(sqrt (- (pow (* (tan (* (- (+ z2 z2) -1/2) PI)) (/ z0 z1)) 2) -1))

Percentage Accurate: 44.8% → 78.4%

Time: 9.4s

Alternatives: 15

Speedup: 1.0×

• 37.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 44.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \left(z2 + z2\right) \cdot \pi\\ t_1 := -\sin t\_0\\ t_2 := \pi \cdot \left(0.5 + 2 \cdot z2\right)\\ t_3 := \sqrt{{\left(\frac{\left(-1 + 1.3333333333333333 \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1}\\ t_4 := 1 + 0.25 \cdot \frac{{z0}^{2} \cdot {\sin t\_2}^{2}}{{z1}^{2} \cdot {\cos t\_2}^{2}}\\ \mathbf{if}\;z2 \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z2 \leq -1.15 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{\left(\left({\left(\frac{\cos t\_0}{t\_1}\right)}^{2} \cdot \frac{z0}{z1}\right) \cdot \frac{1}{z1}\right) \cdot z0 - -1}\\ \mathbf{elif}\;z2 \leq 8.2 \cdot 10^{-148}:\\ \;\;\;\;t\_4 \cdot t\_4\\ \mathbf{elif}\;z2 \leq 45000000:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)\right)}{t\_1}}{t\_1} \cdot \frac{z0}{z1}\right) \cdot \frac{1}{z1}\right) \cdot z0 - -1}\\ \mathbf{elif}\;z2 \leq 1.05 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{{\left(\left(z2 \cdot \left(2 \cdot \pi - \frac{\left(-2 \cdot \pi\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \pi\right)\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right) \cdot \frac{z0}{z1}\right)}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (+ z2 z2) PI))
       (t_1 (- (sin t_0)))
       (t_2 (* PI (+ 0.5 (* 2.0 z2))))
       (t_3
        (sqrt
         (-
          (pow
           (/
            (*
             (+
              -1.0
              (*
               1.3333333333333333
               (* (pow z2 3.0) (* (pow PI 3.0) (cos (* -0.5 PI))))))
             z0)
            (* (cos (* (- (+ z2 z2) -1.5) PI)) z1))
           2.0)
          -1.0)))
       (t_4
        (+
         1.0
         (*
          0.25
          (/
           (* (pow z0 2.0) (pow (sin t_2) 2.0))
           (* (pow z1 2.0) (pow (cos t_2) 2.0)))))))
  (if (<= z2 -7.2e+15)
    t_3
    (if (<= z2 -1.15e-133)
      (sqrt
       (-
        (*
         (* (* (pow (/ (cos t_0) t_1) 2.0) (/ z0 z1)) (/ 1.0 z1))
         z0)
        -1.0))
      (if (<= z2 8.2e-148)
        (* t_4 t_4)
        (if (<= z2 45000000.0)
          (sqrt
           (-
            (*
             (*
              (*
               (/
                (/
                 (-
                  0.5
                  (* 0.5 (cos (* 2.0 (* (- (+ z2 z2) -0.5) PI)))))
                 t_1)
                t_1)
               (/ z0 z1))
              (/ 1.0 z1))
             z0)
            -1.0))
          (if (<= z2 1.05e+97)
            (sqrt
             (-
              (pow
               (*
                (+
                 (*
                  z2
                  (-
                   (* 2.0 PI)
                   (/
                    (*
                     (* -2.0 PI)
                     (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 PI))))))
                    (+ 0.5 (* 0.5 (cos (* 2.0 (* PI -0.5))))))))
                 (/ (sin (* 0.5 PI)) (cos (* 0.5 PI))))
                (/ z0 z1))
               2.0)
              -1.0))
            t_3)))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) * ((double) M_PI);
	double t_1 = -sin(t_0);
	double t_2 = ((double) M_PI) * (0.5 + (2.0 * z2));
	double t_3 = sqrt((pow((((-1.0 + (1.3333333333333333 * (pow(z2, 3.0) * (pow(((double) M_PI), 3.0) * cos((-0.5 * ((double) M_PI))))))) * z0) / (cos((((z2 + z2) - -1.5) * ((double) M_PI))) * z1)), 2.0) - -1.0));
	double t_4 = 1.0 + (0.25 * ((pow(z0, 2.0) * pow(sin(t_2), 2.0)) / (pow(z1, 2.0) * pow(cos(t_2), 2.0))));
	double tmp;
	if (z2 <= -7.2e+15) {
		tmp = t_3;
	} else if (z2 <= -1.15e-133) {
		tmp = sqrt(((((pow((cos(t_0) / t_1), 2.0) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	} else if (z2 <= 8.2e-148) {
		tmp = t_4 * t_4;
	} else if (z2 <= 45000000.0) {
		tmp = sqrt((((((((0.5 - (0.5 * cos((2.0 * (((z2 + z2) - -0.5) * ((double) M_PI)))))) / t_1) / t_1) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	} else if (z2 <= 1.05e+97) {
		tmp = sqrt((pow((((z2 * ((2.0 * ((double) M_PI)) - (((-2.0 * ((double) M_PI)) * (0.5 - (0.5 * cos((2.0 * (0.5 * ((double) M_PI))))))) / (0.5 + (0.5 * cos((2.0 * (((double) M_PI) * -0.5)))))))) + (sin((0.5 * ((double) M_PI))) / cos((0.5 * ((double) M_PI))))) * (z0 / z1)), 2.0) - -1.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) * Math.PI;
	double t_1 = -Math.sin(t_0);
	double t_2 = Math.PI * (0.5 + (2.0 * z2));
	double t_3 = Math.sqrt((Math.pow((((-1.0 + (1.3333333333333333 * (Math.pow(z2, 3.0) * (Math.pow(Math.PI, 3.0) * Math.cos((-0.5 * Math.PI)))))) * z0) / (Math.cos((((z2 + z2) - -1.5) * Math.PI)) * z1)), 2.0) - -1.0));
	double t_4 = 1.0 + (0.25 * ((Math.pow(z0, 2.0) * Math.pow(Math.sin(t_2), 2.0)) / (Math.pow(z1, 2.0) * Math.pow(Math.cos(t_2), 2.0))));
	double tmp;
	if (z2 <= -7.2e+15) {
		tmp = t_3;
	} else if (z2 <= -1.15e-133) {
		tmp = Math.sqrt(((((Math.pow((Math.cos(t_0) / t_1), 2.0) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	} else if (z2 <= 8.2e-148) {
		tmp = t_4 * t_4;
	} else if (z2 <= 45000000.0) {
		tmp = Math.sqrt((((((((0.5 - (0.5 * Math.cos((2.0 * (((z2 + z2) - -0.5) * Math.PI))))) / t_1) / t_1) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	} else if (z2 <= 1.05e+97) {
		tmp = Math.sqrt((Math.pow((((z2 * ((2.0 * Math.PI) - (((-2.0 * Math.PI) * (0.5 - (0.5 * Math.cos((2.0 * (0.5 * Math.PI)))))) / (0.5 + (0.5 * Math.cos((2.0 * (Math.PI * -0.5)))))))) + (Math.sin((0.5 * Math.PI)) / Math.cos((0.5 * Math.PI)))) * (z0 / z1)), 2.0) - -1.0));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (z2 + z2) * math.pi
	t_1 = -math.sin(t_0)
	t_2 = math.pi * (0.5 + (2.0 * z2))
	t_3 = math.sqrt((math.pow((((-1.0 + (1.3333333333333333 * (math.pow(z2, 3.0) * (math.pow(math.pi, 3.0) * math.cos((-0.5 * math.pi)))))) * z0) / (math.cos((((z2 + z2) - -1.5) * math.pi)) * z1)), 2.0) - -1.0))
	t_4 = 1.0 + (0.25 * ((math.pow(z0, 2.0) * math.pow(math.sin(t_2), 2.0)) / (math.pow(z1, 2.0) * math.pow(math.cos(t_2), 2.0))))
	tmp = 0
	if z2 <= -7.2e+15:
		tmp = t_3
	elif z2 <= -1.15e-133:
		tmp = math.sqrt(((((math.pow((math.cos(t_0) / t_1), 2.0) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0))
	elif z2 <= 8.2e-148:
		tmp = t_4 * t_4
	elif z2 <= 45000000.0:
		tmp = math.sqrt((((((((0.5 - (0.5 * math.cos((2.0 * (((z2 + z2) - -0.5) * math.pi))))) / t_1) / t_1) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0))
	elif z2 <= 1.05e+97:
		tmp = math.sqrt((math.pow((((z2 * ((2.0 * math.pi) - (((-2.0 * math.pi) * (0.5 - (0.5 * math.cos((2.0 * (0.5 * math.pi)))))) / (0.5 + (0.5 * math.cos((2.0 * (math.pi * -0.5)))))))) + (math.sin((0.5 * math.pi)) / math.cos((0.5 * math.pi)))) * (z0 / z1)), 2.0) - -1.0))
	else:
		tmp = t_3
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(z2 + z2) * pi)
	t_1 = Float64(-sin(t_0))
	t_2 = Float64(pi * Float64(0.5 + Float64(2.0 * z2)))
	t_3 = sqrt(Float64((Float64(Float64(Float64(-1.0 + Float64(1.3333333333333333 * Float64((z2 ^ 3.0) * Float64((pi ^ 3.0) * cos(Float64(-0.5 * pi)))))) * z0) / Float64(cos(Float64(Float64(Float64(z2 + z2) - -1.5) * pi)) * z1)) ^ 2.0) - -1.0))
	t_4 = Float64(1.0 + Float64(0.25 * Float64(Float64((z0 ^ 2.0) * (sin(t_2) ^ 2.0)) / Float64((z1 ^ 2.0) * (cos(t_2) ^ 2.0)))))
	tmp = 0.0
	if (z2 <= -7.2e+15)
		tmp = t_3;
	elseif (z2 <= -1.15e-133)
		tmp = sqrt(Float64(Float64(Float64(Float64((Float64(cos(t_0) / t_1) ^ 2.0) * Float64(z0 / z1)) * Float64(1.0 / z1)) * z0) - -1.0));
	elseif (z2 <= 8.2e-148)
		tmp = Float64(t_4 * t_4);
	elseif (z2 <= 45000000.0)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(Float64(z2 + z2) - -0.5) * pi))))) / t_1) / t_1) * Float64(z0 / z1)) * Float64(1.0 / z1)) * z0) - -1.0));
	elseif (z2 <= 1.05e+97)
		tmp = sqrt(Float64((Float64(Float64(Float64(z2 * Float64(Float64(2.0 * pi) - Float64(Float64(Float64(-2.0 * pi) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * pi)))))) / Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(pi * -0.5)))))))) + Float64(sin(Float64(0.5 * pi)) / cos(Float64(0.5 * pi)))) * Float64(z0 / z1)) ^ 2.0) - -1.0));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (z2 + z2) * pi;
	t_1 = -sin(t_0);
	t_2 = pi * (0.5 + (2.0 * z2));
	t_3 = sqrt((((((-1.0 + (1.3333333333333333 * ((z2 ^ 3.0) * ((pi ^ 3.0) * cos((-0.5 * pi)))))) * z0) / (cos((((z2 + z2) - -1.5) * pi)) * z1)) ^ 2.0) - -1.0));
	t_4 = 1.0 + (0.25 * (((z0 ^ 2.0) * (sin(t_2) ^ 2.0)) / ((z1 ^ 2.0) * (cos(t_2) ^ 2.0))));
	tmp = 0.0;
	if (z2 <= -7.2e+15)
		tmp = t_3;
	elseif (z2 <= -1.15e-133)
		tmp = sqrt(((((((cos(t_0) / t_1) ^ 2.0) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	elseif (z2 <= 8.2e-148)
		tmp = t_4 * t_4;
	elseif (z2 <= 45000000.0)
		tmp = sqrt((((((((0.5 - (0.5 * cos((2.0 * (((z2 + z2) - -0.5) * pi))))) / t_1) / t_1) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	elseif (z2 <= 1.05e+97)
		tmp = sqrt((((((z2 * ((2.0 * pi) - (((-2.0 * pi) * (0.5 - (0.5 * cos((2.0 * (0.5 * pi)))))) / (0.5 + (0.5 * cos((2.0 * (pi * -0.5)))))))) + (sin((0.5 * pi)) / cos((0.5 * pi)))) * (z0 / z1)) ^ 2.0) - -1.0));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(z2 + z2), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = (-N[Sin[t$95$0], $MachinePrecision])}, Block[{t$95$2 = N[(Pi * N[(0.5 + N[(2.0 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[N[(N[(N[(-1.0 + N[(1.3333333333333333 * N[(N[Power[z2, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[Cos[N[(-0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[(N[Cos[N[(N[(N[(z2 + z2), $MachinePrecision] - -1.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(1.0 + N[(0.25 * N[(N[(N[Power[z0, 2.0], $MachinePrecision] * N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[z1, 2.0], $MachinePrecision] * N[Power[N[Cos[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z2, -7.2e+15], t$95$3, If[LessEqual[z2, -1.15e-133], N[Sqrt[N[(N[(N[(N[(N[Power[N[(N[Cos[t$95$0], $MachinePrecision] / t$95$1), $MachinePrecision], 2.0], $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 8.2e-148], N[(t$95$4 * t$95$4), $MachinePrecision], If[LessEqual[z2, 45000000.0], N[Sqrt[N[(N[(N[(N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / t$95$1), $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 1.05e+97], N[Sqrt[N[(N[Power[N[(N[(N[(z2 * N[(N[(2.0 * Pi), $MachinePrecision] - N[(N[(N[(-2.0 * Pi), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], t$95$3]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z2 < -7.2e15 or 1.0500000000000001e97 < z2

