(- 1 (cos (* (atan (* (tan (* (- (+ z2 z2) -1/2) PI)) (/ z0 z1))) 2)))

Percentage Accurate: 60.1% → 91.6%

Time: 8.4s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (-
 1.0
 (cos (* (atan (* (tan (* (- (+ z2 z2) -0.5) PI)) (/ z0 z1))) 2.0))))

C

double code(double z2, double z0, double z1) {
	return 1.0 - cos((atan((tan((((z2 + z2) - -0.5) * ((double) M_PI))) * (z0 / z1))) * 2.0));
}

Java

public static double code(double z2, double z0, double z1) {
	return 1.0 - Math.cos((Math.atan((Math.tan((((z2 + z2) - -0.5) * Math.PI)) * (z0 / z1))) * 2.0));
}

Python

def code(z2, z0, z1):
	return 1.0 - math.cos((math.atan((math.tan((((z2 + z2) - -0.5) * math.pi)) * (z0 / z1))) * 2.0))

Julia

function code(z2, z0, z1)
	return Float64(1.0 - cos(Float64(atan(Float64(tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) * Float64(z0 / z1))) * 2.0)))
end

MATLAB

function tmp = code(z2, z0, z1)
	tmp = 1.0 - cos((atan((tan((((z2 + z2) - -0.5) * pi)) * (z0 / z1))) * 2.0));
end

Wolfram

code[z2_, z0_, z1_] := N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 60.1% accurate, 1.0× speedup

Math

\[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (-
 1.0
 (cos (* (atan (* (tan (* (- (+ z2 z2) -0.5) PI)) (/ z0 z1))) 2.0))))

C

double code(double z2, double z0, double z1) {
	return 1.0 - cos((atan((tan((((z2 + z2) - -0.5) * ((double) M_PI))) * (z0 / z1))) * 2.0));
}

Java

public static double code(double z2, double z0, double z1) {
	return 1.0 - Math.cos((Math.atan((Math.tan((((z2 + z2) - -0.5) * Math.PI)) * (z0 / z1))) * 2.0));
}

Python

def code(z2, z0, z1):
	return 1.0 - math.cos((math.atan((math.tan((((z2 + z2) - -0.5) * math.pi)) * (z0 / z1))) * 2.0))

Julia

function code(z2, z0, z1)
	return Float64(1.0 - cos(Float64(atan(Float64(tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) * Float64(z0 / z1))) * 2.0)))
end

MATLAB

function tmp = code(z2, z0, z1)
	tmp = 1.0 - cos((atan((tan((((z2 + z2) - -0.5) * pi)) * (z0 / z1))) * 2.0));
end

Wolfram

code[z2_, z0_, z1_] := N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 91.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := 1 - \cos \left(\tan^{-1} \left(\frac{\left(2 \cdot \left(z2 \cdot \left(\pi + \frac{\pi \cdot \left(0.5 - 0.5 \cdot \cos \pi\right)}{0.5 + 0.5 \cdot \cos \left(-1 \cdot \pi\right)}\right)\right) - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right) \cdot 2\right)\\ \mathbf{if}\;z2 \leq -2.8 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq 0.01:\\ \;\;\;\;1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + 0.5 \cdot \pi\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0
        (-
         1.0
         (cos
          (*
           (atan
            (/
             (*
              (-
               (*
                2.0
                (*
                 z2
                 (+
                  PI
                  (/
                   (* PI (- 0.5 (* 0.5 (cos PI))))
                   (+ 0.5 (* 0.5 (cos (* -1.0 PI))))))))
               (tan (* PI -0.5)))
              z0)
             z1))
           2.0)))))
  (if (<= z2 -2.8e+21)
    t_0
    (if (<= z2 0.01)
      (-
       1.0
       (cos
        (*
         (atan
          (/
           (* (sin (+ (* (- PI) (+ z2 z2)) (* 0.5 PI))) z0)
           (* (- (sin (* PI (+ z2 z2)))) z1)))
         2.0)))
      t_0))))

