * ((Math.PI * Math.abs(z0)) - (-0.5 * Math.PI))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0):
	t_0 = math.cos((math.fabs(z0) * (math.pi + math.pi)))
	tmp = 0
	if t_0 <= -0.001:
		tmp = -math.cos((2.0 * ((math.pi * math.fabs(z0)) - (-0.5 * math.pi))))
	else:
		tmp = t_0
	return tmp

Julia

function code(z0)
	t_0 = cos(Float64(abs(z0) * Float64(pi + pi)))
	tmp = 0.0
	if (t_0 <= -0.001)
		tmp = Float64(-cos(Float64(2.0 * Float64(Float64(pi * abs(z0)) - Float64(-0.5 * pi)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = cos((abs(z0) * (pi + pi)));
	tmp = 0.0;
	if (t_0 <= -0.001)
		tmp = -cos((2.0 * ((pi * abs(z0)) - (-0.5 * pi))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[Cos[N[(N[Abs[z0], $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -1152921504606847/1152921504606846976], (-N[Cos[N[(2 * N[(N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision] - N[(-1/2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (cos.f64 (*.f64 z0 (+.f64 (PI.f64) (PI.f64)))) < -1e-3

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-lft-in N/A

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

      5. cos-sum N/A

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

      6. difference-of-squares N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-sin.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower--.f64 N/A

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

   3. Applied rewrites 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift--.f64 N/A

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

      4. difference-of-squares-rev N/A

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

      5. sub-negate-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. lift-sin.f64 N/A

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

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

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

      9. lift-sin.f64 N/A

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

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

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

   5. Applied rewrites 57.0%

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

   if -1e-3 < (cos.f64 (*.f64 z0 (+.f64 (PI.f64) (PI.f64))))

   1. Initial program 56.9%

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

Alternative 6: 59.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

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

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

      4. lower-sin.f64 N/A

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

      5. add-flip N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. count-2 N/A

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

      10. associate-*r* N/A

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

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

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. count-2 N/A

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

      15. lower-+.f64 N/A

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

      16. lower-neg.f64 N/A

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

      17. lift-PI.f64 N/A

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

      18. mult-flip N/A

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

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

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. metadata-eval N/A

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

      23. metadata-eval N/A

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

   3. Applied rewrites 56.9%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

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

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

      5. metadata-eval N/A

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. mult-flip-rev N/A

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

      9. lift-PI.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-PI.f64 N/A

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

      13. mult-flip-rev N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

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

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

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

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

      18. lift-neg.f64 N/A

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

      19. distribute-lft-out N/A

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

      20. lower-*.f64 N/A

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

      21. lower-+.f64 56.9%

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

   5. Applied rewrites 56.9%

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

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

   1. Initial program 56.9%

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

Alternative 7: 59.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 56.9%

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

      1. lift-cos.f64 N/A

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

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

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

      3. lower-sin.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-PI.f64 N/A

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

      6. mult-flip N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. distribute-rgt-out N/A

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

      11. distribute-lft-out N/A

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

      12. lower-*.f64 N/A

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

      13. count-2 N/A

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

      14. *-commutative N/A

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

      15. lower-+.f64 N/A

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

      16. metadata-eval N/A

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

      17. *-commutative N/A

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

      18. count-2 N/A

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

      19. lower-+.f64 56.9%

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

   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 z0 (+.f64 (PI.f64) (PI.f64))))

   1. Initial program 56.9%

      \[\cos \left(z0 \cdot \left(\pi + \pi\right)\right) \]
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 (+ PI PI)))"
  :precision binary64
  (cos (* z0 (+ PI PI))))