   1. Initial program 44.8%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. frac-2neg N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 44.7%

      \[\leadsto \sqrt{{\color{blue}{\left(\frac{\sin \left(\left(-0.5 - \left(z2 + z2\right)\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}}^{2} - -1} \]

   4. Taylor expanded in z2 around 0

      \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\pi \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right) + z2 \cdot \left(-2 \cdot \left({\pi}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \pi\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z2} \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      4. lower-PI.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \color{blue}{\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   6. Applied rewrites 70.9%

      \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(\sin \left(-0.5 \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\pi \cdot \cos \left(-0.5 \cdot \pi\right)\right) + z2 \cdot \left(-2 \cdot \left({\pi}^{2} \cdot \sin \left(-0.5 \cdot \pi\right)\right) + 1.3333333333333333 \cdot \left(z2 \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   7. Taylor expanded in z2 around inf

      \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + \frac{4}{3} \cdot \color{blue}{\left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      6. lower-PI.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      9. lower-PI.f64 71.1%

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + 1.3333333333333333 \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   9. Applied rewrites 71.1%

      \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + 1.3333333333333333 \cdot \color{blue}{\left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   10. Evaluated real constant 71.1%

       \[\leadsto \sqrt{{\left(\frac{\left(-1 + \color{blue}{1.3333333333333333} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   if -7.2e15 < z2 < -1.15e-133

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. tan-quot N/A

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

      8. lift-+.f64 N/A

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

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. count-2-rev N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-sin.f64 N/A

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

   5. Applied rewrites 47.8%

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

   if -1.15e-133 < z2 < 8.2000000000000005e-148

   1. Initial program 44.8%

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

      1. lift-sqrt.f64 N/A

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

      2. sqrt-fabs-rev N/A

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

      3. lift-sqrt.f64 N/A

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

      4. rem-sqrt-square-rev N/A

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

      5. sqrt-prod N/A

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

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

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

   3. Applied rewrites 44.8%

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

   4. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}\right)} \cdot \sqrt{\sqrt{{\left(\frac{z0}{z1} \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)\right)}^{2} - -1}} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \color{blue}{\frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}}\right) \cdot \sqrt{\sqrt{{\left(\frac{z0}{z1} \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)\right)}^{2} - -1}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \color{blue}{\frac{{z0}^{2} \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}}\right) \cdot \sqrt{\sqrt{{\left(\frac{z0}{z1} \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)\right)}^{2} - -1}} \]

      3. lower-/.f64 N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{\color{blue}{{z1}^{2} \cdot {\cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}}\right) \cdot \sqrt{\sqrt{{\left(\frac{z0}{z1} \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)\right)}^{2} - -1}} \]

   6. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\left(1 + 0.25 \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(0.5 + 2 \cdot z2\right)\right)}^{2}}\right)} \cdot \sqrt{\sqrt{{\left(\frac{z0}{z1} \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)\right)}^{2} - -1}} \]

   7. Taylor expanded in z0 around 0

      \[\leadsto \left(1 + 0.25 \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(0.5 + 2 \cdot z2\right)\right)}^{2}}\right) \cdot \color{blue}{\left(1 + \frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}\right) \cdot \left(1 + \color{blue}{\frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}\right) \cdot \left(1 + \frac{1}{4} \cdot \color{blue}{\frac{{z0}^{2} \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto \left(1 + \frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}\right) \cdot \left(1 + \frac{1}{4} \cdot \frac{{z0}^{2} \cdot {\sin \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}{\color{blue}{{z1}^{2} \cdot {\cos \left(\mathsf{PI}\left(\right) \cdot \left(\frac{1}{2} + 2 \cdot z2\right)\right)}^{2}}}\right) \]

   9. Applied rewrites 49.3%

      \[\leadsto \left(1 + 0.25 \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(0.5 + 2 \cdot z2\right)\right)}^{2}}\right) \cdot \color{blue}{\left(1 + 0.25 \cdot \frac{{z0}^{2} \cdot {\sin \left(\pi \cdot \left(0.5 + 2 \cdot z2\right)\right)}^{2}}{{z1}^{2} \cdot {\cos \left(\pi \cdot \left(0.5 + 2 \cdot z2\right)\right)}^{2}}\right)} \]