C

double code(double z2, double z0, double z1) {
	double t_0 = 1.0 - cos((atan(((((2.0 * (z2 * (((double) M_PI) + ((((double) M_PI) * (0.5 - (0.5 * cos(((double) M_PI))))) / (0.5 + (0.5 * cos((-1.0 * ((double) M_PI))))))))) - tan((((double) M_PI) * -0.5))) * z0) / z1)) * 2.0));
	double tmp;
	if (z2 <= -2.8e+21) {
		tmp = t_0;
	} else if (z2 <= 0.01) {
		tmp = 1.0 - cos((atan(((sin(((-((double) M_PI) * (z2 + z2)) + (0.5 * ((double) M_PI)))) * z0) / (-sin((((double) M_PI) * (z2 + z2))) * z1))) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = 1.0 - Math.cos((Math.atan(((((2.0 * (z2 * (Math.PI + ((Math.PI * (0.5 - (0.5 * Math.cos(Math.PI)))) / (0.5 + (0.5 * Math.cos((-1.0 * Math.PI)))))))) - Math.tan((Math.PI * -0.5))) * z0) / z1)) * 2.0));
	double tmp;
	if (z2 <= -2.8e+21) {
		tmp = t_0;
	} else if (z2 <= 0.01) {
		tmp = 1.0 - Math.cos((Math.atan(((Math.sin(((-Math.PI * (z2 + z2)) + (0.5 * Math.PI))) * z0) / (-Math.sin((Math.PI * (z2 + z2))) * z1))) * 2.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = 1.0 - math.cos((math.atan(((((2.0 * (z2 * (math.pi + ((math.pi * (0.5 - (0.5 * math.cos(math.pi)))) / (0.5 + (0.5 * math.cos((-1.0 * math.pi)))))))) - math.tan((math.pi * -0.5))) * z0) / z1)) * 2.0))
	tmp = 0
	if z2 <= -2.8e+21:
		tmp = t_0
	elif z2 <= 0.01:
		tmp = 1.0 - math.cos((math.atan(((math.sin(((-math.pi * (z2 + z2)) + (0.5 * math.pi))) * z0) / (-math.sin((math.pi * (z2 + z2))) * z1))) * 2.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(1.0 - cos(Float64(atan(Float64(Float64(Float64(Float64(2.0 * Float64(z2 * Float64(pi + Float64(Float64(pi * Float64(0.5 - Float64(0.5 * cos(pi)))) / Float64(0.5 + Float64(0.5 * cos(Float64(-1.0 * pi)))))))) - tan(Float64(pi * -0.5))) * z0) / z1)) * 2.0)))
	tmp = 0.0
	if (z2 <= -2.8e+21)
		tmp = t_0;
	elseif (z2 <= 0.01)
		tmp = Float64(1.0 - cos(Float64(atan(Float64(Float64(sin(Float64(Float64(Float64(-pi) * Float64(z2 + z2)) + Float64(0.5 * pi))) * z0) / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * z1))) * 2.0)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = 1.0 - cos((atan(((((2.0 * (z2 * (pi + ((pi * (0.5 - (0.5 * cos(pi)))) / (0.5 + (0.5 * cos((-1.0 * pi)))))))) - tan((pi * -0.5))) * z0) / z1)) * 2.0));
	tmp = 0.0;
	if (z2 <= -2.8e+21)
		tmp = t_0;
	elseif (z2 <= 0.01)
		tmp = 1.0 - cos((atan(((sin(((-pi * (z2 + z2)) + (0.5 * pi))) * z0) / (-sin((pi * (z2 + z2))) * z1))) * 2.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[(N[(N[(2.0 * N[(z2 * N[(Pi + N[(N[(Pi * N[(0.5 - N[(0.5 * N[Cos[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(-1.0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z2, -2.8e+21], t$95$0, If[LessEqual[z2, 0.01], N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[(N[Sin[N[(N[((-Pi) * N[(z2 + z2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * z0), $MachinePrecision] / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z2 < -2.8e21 or 0.01 < z2

   1. Initial program 60.1%

      \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   2. Taylor expanded in z2 around 0

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\left(z2 \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\left(z2 \cdot \left(2 \cdot \mathsf{PI}\left(\right) - -2 \cdot \frac{\mathsf{PI}\left(\right) \cdot {\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}^{2}}\right) + \color{blue}{\frac{\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}{\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   4. Applied rewrites 71.4%

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\left(z2 \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   6. Applied rewrites 71.9%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right)} \cdot 2\right) \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\tan \left(\frac{1}{2} \cdot \pi\right) \cdot \tan \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\tan \left(\frac{1}{2} \cdot \pi\right) \cdot \tan \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      3. tan-quot N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \tan \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \tan \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      5. lift-cos.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \tan \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      6. lift-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \tan \left(\frac{1}{2} \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      7. tan-quot N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)}\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      8. lift-sin.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)}\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      9. lift-cos.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \left(\frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \frac{\sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right)}\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      10. frac-times N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \frac{\sin \left(\frac{1}{2} \cdot \pi\right) \cdot \sin \left(\frac{1}{2} \cdot \pi\right)}{\cos \left(\frac{1}{2} \cdot \pi\right) \cdot \cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      11. unpow2 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \frac{{\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{\cos \left(\frac{1}{2} \cdot \pi\right) \cdot \cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      12. lift-pow.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \frac{{\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{\cos \left(\frac{1}{2} \cdot \pi\right) \cdot \cos \left(\frac{1}{2} \cdot \pi\right)} \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      13. unpow2 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \frac{{\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}} \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      14. lift-pow.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \frac{{\sin \left(\frac{1}{2} \cdot \pi\right)}^{2}}{{\cos \left(\frac{1}{2} \cdot \pi\right)}^{2}} \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

      15. lower-/.f64 71.9%

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \frac{{\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}} \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

   8. Applied rewrites 84.5%

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(\left(2 \cdot \left(\pi + \frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot 0.5\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot -0.5\right)\right)} \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

   9. Taylor expanded in z2 around 0

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(2 \cdot \left(z2 \cdot \left(\pi + \frac{\pi \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \pi\right)}\right)\right) - \tan \color{blue}{\left(\pi \cdot -0.5\right)}\right) \cdot z0}{z1}\right) \cdot 2\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(2 \cdot \left(z2 \cdot \left(\mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) - \tan \left(\pi \cdot \color{blue}{\frac{-1}{2}}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

       2. lower-*.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(2 \cdot \left(z2 \cdot \left(\mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

       3. lower-+.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(2 \cdot \left(z2 \cdot \left(\mathsf{PI}\left(\right) + \frac{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