   if 8.2000000000000005e-148 < z2 < 4.5e7

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

   5. Applied rewrites 47.8%

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

   if 4.5e7 < z2 < 1.0500000000000001e97

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 58.3%

      \[\leadsto \sqrt{{\left(\left(z2 \cdot \left(2 \cdot \pi - \frac{\left(-2 \cdot \pi\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \pi\right)\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right) \cdot \frac{z0}{z1}\right)}^{2} - -1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 78.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \sqrt{{\left(\frac{\left(-1 + 1.3333333333333333 \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1}\\ t_1 := \left(z2 + z2\right) \cdot \pi\\ t_2 := -\sin t\_1\\ t_3 := \left(z2 + z2\right) - -0.5\\ t_4 := {\left(\frac{z0}{z1} \cdot \tan \left(\pi \cdot t\_3\right)\right)}^{2} - -1\\ \mathbf{if}\;z2 \leq -7.2 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq -1 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(\left({\left(\frac{\cos t\_1}{t\_2}\right)}^{2} \cdot \frac{z0}{z1}\right) \cdot \frac{1}{z1}\right) \cdot z0 - -1}\\ \mathbf{elif}\;z2 \leq 2.7 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\sqrt{t\_4 \cdot t\_4}}\\ \mathbf{elif}\;z2 \leq 45000000:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(t\_3 \cdot \pi\right)\right)}{t\_2}}{t\_2} \cdot \frac{z0}{z1}\right) \cdot \frac{1}{z1}\right) \cdot z0 - -1}\\ \mathbf{elif}\;z2 \leq 1.05 \cdot 10^{+97}:\\ \;\;\;\;\sqrt{{\left(\left(z2 \cdot \left(2 \cdot \pi - \frac{\left(-2 \cdot \pi\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \pi\right)\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right) \cdot \frac{z0}{z1}\right)}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0
        (sqrt
         (-
          (pow
           (/
            (*
             (+
              -1.0
              (*
               1.3333333333333333
               (* (pow z2 3.0) (* (pow PI 3.0) (cos (* -0.5 PI))))))
             z0)
            (* (cos (* (- (+ z2 z2) -1.5) PI)) z1))
           2.0)
          -1.0)))
       (t_1 (* (+ z2 z2) PI))
       (t_2 (- (sin t_1)))
       (t_3 (- (+ z2 z2) -0.5))
       (t_4 (- (pow (* (/ z0 z1) (tan (* PI t_3))) 2.0) -1.0)))
  (if (<= z2 -7.2e+15)
    t_0
    (if (<= z2 -1e-159)
      (sqrt
       (-
        (*
         (* (* (pow (/ (cos t_1) t_2) 2.0) (/ z0 z1)) (/ 1.0 z1))
         z0)
        -1.0))
      (if (<= z2 2.7e-125)
        (sqrt (sqrt (* t_4 t_4)))
        (if (<= z2 45000000.0)
          (sqrt
           (-
            (*
             (*
              (*
               (/
                (/ (- 0.5 (* 0.5 (cos (* 2.0 (* t_3 PI))))) t_2)
                t_2)
               (/ z0 z1))
              (/ 1.0 z1))
             z0)
            -1.0))
          (if (<= z2 1.05e+97)
            (sqrt
             (-
              (pow
               (*
                (+
                 (*
                  z2
                  (-
                   (* 2.0 PI)
                   (/
                    (*
                     (* -2.0 PI)
                     (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 PI))))))
                    (+ 0.5 (* 0.5 (cos (* 2.0 (* PI -0.5))))))))
                 (/ (sin (* 0.5 PI)) (cos (* 0.5 PI))))
                (/ z0 z1))
               2.0)
              -1.0))
            t_0)))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = sqrt((pow((((-1.0 + (1.3333333333333333 * (pow(z2, 3.0) * (pow(((double) M_PI), 3.0) * cos((-0.5 * ((double) M_PI))))))) * z0) / (cos((((z2 + z2) - -1.5) * ((double) M_PI))) * z1)), 2.0) - -1.0));
	double t_1 = (z2 + z2) * ((double) M_PI);
	double t_2 = -sin(t_1);
	double t_3 = (z2 + z2) - -0.5;
	double t_4 = pow(((z0 / z1) * tan((((double) M_PI) * t_3))), 2.0) - -1.0;
	double tmp;
	if (z2 <= -7.2e+15) {
		tmp = t_0;
	} else if (z2 <= -1e-159) {
		tmp = sqrt(((((pow((cos(t_1) / t_2), 2.0) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	} else if (z2 <= 2.7e-125) {
		tmp = sqrt(sqrt((t_4 * t_4)));
	} else if (z2 <= 45000000.0) {
		tmp = sqrt((((((((0.5 - (0.5 * cos((2.0 * (t_3 * ((double) M_PI)))))) / t_2) / t_2) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	} else if (z2 <= 1.05e+97) {
		tmp = sqrt((pow((((z2 * ((2.0 * ((double) M_PI)) - (((-2.0 * ((double) M_PI)) * (0.5 - (0.5 * cos((2.0 * (0.5 * ((double) M_PI))))))) / (0.5 + (0.5 * cos((2.0 * (((double) M_PI) * -0.5)))))))) + (sin((0.5 * ((double) M_PI))) / cos((0.5 * ((double) M_PI))))) * (z0 / z1)), 2.0) - -1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.sqrt((Math.pow((((-1.0 + (1.3333333333333333 * (Math.pow(z2, 3.0) * (Math.pow(Math.PI, 3.0) * Math.cos((-0.5 * Math.PI)))))) * z0) / (Math.cos((((z2 + z2) - -1.5) * Math.PI)) * z1)), 2.0) - -1.0));
	double t_1 = (z2 + z2) * Math.PI;
	double t_2 = -Math.sin(t_1);
	double t_3 = (z2 + z2) - -0.5;
	double t_4 = Math.pow(((z0 / z1) * Math.tan((Math.PI * t_3))), 2.0) - -1.0;
	double tmp;
	if (z2 <= -7.2e+15) {
		tmp = t_0;
	} else if (z2 <= -1e-159) {
		tmp = Math.sqrt(((((Math.pow((Math.cos(t_1) / t_2), 2.0) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	} else if (z2 <= 2.7e-125) {
		tmp = Math.sqrt(Math.sqrt((t_4 * t_4)));
	} else if (z2 <= 45000000.0) {
		tmp = Math.sqrt((((((((0.5 - (0.5 * Math.cos((2.0 * (t_3 * Math.PI))))) / t_2) / t_2) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	} else if (z2 <= 1.05e+97) {
		tmp = Math.sqrt((Math.pow((((z2 * ((2.0 * Math.PI) - (((-2.0 * Math.PI) * (0.5 - (0.5 * Math.cos((2.0 * (0.5 * Math.PI)))))) / (0.5 + (0.5 * Math.cos((2.0 * (Math.PI * -0.5)))))))) + (Math.sin((0.5 * Math.PI)) / Math.cos((0.5 * Math.PI)))) * (z0 / z1)), 2.0) - -1.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.sqrt((math.pow((((-1.0 + (1.3333333333333333 * (math.pow(z2, 3.0) * (math.pow(math.pi, 3.0) * math.cos((-0.5 * math.pi)))))) * z0) / (math.cos((((z2 + z2) - -1.5) * math.pi)) * z1)), 2.0) - -1.0))
	t_1 = (z2 + z2) * math.pi
	t_2 = -math.sin(t_1)
	t_3 = (z2 + z2) - -0.5
	t_4 = math.pow(((z0 / z1) * math.tan((math.pi * t_3))), 2.0) - -1.0
	tmp = 0
	if z2 <= -7.2e+15:
		tmp = t_0
	elif z2 <= -1e-159:
		tmp = math.sqrt(((((math.pow((math.cos(t_1) / t_2), 2.0) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0))
	elif z2 <= 2.7e-125:
		tmp = math.sqrt(math.sqrt((t_4 * t_4)))
	elif z2 <= 45000000.0:
		tmp = math.sqrt((((((((0.5 - (0.5 * math.cos((2.0 * (t_3 * math.pi))))) / t_2) / t_2) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0))
	elif z2 <= 1.05e+97:
		tmp = math.sqrt((math.pow((((z2 * ((2.0 * math.pi) - (((-2.0 * math.pi) * (0.5 - (0.5 * math.cos((2.0 * (0.5 * math.pi)))))) / (0.5 + (0.5 * math.cos((2.0 * (math.pi * -0.5)))))))) + (math.sin((0.5 * math.pi)) / math.cos((0.5 * math.pi)))) * (z0 / z1)), 2.0) - -1.0))
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = sqrt(Float64((Float64(Float64(Float64(-1.0 + Float64(1.3333333333333333 * Float64((z2 ^ 3.0) * Float64((pi ^ 3.0) * cos(Float64(-0.5 * pi)))))) * z0) / Float64(cos(Float64(Float64(Float64(z2 + z2) - -1.5) * pi)) * z1)) ^ 2.0) - -1.0))
	t_1 = Float64(Float64(z2 + z2) * pi)
	t_2 = Float64(-sin(t_1))
	t_3 = Float64(Float64(z2 + z2) - -0.5)
	t_4 = Float64((Float64(Float64(z0 / z1) * tan(Float64(pi * t_3))) ^ 2.0) - -1.0)
	tmp = 0.0
	if (z2 <= -7.2e+15)
		tmp = t_0;
	elseif (z2 <= -1e-159)
		tmp = sqrt(Float64(Float64(Float64(Float64((Float64(cos(t_1) / t_2) ^ 2.0) * Float64(z0 / z1)) * Float64(1.0 / z1)) * z0) - -1.0));
	elseif (z2 <= 2.7e-125)
		tmp = sqrt(sqrt(Float64(t_4 * t_4)));
	elseif (z2 <= 45000000.0)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(t_3 * pi))))) / t_2) / t_2) * Float64(z0 / z1)) * Float64(1.0 / z1)) * z0) - -1.0));
	elseif (z2 <= 1.05e+97)
		tmp = sqrt(Float64((Float64(Float64(Float64(z2 * Float64(Float64(2.0 * pi) - Float64(Float64(Float64(-2.0 * pi) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * pi)))))) / Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(pi * -0.5)))))))) + Float64(sin(Float64(0.5 * pi)) / cos(Float64(0.5 * pi)))) * Float64(z0 / z1)) ^ 2.0) - -1.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = sqrt((((((-1.0 + (1.3333333333333333 * ((z2 ^ 3.0) * ((pi ^ 3.0) * cos((-0.5 * pi)))))) * z0) / (cos((((z2 + z2) - -1.5) * pi)) * z1)) ^ 2.0) - -1.0));
	t_1 = (z2 + z2) * pi;
	t_2 = -sin(t_1);
	t_3 = (z2 + z2) - -0.5;
	t_4 = (((z0 / z1) * tan((pi * t_3))) ^ 2.0) - -1.0;
	tmp = 0.0;
	if (z2 <= -7.2e+15)
		tmp = t_0;
	elseif (z2 <= -1e-159)
		tmp = sqrt(((((((cos(t_1) / t_2) ^ 2.0) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	elseif (z2 <= 2.7e-125)
		tmp = sqrt(sqrt((t_4 * t_4)));
	elseif (z2 <= 45000000.0)
		tmp = sqrt((((((((0.5 - (0.5 * cos((2.0 * (t_3 * pi))))) / t_2) / t_2) * (z0 / z1)) * (1.0 / z1)) * z0) - -1.0));
	elseif (z2 <= 1.05e+97)
		tmp = sqrt((((((z2 * ((2.0 * pi) - (((-2.0 * pi) * (0.5 - (0.5 * cos((2.0 * (0.5 * pi)))))) / (0.5 + (0.5 * cos((2.0 * (pi * -0.5)))))))) + (sin((0.5 * pi)) / cos((0.5 * pi)))) * (z0 / z1)) ^ 2.0) - -1.0));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[Sqrt[N[(N[Power[N[(N[(N[(-1.0 + N[(1.3333333333333333 * N[(N[Power[z2, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[Cos[N[(-0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[(N[Cos[N[(N[(N[(z2 + z2), $MachinePrecision] - -1.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(z2 + z2), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = (-N[Sin[t$95$1], $MachinePrecision])}, Block[{t$95$3 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[(N[(z0 / z1), $MachinePrecision] * N[Tan[N[(Pi * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[z2, -7.2e+15], t$95$0, If[LessEqual[z2, -1e-159], N[Sqrt[N[(N[(N[(N[(N[Power[N[(N[Cos[t$95$1], $MachinePrecision] / t$95$2), $MachinePrecision], 2.0], $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 2.7e-125], N[Sqrt[N[Sqrt[N[(t$95$4 * t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 45000000.0], N[Sqrt[N[(N[(N[(N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(t$95$3 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z1), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 1.05e+97], N[Sqrt[N[(N[Power[N[(N[(N[(z2 * N[(N[(2.0 * Pi), $MachinePrecision] - N[(N[(N[(-2.0 * Pi), $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(Pi * -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] / N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], t$95$0]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z2 < -7.2e15 or 1.0500000000000001e97 < z2

   1. Initial program 44.8%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. frac-2neg N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 44.7%

      \[\leadsto \sqrt{{\color{blue}{\left(\frac{\sin \left(\left(-0.5 - \left(z2 + z2\right)\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}}^{2} - -1} \]

   4. Taylor expanded in z2 around 0

      \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\pi \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right) + z2 \cdot \left(-2 \cdot \left({\pi}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \pi\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z2} \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      4. lower-PI.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \color{blue}{\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   6. Applied rewrites 70.9%

      \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(\sin \left(-0.5 \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\pi \cdot \cos \left(-0.5 \cdot \pi\right)\right) + z2 \cdot \left(-2 \cdot \left({\pi}^{2} \cdot \sin \left(-0.5 \cdot \pi\right)\right) + 1.3333333333333333 \cdot \left(z2 \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   7. Taylor expanded in z2 around inf

      \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + \frac{4}{3} \cdot \color{blue}{\left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      6. lower-PI.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      9. lower-PI.f64 71.1%

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + 1.3333333333333333 \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   9. Applied rewrites 71.1%

      \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + 1.3333333333333333 \cdot \color{blue}{\left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   10. Evaluated real constant 71.1%

       \[\leadsto \sqrt{{\left(\frac{\left(-1 + \color{blue}{1.3333333333333333} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   if -7.2e15 < z2 < -9.9999999999999999e-160

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. tan-quot N/A

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

      8. lift-+.f64 N/A

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

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. count-2-rev N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-sin.f64 N/A

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

   5. Applied rewrites 47.8%

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

   if -9.9999999999999999e-160 < z2 < 2.6999999999999998e-125

   1. Initial program 44.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 50.7%

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

   3. Applied rewrites 50.7%

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

   if 2.6999999999999998e-125 < z2 < 4.5e7

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

   5. Applied rewrites 47.8%

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

   if 4.5e7 < z2 < 1.0500000000000001e97

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 57.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 58.3%

      \[\leadsto \sqrt{{\left(\left(z2 \cdot \left(2 \cdot \pi - \frac{\left(-2 \cdot \pi\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \pi\right)\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\pi \cdot -0.5\right)\right)}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right) \cdot \frac{z0}{z1}\right)}^{2} - -1} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 3: 71.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := -\sin \left(\left(z2 + z2\right) \cdot \pi\right)\\ t_1 := \left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\\ \mathbf{if}\;\tan t\_1 \cdot \frac{\left|z0\right|}{\left|z1\right|} \leq 2 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{{\left(\frac{\left(-1 + 1.3333333333333333 \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot \left|z0\right|}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot \left|z1\right|}\right)}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) \cdot \left(\left|z0\right| \cdot \left|z0\right|\right)}{t\_0}}{t\_0}}}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (sin (* (+ z2 z2) PI))))
       (t_1 (* (- (+ z2 z2) -0.5) PI)))
  (if (<= (* (tan t_1) (/ (fabs z0) (fabs z1))) 2e+52)
    (sqrt
     (-
      (pow
       (/
        (*
         (+
          -1.0
          (*
           1.3333333333333333
           (* (pow z2 3.0) (* (pow PI 3.0) (cos (* -0.5 PI))))))
         (fabs z0))
        (* (cos (* (- (+ z2 z2) -1.5) PI)) (fabs z1)))
       2.0)
      -1.0))
    (/
     (sqrt
      (/
       (/
        (* (- 0.5 (* 0.5 (cos (* 2.0 t_1)))) (* (fabs z0) (fabs z0)))
        t_0)
       t_0))
     (fabs z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = -sin(((z2 + z2) * ((double) M_PI)));
	double t_1 = ((z2 + z2) - -0.5) * ((double) M_PI);
	double tmp;
	if ((tan(t_1) * (fabs(z0) / fabs(z1))) <= 2e+52) {
		tmp = sqrt((pow((((-1.0 + (1.3333333333333333 * (pow(z2, 3.0) * (pow(((double) M_PI), 3.0) * cos((-0.5 * ((double) M_PI))))))) * fabs(z0)) / (cos((((z2 + z2) - -1.5) * ((double) M_PI))) * fabs(z1))), 2.0) - -1.0));
	} else {
		tmp = sqrt(((((0.5 - (0.5 * cos((2.0 * t_1)))) * (fabs(z0) * fabs(z0))) / t_0) / t_0)) / fabs(z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = -Math.sin(((z2 + z2) * Math.PI));
	double t_1 = ((z2 + z2) - -0.5) * Math.PI;
	double tmp;
	if ((Math.tan(t_1) * (Math.abs(z0) / Math.abs(z1))) <= 2e+52) {
		tmp = Math.sqrt((Math.pow((((-1.0 + (1.3333333333333333 * (Math.pow(z2, 3.0) * (Math.pow(Math.PI, 3.0) * Math.cos((-0.5 * Math.PI)))))) * Math.abs(z0)) / (Math.cos((((z2 + z2) - -1.5) * Math.PI)) * Math.abs(z1))), 2.0) - -1.0));
	} else {
		tmp = Math.sqrt(((((0.5 - (0.5 * Math.cos((2.0 * t_1)))) * (Math.abs(z0) * Math.abs(z0))) / t_0) / t_0)) / Math.abs(z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = -math.sin(((z2 + z2) * math.pi))
	t_1 = ((z2 + z2) - -0.5) * math.pi
	tmp = 0
	if (math.tan(t_1) * (math.fabs(z0) / math.fabs(z1))) <= 2e+52:
		tmp = math.sqrt((math.pow((((-1.0 + (1.3333333333333333 * (math.pow(z2, 3.0) * (math.pow(math.pi, 3.0) * math.cos((-0.5 * math.pi)))))) * math.fabs(z0)) / (math.cos((((z2 + z2) - -1.5) * math.pi)) * math.fabs(z1))), 2.0) - -1.0))
	else:
		tmp = math.sqrt(((((0.5 - (0.5 * math.cos((2.0 * t_1)))) * (math.fabs(z0) * math.fabs(z0))) / t_0) / t_0)) / math.fabs(z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(-sin(Float64(Float64(z2 + z2) * pi)))
	t_1 = Float64(Float64(Float64(z2 + z2) - -0.5) * pi)
	tmp = 0.0
	if (Float64(tan(t_1) * Float64(abs(z0) / abs(z1))) <= 2e+52)
		tmp = sqrt(Float64((Float64(Float64(Float64(-1.0 + Float64(1.3333333333333333 * Float64((z2 ^ 3.0) * Float64((pi ^ 3.0) * cos(Float64(-0.5 * pi)))))) * abs(z0)) / Float64(cos(Float64(Float64(Float64(z2 + z2) - -1.5) * pi)) * abs(z1))) ^ 2.0) - -1.0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_1)))) * Float64(abs(z0) * abs(z0))) / t_0) / t_0)) / abs(z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = -sin(((z2 + z2) * pi));
	t_1 = ((z2 + z2) - -0.5) * pi;
	tmp = 0.0;
	if ((tan(t_1) * (abs(z0) / abs(z1))) <= 2e+52)
		tmp = sqrt((((((-1.0 + (1.3333333333333333 * ((z2 ^ 3.0) * ((pi ^ 3.0) * cos((-0.5 * pi)))))) * abs(z0)) / (cos((((z2 + z2) - -1.5) * pi)) * abs(z1))) ^ 2.0) - -1.0));
	else
		tmp = sqrt(((((0.5 - (0.5 * cos((2.0 * t_1)))) * (abs(z0) * abs(z0))) / t_0) / t_0)) / abs(z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = (-N[Sin[N[(N[(z2 + z2), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision])}, Block[{t$95$1 = N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+52], N[Sqrt[N[(N[Power[N[(N[(N[(-1.0 + N[(1.3333333333333333 * N[(N[Power[z2, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * N[Cos[N[(-0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[N[(N[(N[(z2 + z2), $MachinePrecision] - -1.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < 2e52

   1. Initial program 44.8%

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

      1. lift-*.f64 N/A

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

      2. lift-tan.f64 N/A

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

      3. tan-quot N/A

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

      4. frac-2neg N/A

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

      5. lift-/.f64 N/A

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

      6. frac-times N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 44.7%

      \[\leadsto \sqrt{{\color{blue}{\left(\frac{\sin \left(\left(-0.5 - \left(z2 + z2\right)\right) \cdot \pi\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}}^{2} - -1} \]

   4. Taylor expanded in z2 around 0

      \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\pi \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right) + z2 \cdot \left(-2 \cdot \left({\pi}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \pi\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]
   5. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      2. lower-sin.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + \color{blue}{z2} \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      3. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right) + z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      4. lower-PI.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      5. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + z2 \cdot \color{blue}{\left(-2 \cdot \left(\mathsf{PI}\left(\right) \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + z2 \cdot \left(-2 \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sin \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right) + \frac{4}{3} \cdot \left(z2 \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   6. Applied rewrites 70.9%

      \[\leadsto \sqrt{{\left(\frac{\color{blue}{\left(\sin \left(-0.5 \cdot \pi\right) + z2 \cdot \left(-2 \cdot \left(\pi \cdot \cos \left(-0.5 \cdot \pi\right)\right) + z2 \cdot \left(-2 \cdot \left({\pi}^{2} \cdot \sin \left(-0.5 \cdot \pi\right)\right) + 1.3333333333333333 \cdot \left(z2 \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right)\right)\right)} \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   7. Taylor expanded in z2 around inf

      \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + \frac{4}{3} \cdot \color{blue}{\left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \pi\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \color{blue}{\left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)}\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \color{blue}{\cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      3. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \color{blue}{\left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)}\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      4. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\mathsf{PI}\left(\right)}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      6. lower-PI.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      7. lower-cos.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      8. lower-*.f64 N/A

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(\frac{-1}{2} \cdot \pi\right) + \frac{4}{3} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(\frac{-1}{2} \cdot \mathsf{PI}\left(\right)\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - \frac{-3}{2}\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

      9. lower-PI.f64 71.1%

         \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + 1.3333333333333333 \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   9. Applied rewrites 71.1%

      \[\leadsto \sqrt{{\left(\frac{\left(\sin \left(-0.5 \cdot \pi\right) + 1.3333333333333333 \cdot \color{blue}{\left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)}\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   10. Evaluated real constant 71.1%

       \[\leadsto \sqrt{{\left(\frac{\left(-1 + \color{blue}{1.3333333333333333} \cdot \left({z2}^{3} \cdot \left({\pi}^{3} \cdot \cos \left(-0.5 \cdot \pi\right)\right)\right)\right) \cdot z0}{\cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \pi\right) \cdot z1}\right)}^{2} - -1} \]

   if 2e52 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1))