       4. lower-PI.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(2 \cdot \left(z2 \cdot \left(\pi + \frac{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

       5. lower-/.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(2 \cdot \left(z2 \cdot \left(\pi + \frac{\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\frac{1}{2} + \frac{1}{2} \cdot \cos \left(-1 \cdot \mathsf{PI}\left(\right)\right)}\right)\right) - \tan \left(\pi \cdot \frac{-1}{2}\right)\right) \cdot z0}{z1}\right) \cdot 2\right) \]

   11. Applied rewrites 84.5%

       \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(2 \cdot \left(z2 \cdot \left(\pi + \frac{\pi \cdot \left(0.5 - 0.5 \cdot \cos \pi\right)}{0.5 + 0.5 \cdot \cos \left(-1 \cdot \pi\right)}\right)\right) - \tan \color{blue}{\left(\pi \cdot -0.5\right)}\right) \cdot z0}{z1}\right) \cdot 2\right) \]

   if -2.8e21 < z2 < 0.01

   1. Initial program 60.1%

      \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} \cdot 2\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      3. tan-quot N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      4. lift-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \color{blue}{\frac{z0}{z1}}\right) \cdot 2\right) \]

      5. frac-times N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

   3. Applied rewrites 69.2%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\cos \left(\pi \cdot \left(z2 + z2\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)} \cdot 2\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\cos \left(\pi \cdot \left(z2 + z2\right)\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      2. cos-neg-rev N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \left(z2 + z2\right)\right)\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

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

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(z2 + z2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(z2 + z2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \left(z2 + z2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z2 + z2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

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

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\color{blue}{\left(\mathsf{neg}\left(\pi\right)\right) \cdot \left(z2 + z2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\color{blue}{\left(\mathsf{neg}\left(\pi\right)\right) \cdot \left(z2 + z2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\color{blue}{\left(-\pi\right)} \cdot \left(z2 + z2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      11. mult-flip N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      12. metadata-eval N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      13. *-commutative N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      14. lower-*.f64 69.2%

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \color{blue}{0.5 \cdot \pi}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

   5. Applied rewrites 69.2%

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + 0.5 \cdot \pi\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 86.2% accurate, 0.6× speedup

Math

\[1 - \cos \left(\tan^{-1} \left(\frac{\left(1 + -2 \cdot \left({z2}^{2} \cdot {\pi}^{2}\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (-
 1.0
 (cos
  (*
   (atan
    (/
     (* (+ 1.0 (* -2.0 (* (pow z2 2.0) (pow PI 2.0)))) z0)
     (* (- (sin (* PI (+ z2 z2)))) z1)))
   2.0))))

C

double code(double z2, double z0, double z1) {
	return 1.0 - cos((atan((((1.0 + (-2.0 * (pow(z2, 2.0) * pow(((double) M_PI), 2.0)))) * z0) / (-sin((((double) M_PI) * (z2 + z2))) * z1))) * 2.0));
}

Java

public static double code(double z2, double z0, double z1) {
	return 1.0 - Math.cos((Math.atan((((1.0 + (-2.0 * (Math.pow(z2, 2.0) * Math.pow(Math.PI, 2.0)))) * z0) / (-Math.sin((Math.PI * (z2 + z2))) * z1))) * 2.0));
}

Python

def code(z2, z0, z1):
	return 1.0 - math.cos((math.atan((((1.0 + (-2.0 * (math.pow(z2, 2.0) * math.pow(math.pi, 2.0)))) * z0) / (-math.sin((math.pi * (z2 + z2))) * z1))) * 2.0))

Julia

function code(z2, z0, z1)
	return Float64(1.0 - cos(Float64(atan(Float64(Float64(Float64(1.0 + Float64(-2.0 * Float64((z2 ^ 2.0) * (pi ^ 2.0)))) * z0) / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * z1))) * 2.0)))
end

MATLAB

function tmp = code(z2, z0, z1)
	tmp = 1.0 - cos((atan((((1.0 + (-2.0 * ((z2 ^ 2.0) * (pi ^ 2.0)))) * z0) / (-sin((pi * (z2 + z2))) * z1))) * 2.0));
end

Wolfram

code[z2_, z0_, z1_] := N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[(N[(1.0 + N[(-2.0 * N[(N[Power[z2, 2.0], $MachinePrecision] * N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

   \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} \cdot 2\right) \]

   2. lift-tan.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   3. tan-quot N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \color{blue}{\frac{z0}{z1}}\right) \cdot 2\right) \]

   5. frac-times N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

   6. lower-/.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

3. Applied rewrites 69.2%

   \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\cos \left(\pi \cdot \left(z2 + z2\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)} \cdot 2\right) \]

4. Taylor expanded in z2 around 0

   \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\left(1 + -2 \cdot \left({z2}^{2} \cdot {\pi}^{2}\right)\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]
5. Step-by-step derivation

   1. lower-+.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(1 + \color{blue}{-2 \cdot \left({z2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

   2. lower-*.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(1 + -2 \cdot \color{blue}{\left({z2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

   3. lower-*.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(1 + -2 \cdot \left({z2}^{2} \cdot \color{blue}{{\mathsf{PI}\left(\right)}^{2}}\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(1 + -2 \cdot \left({z2}^{2} \cdot {\color{blue}{\mathsf{PI}\left(\right)}}^{2}\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(1 + -2 \cdot \left({z2}^{2} \cdot {\mathsf{PI}\left(\right)}^{\color{blue}{2}}\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