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 14.3%

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

   9. Applied rewrites 18.7%

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

Alternative 4: 64.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(0.5 \cdot \pi\right)\\ t_1 := \frac{z0}{\left|z1\right|}\\ t_2 := \frac{1}{\left|z1\right|}\\ t_3 := \left(2 \cdot \left(\pi + \left(t\_0 \cdot t\_0\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\\ t_4 := \left(z2 + z2\right) - -0.5\\ t_5 := {\left(t\_1 \cdot \tan \left(\pi \cdot t\_4\right)\right)}^{2} - -1\\ t_6 := t\_4 \cdot \pi\\ t_7 := \sqrt{\left(z0 \cdot z0\right) \cdot {\tan t\_6}^{2}}\\ t_8 := \left(z2 + z2\right) \cdot \pi\\ t_9 := -\sin t\_8\\ \mathbf{if}\;z2 \leq -2.7 \cdot 10^{+271}:\\ \;\;\;\;\sqrt{{\left(t\_1 \cdot t\_3\right)}^{2} - -1}\\ \mathbf{elif}\;z2 \leq -1.25 \cdot 10^{+21}:\\ \;\;\;\;\frac{t\_7 - \frac{-0.5 \cdot \left(\left|z1\right| \cdot \left|z1\right|\right)}{t\_7}}{\left|z1\right|}\\ \mathbf{elif}\;z2 \leq -1 \cdot 10^{-159}:\\ \;\;\;\;\sqrt{\left(\left({\left(\frac{\cos t\_8}{t\_9}\right)}^{2} \cdot t\_1\right) \cdot t\_2\right) \cdot z0 - -1}\\ \mathbf{elif}\;z2 \leq 2.7 \cdot 10^{-125}:\\ \;\;\;\;\sqrt{\sqrt{t\_5 \cdot t\_5}}\\ \mathbf{elif}\;z2 \leq 3.5 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\left(\left(\frac{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot t\_6\right)}{t\_9}}{t\_9} \cdot t\_1\right) \cdot t\_2\right) \cdot z0 - -1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(\frac{t\_3 \cdot z0}{\left|z1\right|}\right)}^{2} - -1}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (tan (* 0.5 PI)))
       (t_1 (/ z0 (fabs z1)))
       (t_2 (/ 1.0 (fabs z1)))
       (t_3
        (-
         (* (* 2.0 (+ PI (* (* t_0 t_0) PI))) z2)
         (tan (* PI -0.5))))
       (t_4 (- (+ z2 z2) -0.5))
       (t_5 (- (pow (* t_1 (tan (* PI t_4))) 2.0) -1.0))
       (t_6 (* t_4 PI))
       (t_7 (sqrt (* (* z0 z0) (pow (tan t_6) 2.0))))
       (t_8 (* (+ z2 z2) PI))
       (t_9 (- (sin t_8))))
  (if (<= z2 -2.7e+271)
    (sqrt (- (pow (* t_1 t_3) 2.0) -1.0))
    (if (<= z2 -1.25e+21)
      (/ (- t_7 (/ (* -0.5 (* (fabs z1) (fabs z1))) t_7)) (fabs z1))
      (if (<= z2 -1e-159)
        (sqrt
         (- (* (* (* (pow (/ (cos t_8) t_9) 2.0) t_1) t_2) z0) -1.0))
        (if (<= z2 2.7e-125)
          (sqrt (sqrt (* t_5 t_5)))
          (if (<= z2 3.5e-6)
            (sqrt
             (-
              (*
               (*
                (*
                 (/ (/ (- 0.5 (* 0.5 (cos (* 2.0 t_6)))) t_9) t_9)
                 t_1)
                t_2)
               z0)
              -1.0))
            (sqrt (- (pow (/ (* t_3 z0) (fabs z1)) 2.0) -1.0)))))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = tan((0.5 * ((double) M_PI)));
	double t_1 = z0 / fabs(z1);
	double t_2 = 1.0 / fabs(z1);
	double t_3 = ((2.0 * (((double) M_PI) + ((t_0 * t_0) * ((double) M_PI)))) * z2) - tan((((double) M_PI) * -0.5));
	double t_4 = (z2 + z2) - -0.5;
	double t_5 = pow((t_1 * tan((((double) M_PI) * t_4))), 2.0) - -1.0;
	double t_6 = t_4 * ((double) M_PI);
	double t_7 = sqrt(((z0 * z0) * pow(tan(t_6), 2.0)));
	double t_8 = (z2 + z2) * ((double) M_PI);
	double t_9 = -sin(t_8);
	double tmp;
	if (z2 <= -2.7e+271) {
		tmp = sqrt((pow((t_1 * t_3), 2.0) - -1.0));
	} else if (z2 <= -1.25e+21) {
		tmp = (t_7 - ((-0.5 * (fabs(z1) * fabs(z1))) / t_7)) / fabs(z1);
	} else if (z2 <= -1e-159) {
		tmp = sqrt(((((pow((cos(t_8) / t_9), 2.0) * t_1) * t_2) * z0) - -1.0));
	} else if (z2 <= 2.7e-125) {
		tmp = sqrt(sqrt((t_5 * t_5)));
	} else if (z2 <= 3.5e-6) {
		tmp = sqrt((((((((0.5 - (0.5 * cos((2.0 * t_6)))) / t_9) / t_9) * t_1) * t_2) * z0) - -1.0));
	} else {
		tmp = sqrt((pow(((t_3 * z0) / fabs(z1)), 2.0) - -1.0));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.tan((0.5 * Math.PI));
	double t_1 = z0 / Math.abs(z1);
	double t_2 = 1.0 / Math.abs(z1);
	double t_3 = ((2.0 * (Math.PI + ((t_0 * t_0) * Math.PI))) * z2) - Math.tan((Math.PI * -0.5));
	double t_4 = (z2 + z2) - -0.5;
	double t_5 = Math.pow((t_1 * Math.tan((Math.PI * t_4))), 2.0) - -1.0;
	double t_6 = t_4 * Math.PI;
	double t_7 = Math.sqrt(((z0 * z0) * Math.pow(Math.tan(t_6), 2.0)));
	double t_8 = (z2 + z2) * Math.PI;
	double t_9 = -Math.sin(t_8);
	double tmp;
	if (z2 <= -2.7e+271) {
		tmp = Math.sqrt((Math.pow((t_1 * t_3), 2.0) - -1.0));
	} else if (z2 <= -1.25e+21) {
		tmp = (t_7 - ((-0.5 * (Math.abs(z1) * Math.abs(z1))) / t_7)) / Math.abs(z1);
	} else if (z2 <= -1e-159) {
		tmp = Math.sqrt(((((Math.pow((Math.cos(t_8) / t_9), 2.0) * t_1) * t_2) * z0) - -1.0));
	} else if (z2 <= 2.7e-125) {
		tmp = Math.sqrt(Math.sqrt((t_5 * t_5)));
	} else if (z2 <= 3.5e-6) {
		tmp = Math.sqrt((((((((0.5 - (0.5 * Math.cos((2.0 * t_6)))) / t_9) / t_9) * t_1) * t_2) * z0) - -1.0));
	} else {
		tmp = Math.sqrt((Math.pow(((t_3 * z0) / Math.abs(z1)), 2.0) - -1.0));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.tan((0.5 * math.pi))
	t_1 = z0 / math.fabs(z1)
	t_2 = 1.0 / math.fabs(z1)
	t_3 = ((2.0 * (math.pi + ((t_0 * t_0) * math.pi))) * z2) - math.tan((math.pi * -0.5))
	t_4 = (z2 + z2) - -0.5
	t_5 = math.pow((t_1 * math.tan((math.pi * t_4))), 2.0) - -1.0
	t_6 = t_4 * math.pi
	t_7 = math.sqrt(((z0 * z0) * math.pow(math.tan(t_6), 2.0)))
	t_8 = (z2 + z2) * math.pi
	t_9 = -math.sin(t_8)
	tmp = 0
	if z2 <= -2.7e+271:
		tmp = math.sqrt((math.pow((t_1 * t_3), 2.0) - -1.0))
	elif z2 <= -1.25e+21:
		tmp = (t_7 - ((-0.5 * (math.fabs(z1) * math.fabs(z1))) / t_7)) / math.fabs(z1)
	elif z2 <= -1e-159:
		tmp = math.sqrt(((((math.pow((math.cos(t_8) / t_9), 2.0) * t_1) * t_2) * z0) - -1.0))
	elif z2 <= 2.7e-125:
		tmp = math.sqrt(math.sqrt((t_5 * t_5)))
	elif z2 <= 3.5e-6:
		tmp = math.sqrt((((((((0.5 - (0.5 * math.cos((2.0 * t_6)))) / t_9) / t_9) * t_1) * t_2) * z0) - -1.0))
	else:
		tmp = math.sqrt((math.pow(((t_3 * z0) / math.fabs(z1)), 2.0) - -1.0))
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = tan(Float64(0.5 * pi))
	t_1 = Float64(z0 / abs(z1))
	t_2 = Float64(1.0 / abs(z1))
	t_3 = Float64(Float64(Float64(2.0 * Float64(pi + Float64(Float64(t_0 * t_0) * pi))) * z2) - tan(Float64(pi * -0.5)))
	t_4 = Float64(Float64(z2 + z2) - -0.5)
	t_5 = Float64((Float64(t_1 * tan(Float64(pi * t_4))) ^ 2.0) - -1.0)
	t_6 = Float64(t_4 * pi)
	t_7 = sqrt(Float64(Float64(z0 * z0) * (tan(t_6) ^ 2.0)))
	t_8 = Float64(Float64(z2 + z2) * pi)
	t_9 = Float64(-sin(t_8))
	tmp = 0.0
	if (z2 <= -2.7e+271)
		tmp = sqrt(Float64((Float64(t_1 * t_3) ^ 2.0) - -1.0));
	elseif (z2 <= -1.25e+21)
		tmp = Float64(Float64(t_7 - Float64(Float64(-0.5 * Float64(abs(z1) * abs(z1))) / t_7)) / abs(z1));
	elseif (z2 <= -1e-159)
		tmp = sqrt(Float64(Float64(Float64(Float64((Float64(cos(t_8) / t_9) ^ 2.0) * t_1) * t_2) * z0) - -1.0));
	elseif (z2 <= 2.7e-125)
		tmp = sqrt(sqrt(Float64(t_5 * t_5)));
	elseif (z2 <= 3.5e-6)
		tmp = sqrt(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_6)))) / t_9) / t_9) * t_1) * t_2) * z0) - -1.0));
	else
		tmp = sqrt(Float64((Float64(Float64(t_3 * z0) / abs(z1)) ^ 2.0) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = tan((0.5 * pi));
	t_1 = z0 / abs(z1);
	t_2 = 1.0 / abs(z1);
	t_3 = ((2.0 * (pi + ((t_0 * t_0) * pi))) * z2) - tan((pi * -0.5));
	t_4 = (z2 + z2) - -0.5;
	t_5 = ((t_1 * tan((pi * t_4))) ^ 2.0) - -1.0;
	t_6 = t_4 * pi;
	t_7 = sqrt(((z0 * z0) * (tan(t_6) ^ 2.0)));
	t_8 = (z2 + z2) * pi;
	t_9 = -sin(t_8);
	tmp = 0.0;
	if (z2 <= -2.7e+271)
		tmp = sqrt((((t_1 * t_3) ^ 2.0) - -1.0));
	elseif (z2 <= -1.25e+21)
		tmp = (t_7 - ((-0.5 * (abs(z1) * abs(z1))) / t_7)) / abs(z1);
	elseif (z2 <= -1e-159)
		tmp = sqrt(((((((cos(t_8) / t_9) ^ 2.0) * t_1) * t_2) * z0) - -1.0));
	elseif (z2 <= 2.7e-125)
		tmp = sqrt(sqrt((t_5 * t_5)));
	elseif (z2 <= 3.5e-6)
		tmp = sqrt((((((((0.5 - (0.5 * cos((2.0 * t_6)))) / t_9) / t_9) * t_1) * t_2) * z0) - -1.0));
	else
		tmp = sqrt(((((t_3 * z0) / abs(z1)) ^ 2.0) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(2.0 * N[(Pi + N[(N[(t$95$0 * t$95$0), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$5 = N[(N[Power[N[(t$95$1 * N[Tan[N[(Pi * t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * Pi), $MachinePrecision]}, Block[{t$95$7 = N[Sqrt[N[(N[(z0 * z0), $MachinePrecision] * N[Power[N[Tan[t$95$6], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[(N[(z2 + z2), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$9 = (-N[Sin[t$95$8], $MachinePrecision])}, If[LessEqual[z2, -2.7e+271], N[Sqrt[N[(N[Power[N[(t$95$1 * t$95$3), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, -1.25e+21], N[(N[(t$95$7 - N[(N[(-0.5 * N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, -1e-159], N[Sqrt[N[(N[(N[(N[(N[Power[N[(N[Cos[t$95$8], $MachinePrecision] / t$95$9), $MachinePrecision], 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 2.7e-125], N[Sqrt[N[Sqrt[N[(t$95$5 * t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[z2, 3.5e-6], N[Sqrt[N[(N[(N[(N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$6), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$9), $MachinePrecision] / t$95$9), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(t$95$3 * z0), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]]]]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if z2 < -2.6999999999999999e271

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if -2.6999999999999999e271 < z2 < -1.25e21

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 19.7%

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

   9. Applied rewrites 19.7%

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

   if -1.25e21 < z2 < -9.9999999999999999e-160

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. tan-quot N/A

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

      8. lift-+.f64 N/A

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

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. count-2-rev N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-sin.f64 N/A