   6. lower-PI.f64 86.2%

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\left(1 + -2 \cdot \left({z2}^{2} \cdot {\pi}^{2}\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

6. Applied rewrites 86.2%

   \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\left(1 + -2 \cdot \left({z2}^{2} \cdot {\pi}^{2}\right)\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]
7. Add Preprocessing

Alternative 3: 80.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|z0\right| \cdot \pi}{\left|z1\right|}\\ \mathbf{if}\;\frac{\left|z0\right|}{\left|z1\right|} \leq 5:\\ \;\;\;\;1 - \cos \left(\tan^{-1} \left(\frac{-1 \cdot \left({z2}^{2} \cdot \left(-1 \cdot t\_0 - -0.3333333333333333 \cdot t\_0\right)\right) + -0.5 \cdot \frac{\left|z0\right|}{\left|z1\right| \cdot \pi}}{z2}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + 0.5 \cdot \pi\right) \cdot \left|z0\right|}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot \left|z1\right|}\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ (* (fabs z0) PI) (fabs z1))))
  (if (<= (/ (fabs z0) (fabs z1)) 5.0)
    (-
     1.0
     (cos
      (*
       (atan
        (/
         (+
          (*
           -1.0
           (*
            (pow z2 2.0)
            (- (* -1.0 t_0) (* -0.3333333333333333 t_0))))
          (* -0.5 (/ (fabs z0) (* (fabs z1) PI))))
         z2))
       2.0)))
    (-
     1.0
     (cos
      (*
       (atan
        (/
         (* (sin (+ (* (- PI) (+ z2 z2)) (* 0.5 PI))) (fabs z0))
         (* (- (sin (* PI (+ z2 z2)))) (fabs z1))))
       2.0))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (fabs(z0) * ((double) M_PI)) / fabs(z1);
	double tmp;
	if ((fabs(z0) / fabs(z1)) <= 5.0) {
		tmp = 1.0 - cos((atan((((-1.0 * (pow(z2, 2.0) * ((-1.0 * t_0) - (-0.3333333333333333 * t_0)))) + (-0.5 * (fabs(z0) / (fabs(z1) * ((double) M_PI))))) / z2)) * 2.0));
	} else {
		tmp = 1.0 - cos((atan(((sin(((-((double) M_PI) * (z2 + z2)) + (0.5 * ((double) M_PI)))) * fabs(z0)) / (-sin((((double) M_PI) * (z2 + z2))) * fabs(z1)))) * 2.0));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (Math.abs(z0) * Math.PI) / Math.abs(z1);
	double tmp;
	if ((Math.abs(z0) / Math.abs(z1)) <= 5.0) {
		tmp = 1.0 - Math.cos((Math.atan((((-1.0 * (Math.pow(z2, 2.0) * ((-1.0 * t_0) - (-0.3333333333333333 * t_0)))) + (-0.5 * (Math.abs(z0) / (Math.abs(z1) * Math.PI)))) / z2)) * 2.0));
	} else {
		tmp = 1.0 - Math.cos((Math.atan(((Math.sin(((-Math.PI * (z2 + z2)) + (0.5 * Math.PI))) * Math.abs(z0)) / (-Math.sin((Math.PI * (z2 + z2))) * Math.abs(z1)))) * 2.0));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (math.fabs(z0) * math.pi) / math.fabs(z1)
	tmp = 0
	if (math.fabs(z0) / math.fabs(z1)) <= 5.0:
		tmp = 1.0 - math.cos((math.atan((((-1.0 * (math.pow(z2, 2.0) * ((-1.0 * t_0) - (-0.3333333333333333 * t_0)))) + (-0.5 * (math.fabs(z0) / (math.fabs(z1) * math.pi)))) / z2)) * 2.0))
	else:
		tmp = 1.0 - math.cos((math.atan(((math.sin(((-math.pi * (z2 + z2)) + (0.5 * math.pi))) * math.fabs(z0)) / (-math.sin((math.pi * (z2 + z2))) * math.fabs(z1)))) * 2.0))
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(abs(z0) * pi) / abs(z1))
	tmp = 0.0
	if (Float64(abs(z0) / abs(z1)) <= 5.0)
		tmp = Float64(1.0 - cos(Float64(atan(Float64(Float64(Float64(-1.0 * Float64((z2 ^ 2.0) * Float64(Float64(-1.0 * t_0) - Float64(-0.3333333333333333 * t_0)))) + Float64(-0.5 * Float64(abs(z0) / Float64(abs(z1) * pi)))) / z2)) * 2.0)));
	else
		tmp = Float64(1.0 - cos(Float64(atan(Float64(Float64(sin(Float64(Float64(Float64(-pi) * Float64(z2 + z2)) + Float64(0.5 * pi))) * abs(z0)) / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * abs(z1)))) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (abs(z0) * pi) / abs(z1);
	tmp = 0.0;
	if ((abs(z0) / abs(z1)) <= 5.0)
		tmp = 1.0 - cos((atan((((-1.0 * ((z2 ^ 2.0) * ((-1.0 * t_0) - (-0.3333333333333333 * t_0)))) + (-0.5 * (abs(z0) / (abs(z1) * pi)))) / z2)) * 2.0));
	else
		tmp = 1.0 - cos((atan(((sin(((-pi * (z2 + z2)) + (0.5 * pi))) * abs(z0)) / (-sin((pi * (z2 + z2))) * abs(z1)))) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision], 5.0], N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[(N[(-1.0 * N[(N[Power[z2, 2.0], $MachinePrecision] * N[(N[(-1.0 * t$95$0), $MachinePrecision] - N[(-0.3333333333333333 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Abs[z0], $MachinePrecision] / N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[(N[Sin[N[(N[((-Pi) * N[(z2 + z2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z0 z1) < 5

   1. Initial program 60.1%

      \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} \cdot 2\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      3. tan-quot N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      4. lift-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \color{blue}{\frac{z0}{z1}}\right) \cdot 2\right) \]

      5. frac-times N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

   3. Applied rewrites 69.2%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\cos \left(\pi \cdot \left(z2 + z2\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)} \cdot 2\right) \]

   4. Taylor expanded in z2 around 0

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left({z2}^{2} \cdot \left(-1 \cdot \frac{z0 \cdot \pi}{z1} - \frac{-1}{3} \cdot \frac{z0 \cdot \pi}{z1}\right)\right) + \frac{-1}{2} \cdot \frac{z0}{z1 \cdot \pi}}{z2}\right)} \cdot 2\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1 \cdot \left({z2}^{2} \cdot \left(-1 \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1} - \frac{-1}{3} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right)\right) + \frac{-1}{2} \cdot \frac{z0}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z2}}\right) \cdot 2\right) \]

   6. Applied rewrites 68.4%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left({z2}^{2} \cdot \left(-1 \cdot \frac{z0 \cdot \pi}{z1} - -0.3333333333333333 \cdot \frac{z0 \cdot \pi}{z1}\right)\right) + -0.5 \cdot \frac{z0}{z1 \cdot \pi}}{z2}\right)} \cdot 2\right) \]

   if 5 < (/.f64 z0 z1)

   1. Initial program 60.1%

      \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} \cdot 2\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      3. tan-quot N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      4. lift-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \color{blue}{\frac{z0}{z1}}\right) \cdot 2\right) \]

      5. frac-times N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

   3. Applied rewrites 69.2%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\cos \left(\pi \cdot \left(z2 + z2\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)} \cdot 2\right) \]
   4. Step-by-step derivation

      1. lift-cos.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\cos \left(\pi \cdot \left(z2 + z2\right)\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      2. cos-neg-rev N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\cos \left(\mathsf{neg}\left(\pi \cdot \left(z2 + z2\right)\right)\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

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

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(z2 + z2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\sin \left(\left(\mathsf{neg}\left(\pi \cdot \left(z2 + z2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      5. lower-+.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \color{blue}{\left(\left(\mathsf{neg}\left(\pi \cdot \left(z2 + z2\right)\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      6. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\mathsf{neg}\left(\color{blue}{\pi \cdot \left(z2 + z2\right)}\right)\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

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

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\color{blue}{\left(\mathsf{neg}\left(\pi\right)\right) \cdot \left(z2 + z2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      8. lower-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\color{blue}{\left(\mathsf{neg}\left(\pi\right)\right) \cdot \left(z2 + z2\right)} + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\color{blue}{\left(-\pi\right)} \cdot \left(z2 + z2\right) + \frac{\mathsf{PI}\left(\right)}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      10. lift-PI.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \frac{\color{blue}{\pi}}{2}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      11. mult-flip N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \color{blue}{\pi \cdot \frac{1}{2}}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      12. metadata-eval N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \pi \cdot \color{blue}{\frac{1}{2}}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      13. *-commutative N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \color{blue}{\frac{1}{2} \cdot \pi}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

      14. lower-*.f64 69.2%

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + \color{blue}{0.5 \cdot \pi}\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