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

   5. Applied rewrites 47.8%

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

   if -9.9999999999999999e-160 < z2 < 2.6999999999999998e-125

   1. Initial program 44.8%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 50.7%

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

   3. Applied rewrites 50.7%

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

   if 2.6999999999999998e-125 < z2 < 3.4999999999999999e-6

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

   5. Applied rewrites 47.8%

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

   if 3.4999999999999999e-6 < z2

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 59.2%

      \[\leadsto \sqrt{{\color{blue}{\left(\frac{\left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right)}}^{2} - -1} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 5: 59.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\\ t_1 := \tan \left(0.5 \cdot \pi\right)\\ t_2 := -\sin \left(\left(z2 + z2\right) \cdot \pi\right)\\ \mathbf{if}\;\tan t\_0 \cdot \frac{\left|z0\right|}{\left|z1\right|} \leq 10^{-34}:\\ \;\;\;\;\sqrt{{\left(\frac{\left(\left(2 \cdot \left(\pi + \left(t\_1 \cdot t\_1\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot \left|z0\right|}{\left|z1\right|}\right)}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) \cdot \left(\left|z0\right| \cdot \left|z0\right|\right)}{t\_2}}{t\_2}}}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (- (+ z2 z2) -0.5) PI))
       (t_1 (tan (* 0.5 PI)))
       (t_2 (- (sin (* (+ z2 z2) PI)))))
  (if (<= (* (tan t_0) (/ (fabs z0) (fabs z1))) 1e-34)
    (sqrt
     (-
      (pow
       (/
        (*
         (-
          (* (* 2.0 (+ PI (* (* t_1 t_1) PI))) z2)
          (tan (* PI -0.5)))
         (fabs z0))
        (fabs z1))
       2.0)
      -1.0))
    (/
     (sqrt
      (/
       (/
        (* (- 0.5 (* 0.5 (cos (* 2.0 t_0)))) (* (fabs z0) (fabs z0)))
        t_2)
       t_2))
     (fabs z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((z2 + z2) - -0.5) * ((double) M_PI);
	double t_1 = tan((0.5 * ((double) M_PI)));
	double t_2 = -sin(((z2 + z2) * ((double) M_PI)));
	double tmp;
	if ((tan(t_0) * (fabs(z0) / fabs(z1))) <= 1e-34) {
		tmp = sqrt((pow((((((2.0 * (((double) M_PI) + ((t_1 * t_1) * ((double) M_PI)))) * z2) - tan((((double) M_PI) * -0.5))) * fabs(z0)) / fabs(z1)), 2.0) - -1.0));
	} else {
		tmp = sqrt(((((0.5 - (0.5 * cos((2.0 * t_0)))) * (fabs(z0) * fabs(z0))) / t_2) / t_2)) / fabs(z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = ((z2 + z2) - -0.5) * Math.PI;
	double t_1 = Math.tan((0.5 * Math.PI));
	double t_2 = -Math.sin(((z2 + z2) * Math.PI));
	double tmp;
	if ((Math.tan(t_0) * (Math.abs(z0) / Math.abs(z1))) <= 1e-34) {
		tmp = Math.sqrt((Math.pow((((((2.0 * (Math.PI + ((t_1 * t_1) * Math.PI))) * z2) - Math.tan((Math.PI * -0.5))) * Math.abs(z0)) / Math.abs(z1)), 2.0) - -1.0));
	} else {
		tmp = Math.sqrt(((((0.5 - (0.5 * Math.cos((2.0 * t_0)))) * (Math.abs(z0) * Math.abs(z0))) / t_2) / t_2)) / Math.abs(z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((z2 + z2) - -0.5) * math.pi
	t_1 = math.tan((0.5 * math.pi))
	t_2 = -math.sin(((z2 + z2) * math.pi))
	tmp = 0
	if (math.tan(t_0) * (math.fabs(z0) / math.fabs(z1))) <= 1e-34:
		tmp = math.sqrt((math.pow((((((2.0 * (math.pi + ((t_1 * t_1) * math.pi))) * z2) - math.tan((math.pi * -0.5))) * math.fabs(z0)) / math.fabs(z1)), 2.0) - -1.0))
	else:
		tmp = math.sqrt(((((0.5 - (0.5 * math.cos((2.0 * t_0)))) * (math.fabs(z0) * math.fabs(z0))) / t_2) / t_2)) / math.fabs(z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(z2 + z2) - -0.5) * pi)
	t_1 = tan(Float64(0.5 * pi))
	t_2 = Float64(-sin(Float64(Float64(z2 + z2) * pi)))
	tmp = 0.0
	if (Float64(tan(t_0) * Float64(abs(z0) / abs(z1))) <= 1e-34)
		tmp = sqrt(Float64((Float64(Float64(Float64(Float64(Float64(2.0 * Float64(pi + Float64(Float64(t_1 * t_1) * pi))) * z2) - tan(Float64(pi * -0.5))) * abs(z0)) / abs(z1)) ^ 2.0) - -1.0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0)))) * Float64(abs(z0) * abs(z0))) / t_2) / t_2)) / abs(z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = ((z2 + z2) - -0.5) * pi;
	t_1 = tan((0.5 * pi));
	t_2 = -sin(((z2 + z2) * pi));
	tmp = 0.0;
	if ((tan(t_0) * (abs(z0) / abs(z1))) <= 1e-34)
		tmp = sqrt((((((((2.0 * (pi + ((t_1 * t_1) * pi))) * z2) - tan((pi * -0.5))) * abs(z0)) / abs(z1)) ^ 2.0) - -1.0));
	else
		tmp = sqrt(((((0.5 - (0.5 * cos((2.0 * t_0)))) * (abs(z0) * abs(z0))) / t_2) / t_2)) / abs(z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = (-N[Sin[N[(N[(z2 + z2), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-34], N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[(2.0 * N[(Pi + N[(N[(t$95$1 * t$95$1), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < 9.9999999999999993e-35

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 59.2%

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

   if 9.9999999999999993e-35 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1))

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 14.3%

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

   9. Applied rewrites 18.7%

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

Alternative 6: 57.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\\ t_1 := \frac{\left|z0\right|}{\left|z1\right|}\\ t_2 := \tan \left(0.5 \cdot \pi\right)\\ t_3 := -\sin \left(\left(z2 + z2\right) \cdot \pi\right)\\ \mathbf{if}\;\tan t\_0 \cdot t\_1 \leq 10^{-34}:\\ \;\;\;\;\sqrt{{\left(t\_1 \cdot \left(\left(2 \cdot \left(\pi + \left(t\_2 \cdot t\_2\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)\right)}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_0\right)\right) \cdot \left(\left|z0\right| \cdot \left|z0\right|\right)}{t\_3}}{t\_3}}}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (- (+ z2 z2) -0.5) PI))
       (t_1 (/ (fabs z0) (fabs z1)))
       (t_2 (tan (* 0.5 PI)))
       (t_3 (- (sin (* (+ z2 z2) PI)))))
  (if (<= (* (tan t_0) t_1) 1e-34)
    (sqrt
     (-
      (pow
       (*
        t_1
        (-
         (* (* 2.0 (+ PI (* (* t_2 t_2) PI))) z2)
         (tan (* PI -0.5))))
       2.0)
      -1.0))
    (/
     (sqrt
      (/
       (/
        (* (- 0.5 (* 0.5 (cos (* 2.0 t_0)))) (* (fabs z0) (fabs z0)))
        t_3)
       t_3))
     (fabs z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((z2 + z2) - -0.5) * ((double) M_PI);
	double t_1 = fabs(z0) / fabs(z1);
	double t_2 = tan((0.5 * ((double) M_PI)));
	double t_3 = -sin(((z2 + z2) * ((double) M_PI)));
	double tmp;
	if ((tan(t_0) * t_1) <= 1e-34) {
		tmp = sqrt((pow((t_1 * (((2.0 * (((double) M_PI) + ((t_2 * t_2) * ((double) M_PI)))) * z2) - tan((((double) M_PI) * -0.5)))), 2.0) - -1.0));
	} else {
		tmp = sqrt(((((0.5 - (0.5 * cos((2.0 * t_0)))) * (fabs(z0) * fabs(z0))) / t_3) / t_3)) / fabs(z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = ((z2 + z2) - -0.5) * Math.PI;
	double t_1 = Math.abs(z0) / Math.abs(z1);
	double t_2 = Math.tan((0.5 * Math.PI));
	double t_3 = -Math.sin(((z2 + z2) * Math.PI));
	double tmp;
	if ((Math.tan(t_0) * t_1) <= 1e-34) {
		tmp = Math.sqrt((Math.pow((t_1 * (((2.0 * (Math.PI + ((t_2 * t_2) * Math.PI))) * z2) - Math.tan((Math.PI * -0.5)))), 2.0) - -1.0));
	} else {
		tmp = Math.sqrt(((((0.5 - (0.5 * Math.cos((2.0 * t_0)))) * (Math.abs(z0) * Math.abs(z0))) / t_3) / t_3)) / Math.abs(z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((z2 + z2) - -0.5) * math.pi
	t_1 = math.fabs(z0) / math.fabs(z1)
	t_2 = math.tan((0.5 * math.pi))
	t_3 = -math.sin(((z2 + z2) * math.pi))
	tmp = 0
	if (math.tan(t_0) * t_1) <= 1e-34:
		tmp = math.sqrt((math.pow((t_1 * (((2.0 * (math.pi + ((t_2 * t_2) * math.pi))) * z2) - math.tan((math.pi * -0.5)))), 2.0) - -1.0))
	else:
		tmp = math.sqrt(((((0.5 - (0.5 * math.cos((2.0 * t_0)))) * (math.fabs(z0) * math.fabs(z0))) / t_3) / t_3)) / math.fabs(z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(z2 + z2) - -0.5) * pi)
	t_1 = Float64(abs(z0) / abs(z1))
	t_2 = tan(Float64(0.5 * pi))
	t_3 = Float64(-sin(Float64(Float64(z2 + z2) * pi)))
	tmp = 0.0
	if (Float64(tan(t_0) * t_1) <= 1e-34)
		tmp = sqrt(Float64((Float64(t_1 * Float64(Float64(Float64(2.0 * Float64(pi + Float64(Float64(t_2 * t_2) * pi))) * z2) - tan(Float64(pi * -0.5)))) ^ 2.0) - -1.0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_0)))) * Float64(abs(z0) * abs(z0))) / t_3) / t_3)) / abs(z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = ((z2 + z2) - -0.5) * pi;
	t_1 = abs(z0) / abs(z1);
	t_2 = tan((0.5 * pi));
	t_3 = -sin(((z2 + z2) * pi));
	tmp = 0.0;
	if ((tan(t_0) * t_1) <= 1e-34)
		tmp = sqrt((((t_1 * (((2.0 * (pi + ((t_2 * t_2) * pi))) * z2) - tan((pi * -0.5)))) ^ 2.0) - -1.0));
	else
		tmp = sqrt(((((0.5 - (0.5 * cos((2.0 * t_0)))) * (abs(z0) * abs(z0))) / t_3) / t_3)) / abs(z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = (-N[Sin[N[(N[(z2 + z2), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[N[(N[Tan[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision], 1e-34], N[Sqrt[N[(N[Power[N[(t$95$1 * N[(N[(N[(2.0 * N[(Pi + N[(N[(t$95$2 * t$95$2), $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision] / t$95$3), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < 9.9999999999999993e-35

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 57.6%

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

   6. Applied rewrites 57.6%

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

   if 9.9999999999999993e-35 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1))

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 14.3%

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

   9. Applied rewrites 18.7%

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

Alternative 7: 53.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \left(z2 + z2\right) - -0.5\\ t_1 := t\_0 \cdot \pi\\ t_2 := -\sin \left(\left(z2 + z2\right) \cdot \pi\right)\\ \mathbf{if}\;\tan t\_1 \cdot \frac{\left|z0\right|}{\left|z1\right|} \leq 10^{-34}:\\ \;\;\;\;\sqrt{\frac{\frac{{\left(\left|z0\right| \cdot \tan \left(\pi \cdot t\_0\right)\right)}^{2}}{\left|z1\right|}}{\left|z1\right|} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{\frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot t\_1\right)\right) \cdot \left(\left|z0\right| \cdot \left|z0\right|\right)}{t\_2}}{t\_2}}}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (+ z2 z2) -0.5))
       (t_1 (* t_0 PI))
       (t_2 (- (sin (* (+ z2 z2) PI)))))
  (if (<= (* (tan t_1) (/ (fabs z0) (fabs z1))) 1e-34)
    (sqrt
     (-
      (/
       (/ (pow (* (fabs z0) (tan (* PI t_0))) 2.0) (fabs z1))
       (fabs z1))
      -1.0))
    (/
     (sqrt
      (/
       (/
        (* (- 0.5 (* 0.5 (cos (* 2.0 t_1)))) (* (fabs z0) (fabs z0)))
        t_2)
       t_2))
     (fabs z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double t_1 = t_0 * ((double) M_PI);
	double t_2 = -sin(((z2 + z2) * ((double) M_PI)));
	double tmp;
	if ((tan(t_1) * (fabs(z0) / fabs(z1))) <= 1e-34) {
		tmp = sqrt((((pow((fabs(z0) * tan((((double) M_PI) * t_0))), 2.0) / fabs(z1)) / fabs(z1)) - -1.0));
	} else {
		tmp = sqrt(((((0.5 - (0.5 * cos((2.0 * t_1)))) * (fabs(z0) * fabs(z0))) / t_2) / t_2)) / fabs(z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double t_1 = t_0 * Math.PI;
	double t_2 = -Math.sin(((z2 + z2) * Math.PI));
	double tmp;
	if ((Math.tan(t_1) * (Math.abs(z0) / Math.abs(z1))) <= 1e-34) {
		tmp = Math.sqrt((((Math.pow((Math.abs(z0) * Math.tan((Math.PI * t_0))), 2.0) / Math.abs(z1)) / Math.abs(z1)) - -1.0));
	} else {
		tmp = Math.sqrt(((((0.5 - (0.5 * Math.cos((2.0 * t_1)))) * (Math.abs(z0) * Math.abs(z0))) / t_2) / t_2)) / Math.abs(z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (z2 + z2) - -0.5
	t_1 = t_0 * math.pi
	t_2 = -math.sin(((z2 + z2) * math.pi))
	tmp = 0
	if (math.tan(t_1) * (math.fabs(z0) / math.fabs(z1))) <= 1e-34:
		tmp = math.sqrt((((math.pow((math.fabs(z0) * math.tan((math.pi * t_0))), 2.0) / math.fabs(z1)) / math.fabs(z1)) - -1.0))
	else:
		tmp = math.sqrt(((((0.5 - (0.5 * math.cos((2.0 * t_1)))) * (math.fabs(z0) * math.fabs(z0))) / t_2) / t_2)) / math.fabs(z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(z2 + z2) - -0.5)
	t_1 = Float64(t_0 * pi)
	t_2 = Float64(-sin(Float64(Float64(z2 + z2) * pi)))
	tmp = 0.0
	if (Float64(tan(t_1) * Float64(abs(z0) / abs(z1))) <= 1e-34)
		tmp = sqrt(Float64(Float64(Float64((Float64(abs(z0) * tan(Float64(pi * t_0))) ^ 2.0) / abs(z1)) / abs(z1)) - -1.0));
	else
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * t_1)))) * Float64(abs(z0) * abs(z0))) / t_2) / t_2)) / abs(z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (z2 + z2) - -0.5;
	t_1 = t_0 * pi;
	t_2 = -sin(((z2 + z2) * pi));
	tmp = 0.0;
	if ((tan(t_1) * (abs(z0) / abs(z1))) <= 1e-34)
		tmp = sqrt((((((abs(z0) * tan((pi * t_0))) ^ 2.0) / abs(z1)) / abs(z1)) - -1.0));
	else
		tmp = sqrt(((((0.5 - (0.5 * cos((2.0 * t_1)))) * (abs(z0) * abs(z0))) / t_2) / t_2)) / abs(z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * Pi), $MachinePrecision]}, Block[{t$95$2 = (-N[Sin[N[(N[(z2 + z2), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision])}, If[LessEqual[N[(N[Tan[t$95$1], $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e-34], N[Sqrt[N[(N[(N[(N[Power[N[(N[Abs[z0], $MachinePrecision] * N[Tan[N[(Pi * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] / t$95$2), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < 9.9999999999999993e-35

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 48.8%

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

   if 9.9999999999999993e-35 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1))

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 14.3%

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

   9. Applied rewrites 18.7%

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

Alternative 8: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(z2 + z2\right) \cdot \pi\\ t_1 := \left(z2 + z2\right) - -0.5\\ t_2 := \frac{\left|z0\right|}{\left|z1\right|}\\ \mathbf{if}\;\tan \left(t\_1 \cdot \pi\right) \cdot t\_2 \leq 2 \cdot 10^{-94}:\\ \;\;\;\;\sqrt{\frac{\frac{{\left(\left|z0\right| \cdot \tan \left(\pi \cdot t\_1\right)\right)}^{2}}{\left|z1\right|}}{\left|z1\right|} - -1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left({\left(\frac{\cos t\_0}{-\sin t\_0}\right)}^{2} \cdot t\_2\right) \cdot \frac{1}{\left|z1\right|}\right) \cdot \left|z0\right| - -1}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (+ z2 z2) PI))
       (t_1 (- (+ z2 z2) -0.5))
       (t_2 (/ (fabs z0) (fabs z1))))
  (if (<= (* (tan (* t_1 PI)) t_2) 2e-94)
    (sqrt
     (-
      (/
       (/ (pow (* (fabs z0) (tan (* PI t_1))) 2.0) (fabs z1))
       (fabs z1))
      -1.0))
    (sqrt
     (-
      (*
       (*
        (* (pow (/ (cos t_0) (- (sin t_0))) 2.0) t_2)
        (/ 1.0 (fabs z1)))
       (fabs z0))
      -1.0)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) * ((double) M_PI);
	double t_1 = (z2 + z2) - -0.5;
	double t_2 = fabs(z0) / fabs(z1);
	double tmp;
	if ((tan((t_1 * ((double) M_PI))) * t_2) <= 2e-94) {
		tmp = sqrt((((pow((fabs(z0) * tan((((double) M_PI) * t_1))), 2.0) / fabs(z1)) / fabs(z1)) - -1.0));
	} else {
		tmp = sqrt(((((pow((cos(t_0) / -sin(t_0)), 2.0) * t_2) * (1.0 / fabs(z1))) * fabs(z0)) - -1.0));
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) * Math.PI;
	double t_1 = (z2 + z2) - -0.5;
	double t_2 = Math.abs(z0) / Math.abs(z1);
	double tmp;
	if ((Math.tan((t_1 * Math.PI)) * t_2) <= 2e-94) {
		tmp = Math.sqrt((((Math.pow((Math.abs(z0) * Math.tan((Math.PI * t_1))), 2.0) / Math.abs(z1)) / Math.abs(z1)) - -1.0));
	} else {
		tmp = Math.sqrt(((((Math.pow((Math.cos(t_0) / -Math.sin(t_0)), 2.0) * t_2) * (1.0 / Math.abs(z1))) * Math.abs(z0)) - -1.0));
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (z2 + z2) * math.pi
	t_1 = (z2 + z2) - -0.5
	t_2 = math.fabs(z0) / math.fabs(z1)
	tmp = 0
	if (math.tan((t_1 * math.pi)) * t_2) <= 2e-94:
		tmp = math.sqrt((((math.pow((math.fabs(z0) * math.tan((math.pi * t_1))), 2.0) / math.fabs(z1)) / math.fabs(z1)) - -1.0))
	else:
		tmp = math.sqrt(((((math.pow((math.cos(t_0) / -math.sin(t_0)), 2.0) * t_2) * (1.0 / math.fabs(z1))) * math.fabs(z0)) - -1.0))
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(z2 + z2) * pi)
	t_1 = Float64(Float64(z2 + z2) - -0.5)
	t_2 = Float64(abs(z0) / abs(z1))
	tmp = 0.0
	if (Float64(tan(Float64(t_1 * pi)) * t_2) <= 2e-94)
		tmp = sqrt(Float64(Float64(Float64((Float64(abs(z0) * tan(Float64(pi * t_1))) ^ 2.0) / abs(z1)) / abs(z1)) - -1.0));
	else
		tmp = sqrt(Float64(Float64(Float64(Float64((Float64(cos(t_0) / Float64(-sin(t_0))) ^ 2.0) * t_2) * Float64(1.0 / abs(z1))) * abs(z0)) - -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (z2 + z2) * pi;
	t_1 = (z2 + z2) - -0.5;
	t_2 = abs(z0) / abs(z1);
	tmp = 0.0;
	if ((tan((t_1 * pi)) * t_2) <= 2e-94)
		tmp = sqrt((((((abs(z0) * tan((pi * t_1))) ^ 2.0) / abs(z1)) / abs(z1)) - -1.0));
	else
		tmp = sqrt(((((((cos(t_0) / -sin(t_0)) ^ 2.0) * t_2) * (1.0 / abs(z1))) * abs(z0)) - -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(z2 + z2), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[Tan[N[(t$95$1 * Pi), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], 2e-94], N[Sqrt[N[(N[(N[(N[Power[N[(N[Abs[z0], $MachinePrecision] * N[Tan[N[(Pi * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(N[(N[Power[N[(N[Cos[t$95$0], $MachinePrecision] / (-N[Sin[t$95$0], $MachinePrecision])), $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision] * N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < 1.9999999999999999e-94

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 48.8%

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

   if 1.9999999999999999e-94 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1))

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

      1. lift-tan.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift--.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. tan-quot N/A