   5. Applied rewrites 69.2%

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{\sin \left(\left(-\pi\right) \cdot \left(z2 + z2\right) + 0.5 \cdot \pi\right)} \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|z0\right|}{\left|z1\right|}\\ t_1 := \frac{\left|z0\right| \cdot \pi}{\left|z1\right|}\\ \mathbf{if}\;t\_0 \leq 5:\\ \;\;\;\;1 - \cos \left(\tan^{-1} \left(\frac{-1 \cdot \left({z2}^{2} \cdot \left(-1 \cdot t\_1 - -0.3333333333333333 \cdot t\_1\right)\right) + -0.5 \cdot \frac{\left|z0\right|}{\left|z1\right| \cdot \pi}}{z2}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot t\_0\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ (fabs z0) (fabs z1)))
       (t_1 (/ (* (fabs z0) PI) (fabs z1))))
  (if (<= t_0 5.0)
    (-
     1.0
     (cos
      (*
       (atan
        (/
         (+
          (*
           -1.0
           (*
            (pow z2 2.0)
            (- (* -1.0 t_1) (* -0.3333333333333333 t_1))))
          (* -0.5 (/ (fabs z0) (* (fabs z1) PI))))
         z2))
       2.0)))
    (-
     1.0
     (cos (* (atan (* (tan (* (- (+ z2 z2) -1.5) PI)) t_0)) 2.0))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = fabs(z0) / fabs(z1);
	double t_1 = (fabs(z0) * ((double) M_PI)) / fabs(z1);
	double tmp;
	if (t_0 <= 5.0) {
		tmp = 1.0 - cos((atan((((-1.0 * (pow(z2, 2.0) * ((-1.0 * t_1) - (-0.3333333333333333 * t_1)))) + (-0.5 * (fabs(z0) / (fabs(z1) * ((double) M_PI))))) / z2)) * 2.0));
	} else {
		tmp = 1.0 - cos((atan((tan((((z2 + z2) - -1.5) * ((double) M_PI))) * t_0)) * 2.0));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.abs(z0) / Math.abs(z1);
	double t_1 = (Math.abs(z0) * Math.PI) / Math.abs(z1);
	double tmp;
	if (t_0 <= 5.0) {
		tmp = 1.0 - Math.cos((Math.atan((((-1.0 * (Math.pow(z2, 2.0) * ((-1.0 * t_1) - (-0.3333333333333333 * t_1)))) + (-0.5 * (Math.abs(z0) / (Math.abs(z1) * Math.PI)))) / z2)) * 2.0));
	} else {
		tmp = 1.0 - Math.cos((Math.atan((Math.tan((((z2 + z2) - -1.5) * Math.PI)) * t_0)) * 2.0));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.fabs(z0) / math.fabs(z1)
	t_1 = (math.fabs(z0) * math.pi) / math.fabs(z1)
	tmp = 0
	if t_0 <= 5.0:
		tmp = 1.0 - math.cos((math.atan((((-1.0 * (math.pow(z2, 2.0) * ((-1.0 * t_1) - (-0.3333333333333333 * t_1)))) + (-0.5 * (math.fabs(z0) / (math.fabs(z1) * math.pi)))) / z2)) * 2.0))
	else:
		tmp = 1.0 - math.cos((math.atan((math.tan((((z2 + z2) - -1.5) * math.pi)) * t_0)) * 2.0))
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(abs(z0) / abs(z1))
	t_1 = Float64(Float64(abs(z0) * pi) / abs(z1))
	tmp = 0.0
	if (t_0 <= 5.0)
		tmp = Float64(1.0 - cos(Float64(atan(Float64(Float64(Float64(-1.0 * Float64((z2 ^ 2.0) * Float64(Float64(-1.0 * t_1) - Float64(-0.3333333333333333 * t_1)))) + Float64(-0.5 * Float64(abs(z0) / Float64(abs(z1) * pi)))) / z2)) * 2.0)));
	else
		tmp = Float64(1.0 - cos(Float64(atan(Float64(tan(Float64(Float64(Float64(z2 + z2) - -1.5) * pi)) * t_0)) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = abs(z0) / abs(z1);
	t_1 = (abs(z0) * pi) / abs(z1);
	tmp = 0.0;
	if (t_0 <= 5.0)
		tmp = 1.0 - cos((atan((((-1.0 * ((z2 ^ 2.0) * ((-1.0 * t_1) - (-0.3333333333333333 * t_1)))) + (-0.5 * (abs(z0) / (abs(z1) * pi)))) / z2)) * 2.0));
	else
		tmp = 1.0 - cos((atan((tan((((z2 + z2) - -1.5) * pi)) * t_0)) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 5.0], N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[(N[(-1.0 * N[(N[Power[z2, 2.0], $MachinePrecision] * N[(N[(-1.0 * t$95$1), $MachinePrecision] - N[(-0.3333333333333333 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Abs[z0], $MachinePrecision] / N[(N[Abs[z1], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z2), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -1.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z0 z1) < 5

   1. Initial program 60.1%

      \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} \cdot 2\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      3. tan-quot N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      4. lift-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \color{blue}{\frac{z0}{z1}}\right) \cdot 2\right) \]

      5. frac-times N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

   3. Applied rewrites 69.2%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\cos \left(\pi \cdot \left(z2 + z2\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)} \cdot 2\right) \]

   4. Taylor expanded in z2 around 0

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left({z2}^{2} \cdot \left(-1 \cdot \frac{z0 \cdot \pi}{z1} - \frac{-1}{3} \cdot \frac{z0 \cdot \pi}{z1}\right)\right) + \frac{-1}{2} \cdot \frac{z0}{z1 \cdot \pi}}{z2}\right)} \cdot 2\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1 \cdot \left({z2}^{2} \cdot \left(-1 \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1} - \frac{-1}{3} \cdot \frac{z0 \cdot \mathsf{PI}\left(\right)}{z1}\right)\right) + \frac{-1}{2} \cdot \frac{z0}{z1 \cdot \mathsf{PI}\left(\right)}}{\color{blue}{z2}}\right) \cdot 2\right) \]

   6. Applied rewrites 68.4%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left({z2}^{2} \cdot \left(-1 \cdot \frac{z0 \cdot \pi}{z1} - -0.3333333333333333 \cdot \frac{z0 \cdot \pi}{z1}\right)\right) + -0.5 \cdot \frac{z0}{z1 \cdot \pi}}{z2}\right)} \cdot 2\right) \]

   if 5 < (/.f64 z0 z1)

   1. Initial program 60.1%

      \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
   2. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      2. tan-+PI-rev N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi + \mathsf{PI}\left(\right)\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      3. lower-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi + \mathsf{PI}\left(\right)\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      4. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \left(\color{blue}{\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi} + \mathsf{PI}\left(\right)\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      5. lift-PI.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi + \color{blue}{\pi}\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      6. distribute-lft1-in N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \color{blue}{\left(\left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) + 1\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      7. lower-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \color{blue}{\left(\left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) + 1\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      8. lift--.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\color{blue}{\left(\left(z2 + z2\right) - \frac{-1}{2}\right)} + 1\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      9. associate-+l- N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \left(\color{blue}{\left(\left(z2 + z2\right) - \left(\frac{-1}{2} - 1\right)\right)} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      10. lower--.f64 N/A

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \left(\color{blue}{\left(\left(z2 + z2\right) - \left(\frac{-1}{2} - 1\right)\right)} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      11. metadata-eval 60.1%

          \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - \color{blue}{-1.5}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \leq 2 \cdot 10^{+14}:\\ \;\;\;\;1 - \cos \left(\tan^{-1} \left(\tan \left(0.5 \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \cos \left(\tan^{-1} \left(-0.5 \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right) \cdot 2\right)\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (if (<= (tan (* (- (+ z2 z2) -0.5) PI)) 2e+14)
  (- 1.0 (cos (* (atan (* (tan (* 0.5 PI)) (/ z0 z1))) 2.0)))
  (- 1.0 (cos (* (atan (* -0.5 (/ z0 (* z1 (* z2 PI))))) 2.0)))))

C

double code(double z2, double z0, double z1) {
	double tmp;
	if (tan((((z2 + z2) - -0.5) * ((double) M_PI))) <= 2e+14) {
		tmp = 1.0 - cos((atan((tan((0.5 * ((double) M_PI))) * (z0 / z1))) * 2.0));
	} else {
		tmp = 1.0 - cos((atan((-0.5 * (z0 / (z1 * (z2 * ((double) M_PI)))))) * 2.0));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double tmp;
	if (Math.tan((((z2 + z2) - -0.5) * Math.PI)) <= 2e+14) {
		tmp = 1.0 - Math.cos((Math.atan((Math.tan((0.5 * Math.PI)) * (z0 / z1))) * 2.0));
	} else {
		tmp = 1.0 - Math.cos((Math.atan((-0.5 * (z0 / (z1 * (z2 * Math.PI))))) * 2.0));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	tmp = 0
	if math.tan((((z2 + z2) - -0.5) * math.pi)) <= 2e+14:
		tmp = 1.0 - math.cos((math.atan((math.tan((0.5 * math.pi)) * (z0 / z1))) * 2.0))
	else:
		tmp = 1.0 - math.cos((math.atan((-0.5 * (z0 / (z1 * (z2 * math.pi))))) * 2.0))
	return tmp

Julia

function code(z2, z0, z1)
	tmp = 0.0
	if (tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) <= 2e+14)
		tmp = Float64(1.0 - cos(Float64(atan(Float64(tan(Float64(0.5 * pi)) * Float64(z0 / z1))) * 2.0)));
	else
		tmp = Float64(1.0 - cos(Float64(atan(Float64(-0.5 * Float64(z0 / Float64(z1 * Float64(z2 * pi))))) * 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	tmp = 0.0;
	if (tan((((z2 + z2) - -0.5) * pi)) <= 2e+14)
		tmp = 1.0 - cos((atan((tan((0.5 * pi)) * (z0 / z1))) * 2.0));
	else
		tmp = 1.0 - cos((atan((-0.5 * (z0 / (z1 * (z2 * pi))))) * 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := If[LessEqual[N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], 2e+14], N[(1.0 - N[Cos[N[(N[ArcTan[N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[Cos[N[(N[ArcTan[N[(-0.5 * N[(z0 / N[(z1 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) < 2e14

   1. Initial program 60.1%

      \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   2. Taylor expanded in z2 around 0

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \left(\color{blue}{\frac{1}{2}} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
   3. Step-by-step derivation

   4. Applied rewrites 61.4%

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\tan \left(\color{blue}{0.5} \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   if 2e14 < (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)))

   1. Initial program 60.1%

      \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} \cdot 2\right) \]

      2. lift-tan.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      3. tan-quot N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

      4. lift-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \color{blue}{\frac{z0}{z1}}\right) \cdot 2\right) \]

      5. frac-times N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

      6. lower-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

   3. Applied rewrites 69.2%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\cos \left(\pi \cdot \left(z2 + z2\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)} \cdot 2\right) \]

   4. Taylor expanded in z2 around 0

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right)} \cdot 2\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z0}{z1 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot 2\right) \]

      2. lower-/.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1}{2} \cdot \frac{z0}{\color{blue}{z1 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot 2\right) \]

      3. lower-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1}{2} \cdot \frac{z0}{z1 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot 2\right) \]

      4. lower-*.f64 N/A

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1}{2} \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \cdot 2\right) \]

      5. lower-PI.f64 59.0%

         \[\leadsto 1 - \cos \left(\tan^{-1} \left(-0.5 \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right) \cdot 2\right) \]

   6. Applied rewrites 59.0%

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right)} \cdot 2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 69.2% accurate, 1.0× speedup

Math

\[1 - \cos \left(\tan^{-1} \left(\frac{z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (- 1.0 (cos (* (atan (/ z0 (* (- (sin (* PI (+ z2 z2)))) z1))) 2.0))))

C

double code(double z2, double z0, double z1) {
	return 1.0 - cos((atan((z0 / (-sin((((double) M_PI) * (z2 + z2))) * z1))) * 2.0));
}

Java

public static double code(double z2, double z0, double z1) {
	return 1.0 - Math.cos((Math.atan((z0 / (-Math.sin((Math.PI * (z2 + z2))) * z1))) * 2.0));
}

Python

def code(z2, z0, z1):
	return 1.0 - math.cos((math.atan((z0 / (-math.sin((math.pi * (z2 + z2))) * z1))) * 2.0))

Julia

function code(z2, z0, z1)
	return Float64(1.0 - cos(Float64(atan(Float64(z0 / Float64(Float64(-sin(Float64(pi * Float64(z2 + z2)))) * z1))) * 2.0)))
end

MATLAB

function tmp = code(z2, z0, z1)
	tmp = 1.0 - cos((atan((z0 / (-sin((pi * (z2 + z2))) * z1))) * 2.0));
end

Wolfram

code[z2_, z0_, z1_] := N[(1.0 - N[Cos[N[(N[ArcTan[N[(z0 / N[((-N[Sin[N[(Pi * N[(z2 + z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) * z1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

   \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} \cdot 2\right) \]

   2. lift-tan.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   3. tan-quot N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \color{blue}{\frac{z0}{z1}}\right) \cdot 2\right) \]

   5. frac-times N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

   6. lower-/.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

3. Applied rewrites 69.2%

   \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\cos \left(\pi \cdot \left(z2 + z2\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)} \cdot 2\right) \]

4. Taylor expanded in z2 around 0

   \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{z0}}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]
5. Step-by-step derivation

6. Applied rewrites 69.2%

   \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\color{blue}{z0}}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right) \cdot 2\right) \]
7. Add Preprocessing

Alternative 7: 59.0% accurate, 1.4× speedup

Math

\[1 - \cos \left(\tan^{-1} \left(-0.5 \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right) \cdot 2\right) \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (- 1.0 (cos (* (atan (* -0.5 (/ z0 (* z1 (* z2 PI))))) 2.0))))

C

double code(double z2, double z0, double z1) {
	return 1.0 - cos((atan((-0.5 * (z0 / (z1 * (z2 * ((double) M_PI)))))) * 2.0));
}

Java

public static double code(double z2, double z0, double z1) {
	return 1.0 - Math.cos((Math.atan((-0.5 * (z0 / (z1 * (z2 * Math.PI))))) * 2.0));
}

Python

def code(z2, z0, z1):
	return 1.0 - math.cos((math.atan((-0.5 * (z0 / (z1 * (z2 * math.pi))))) * 2.0))

Julia

function code(z2, z0, z1)
	return Float64(1.0 - cos(Float64(atan(Float64(-0.5 * Float64(z0 / Float64(z1 * Float64(z2 * pi))))) * 2.0)))
end

MATLAB

function tmp = code(z2, z0, z1)
	tmp = 1.0 - cos((atan((-0.5 * (z0 / (z1 * (z2 * pi))))) * 2.0));
end

Wolfram

code[z2_, z0_, z1_] := N[(1.0 - N[Cos[N[(N[ArcTan[N[(-0.5 * N[(z0 / N[(z1 * N[(z2 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.1%

   \[1 - \cos \left(\tan^{-1} \left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot 2\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} \cdot 2\right) \]

   2. lift-tan.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   3. tan-quot N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\color{blue}{\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}} \cdot \frac{z0}{z1}\right) \cdot 2\right) \]

   4. lift-/.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)} \cdot \color{blue}{\frac{z0}{z1}}\right) \cdot 2\right) \]

   5. frac-times N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

   6. lower-/.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\sin \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot z1}\right)} \cdot 2\right) \]

3. Applied rewrites 69.2%

   \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{\cos \left(\pi \cdot \left(z2 + z2\right)\right) \cdot z0}{\left(-\sin \left(\pi \cdot \left(z2 + z2\right)\right)\right) \cdot z1}\right)} \cdot 2\right) \]

4. Taylor expanded in z2 around 0

   \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(\frac{-1}{2} \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right)} \cdot 2\right) \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1}{2} \cdot \color{blue}{\frac{z0}{z1 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot 2\right) \]

   2. lower-/.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1}{2} \cdot \frac{z0}{\color{blue}{z1 \cdot \left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot 2\right) \]

   3. lower-*.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1}{2} \cdot \frac{z0}{z1 \cdot \color{blue}{\left(z2 \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot 2\right) \]

   4. lower-*.f64 N/A

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(\frac{-1}{2} \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}\right) \cdot 2\right) \]

   5. lower-PI.f64 59.0%

      \[\leadsto 1 - \cos \left(\tan^{-1} \left(-0.5 \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right) \cdot 2\right) \]

6. Applied rewrites 59.0%

   \[\leadsto 1 - \cos \left(\tan^{-1} \color{blue}{\left(-0.5 \cdot \frac{z0}{z1 \cdot \left(z2 \cdot \pi\right)}\right)} \cdot 2\right) \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z2 z0 z1)
  :name "(- 1 (cos (* (atan (* (tan (* (- (+ z2 z2) -1/2) PI)) (/ z0 z1))) 2)))"
  :precision binary64
  (- 1.0 (cos (* (atan (* (tan (* (- (+ z2 z2) -0.5) PI)) (/ z0 z1))) 2.0))))