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

      8. lift-+.f64 N/A

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

      9. sub-flip N/A

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

      10. metadata-eval N/A

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

      11. +-commutative N/A

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

      12. lift-+.f64 N/A

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

      13. count-2-rev N/A

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

      14. lift-*.f64 N/A

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

      15. lift-+.f64 N/A

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

      16. lift-PI.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-sin.f64 N/A

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

   5. Applied rewrites 47.8%

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

Alternative 9: 52.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(z2 + z2\right) - -0.5\\ \mathbf{if}\;\left|z0\right| \leq 6.8 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\left(\left({\tan \left(\pi \cdot t\_0\right)}^{2} \cdot \frac{\left|z0\right|}{\left|z1\right|}\right) \cdot \frac{1}{\left|z1\right|}\right) \cdot \left|z0\right| - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\tan \left(t\_0 \cdot \pi\right)}^{2} \cdot \left(\left|z0\right| \cdot \left|z0\right|\right)}}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (+ z2 z2) -0.5)))
  (if (<= (fabs z0) 6.8e+156)
    (sqrt
     (-
      (*
       (*
        (* (pow (tan (* PI t_0)) 2.0) (/ (fabs z0) (fabs z1)))
        (/ 1.0 (fabs z1)))
       (fabs z0))
      -1.0))
    (/
     (sqrt (* (pow (tan (* t_0 PI)) 2.0) (* (fabs z0) (fabs z0))))
     (fabs z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double tmp;
	if (fabs(z0) <= 6.8e+156) {
		tmp = sqrt(((((pow(tan((((double) M_PI) * t_0)), 2.0) * (fabs(z0) / fabs(z1))) * (1.0 / fabs(z1))) * fabs(z0)) - -1.0));
	} else {
		tmp = sqrt((pow(tan((t_0 * ((double) M_PI))), 2.0) * (fabs(z0) * fabs(z0)))) / fabs(z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double tmp;
	if (Math.abs(z0) <= 6.8e+156) {
		tmp = Math.sqrt(((((Math.pow(Math.tan((Math.PI * t_0)), 2.0) * (Math.abs(z0) / Math.abs(z1))) * (1.0 / Math.abs(z1))) * Math.abs(z0)) - -1.0));
	} else {
		tmp = Math.sqrt((Math.pow(Math.tan((t_0 * Math.PI)), 2.0) * (Math.abs(z0) * Math.abs(z0)))) / Math.abs(z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (z2 + z2) - -0.5
	tmp = 0
	if math.fabs(z0) <= 6.8e+156:
		tmp = math.sqrt(((((math.pow(math.tan((math.pi * t_0)), 2.0) * (math.fabs(z0) / math.fabs(z1))) * (1.0 / math.fabs(z1))) * math.fabs(z0)) - -1.0))
	else:
		tmp = math.sqrt((math.pow(math.tan((t_0 * math.pi)), 2.0) * (math.fabs(z0) * math.fabs(z0)))) / math.fabs(z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(z2 + z2) - -0.5)
	tmp = 0.0
	if (abs(z0) <= 6.8e+156)
		tmp = sqrt(Float64(Float64(Float64(Float64((tan(Float64(pi * t_0)) ^ 2.0) * Float64(abs(z0) / abs(z1))) * Float64(1.0 / abs(z1))) * abs(z0)) - -1.0));
	else
		tmp = Float64(sqrt(Float64((tan(Float64(t_0 * pi)) ^ 2.0) * Float64(abs(z0) * abs(z0)))) / abs(z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (z2 + z2) - -0.5;
	tmp = 0.0;
	if (abs(z0) <= 6.8e+156)
		tmp = sqrt((((((tan((pi * t_0)) ^ 2.0) * (abs(z0) / abs(z1))) * (1.0 / abs(z1))) * abs(z0)) - -1.0));
	else
		tmp = sqrt(((tan((t_0 * pi)) ^ 2.0) * (abs(z0) * abs(z0)))) / abs(z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 6.8e+156], N[Sqrt[N[(N[(N[(N[(N[Power[N[Tan[N[(Pi * t$95$0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[Power[N[Tan[N[(t$95$0 * Pi), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 6.8000000000000002e156

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

   if 6.8000000000000002e156 < z0

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 14.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. frac-times N/A

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

   9. Applied rewrites 14.3%

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

Alternative 10: 50.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := {\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)}^{2}\\ \mathbf{if}\;\left|z0\right| \leq 6.8 \cdot 10^{+157}:\\ \;\;\;\;\sqrt{\frac{t\_0 \cdot \left|z0\right|}{\left|z1\right| \cdot \left|z1\right|} \cdot \left|z0\right| - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(\left|z0\right| \cdot \left|z0\right|\right)}}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (pow (tan (* (- (+ z2 z2) -0.5) PI)) 2.0)))
  (if (<= (fabs z0) 6.8e+157)
    (sqrt
     (-
      (* (/ (* t_0 (fabs z0)) (* (fabs z1) (fabs z1))) (fabs z0))
      -1.0))
    (/ (sqrt (* t_0 (* (fabs z0) (fabs z0)))) (fabs z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = pow(tan((((z2 + z2) - -0.5) * ((double) M_PI))), 2.0);
	double tmp;
	if (fabs(z0) <= 6.8e+157) {
		tmp = sqrt(((((t_0 * fabs(z0)) / (fabs(z1) * fabs(z1))) * fabs(z0)) - -1.0));
	} else {
		tmp = sqrt((t_0 * (fabs(z0) * fabs(z0)))) / fabs(z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.pow(Math.tan((((z2 + z2) - -0.5) * Math.PI)), 2.0);
	double tmp;
	if (Math.abs(z0) <= 6.8e+157) {
		tmp = Math.sqrt(((((t_0 * Math.abs(z0)) / (Math.abs(z1) * Math.abs(z1))) * Math.abs(z0)) - -1.0));
	} else {
		tmp = Math.sqrt((t_0 * (Math.abs(z0) * Math.abs(z0)))) / Math.abs(z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.pow(math.tan((((z2 + z2) - -0.5) * math.pi)), 2.0)
	tmp = 0
	if math.fabs(z0) <= 6.8e+157:
		tmp = math.sqrt(((((t_0 * math.fabs(z0)) / (math.fabs(z1) * math.fabs(z1))) * math.fabs(z0)) - -1.0))
	else:
		tmp = math.sqrt((t_0 * (math.fabs(z0) * math.fabs(z0)))) / math.fabs(z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) ^ 2.0
	tmp = 0.0
	if (abs(z0) <= 6.8e+157)
		tmp = sqrt(Float64(Float64(Float64(Float64(t_0 * abs(z0)) / Float64(abs(z1) * abs(z1))) * abs(z0)) - -1.0));
	else
		tmp = Float64(sqrt(Float64(t_0 * Float64(abs(z0) * abs(z0)))) / abs(z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = tan((((z2 + z2) - -0.5) * pi)) ^ 2.0;
	tmp = 0.0;
	if (abs(z0) <= 6.8e+157)
		tmp = sqrt(((((t_0 * abs(z0)) / (abs(z1) * abs(z1))) * abs(z0)) - -1.0));
	else
		tmp = sqrt((t_0 * (abs(z0) * abs(z0)))) / abs(z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[Power[N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 6.8e+157], N[Sqrt[N[(N[(N[(N[(t$95$0 * N[Abs[z0], $MachinePrecision]), $MachinePrecision] / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(t$95$0 * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 6.7999999999999996e157

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 46.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r/ N/A

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

      7. associate-/l/ N/A

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

      8. unpow2 N/A

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

      9. lift-pow.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 48.7%

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 48.7%

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

   5. Applied rewrites 48.7%

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

   if 6.7999999999999996e157 < z0

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 14.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. frac-times N/A

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

   9. Applied rewrites 14.3%

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

Alternative 11: 50.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(z2 + z2\right) - -0.5\\ \mathbf{if}\;\left|z0\right| \leq 6.2 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{\frac{{\left(\left|z0\right| \cdot \tan \left(\pi \cdot t\_0\right)\right)}^{2}}{\left|z1\right| \cdot \left|z1\right|} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\tan \left(t\_0 \cdot \pi\right)}^{2} \cdot \left(\left|z0\right| \cdot \left|z0\right|\right)}}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (+ z2 z2) -0.5)))
  (if (<= (fabs z0) 6.2e+150)
    (sqrt
     (-
      (/
       (pow (* (fabs z0) (tan (* PI t_0))) 2.0)
       (* (fabs z1) (fabs z1)))
      -1.0))
    (/
     (sqrt (* (pow (tan (* t_0 PI)) 2.0) (* (fabs z0) (fabs z0))))
     (fabs z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double tmp;
	if (fabs(z0) <= 6.2e+150) {
		tmp = sqrt(((pow((fabs(z0) * tan((((double) M_PI) * t_0))), 2.0) / (fabs(z1) * fabs(z1))) - -1.0));
	} else {
		tmp = sqrt((pow(tan((t_0 * ((double) M_PI))), 2.0) * (fabs(z0) * fabs(z0)))) / fabs(z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double tmp;
	if (Math.abs(z0) <= 6.2e+150) {
		tmp = Math.sqrt(((Math.pow((Math.abs(z0) * Math.tan((Math.PI * t_0))), 2.0) / (Math.abs(z1) * Math.abs(z1))) - -1.0));
	} else {
		tmp = Math.sqrt((Math.pow(Math.tan((t_0 * Math.PI)), 2.0) * (Math.abs(z0) * Math.abs(z0)))) / Math.abs(z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (z2 + z2) - -0.5
	tmp = 0
	if math.fabs(z0) <= 6.2e+150:
		tmp = math.sqrt(((math.pow((math.fabs(z0) * math.tan((math.pi * t_0))), 2.0) / (math.fabs(z1) * math.fabs(z1))) - -1.0))
	else:
		tmp = math.sqrt((math.pow(math.tan((t_0 * math.pi)), 2.0) * (math.fabs(z0) * math.fabs(z0)))) / math.fabs(z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(z2 + z2) - -0.5)
	tmp = 0.0
	if (abs(z0) <= 6.2e+150)
		tmp = sqrt(Float64(Float64((Float64(abs(z0) * tan(Float64(pi * t_0))) ^ 2.0) / Float64(abs(z1) * abs(z1))) - -1.0));
	else
		tmp = Float64(sqrt(Float64((tan(Float64(t_0 * pi)) ^ 2.0) * Float64(abs(z0) * abs(z0)))) / abs(z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (z2 + z2) - -0.5;
	tmp = 0.0;
	if (abs(z0) <= 6.2e+150)
		tmp = sqrt(((((abs(z0) * tan((pi * t_0))) ^ 2.0) / (abs(z1) * abs(z1))) - -1.0));
	else
		tmp = sqrt(((tan((t_0 * pi)) ^ 2.0) * (abs(z0) * abs(z0)))) / abs(z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, If[LessEqual[N[Abs[z0], $MachinePrecision], 6.2e+150], N[Sqrt[N[(N[(N[Power[N[(N[Abs[z0], $MachinePrecision] * N[Tan[N[(Pi * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[Power[N[Tan[N[(t$95$0 * Pi), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < 6.2000000000000003e150

   1. Initial program 44.8%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   if 6.2000000000000003e150 < z0

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 14.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. frac-times N/A

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

   9. Applied rewrites 14.3%

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

Alternative 12: 48.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 6.8 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{{\left(\tan \left(\left(z2 - \left(-0.5 - z2\right)\right) \cdot \pi\right) \cdot \frac{\left|z0\right|}{\left|z1\right|}\right)}^{2} - -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{{\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)}^{2} \cdot \left(\left|z0\right| \cdot \left|z0\right|\right)}}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (if (<= (fabs z0) 6.8e+156)
  (sqrt
   (-
    (pow
     (* (tan (* (- z2 (- -0.5 z2)) PI)) (/ (fabs z0) (fabs z1)))
     2.0)
    -1.0))
  (/
   (sqrt
    (*
     (pow (tan (* (- (+ z2 z2) -0.5) PI)) 2.0)
     (* (fabs z0) (fabs z0))))
   (fabs z1))))

C

double code(double z2, double z0, double z1) {
	double tmp;
	if (fabs(z0) <= 6.8e+156) {
		tmp = sqrt((pow((tan(((z2 - (-0.5 - z2)) * ((double) M_PI))) * (fabs(z0) / fabs(z1))), 2.0) - -1.0));
	} else {
		tmp = sqrt((pow(tan((((z2 + z2) - -0.5) * ((double) M_PI))), 2.0) * (fabs(z0) * fabs(z0)))) / fabs(z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double tmp;
	if (Math.abs(z0) <= 6.8e+156) {
		tmp = Math.sqrt((Math.pow((Math.tan(((z2 - (-0.5 - z2)) * Math.PI)) * (Math.abs(z0) / Math.abs(z1))), 2.0) - -1.0));
	} else {
		tmp = Math.sqrt((Math.pow(Math.tan((((z2 + z2) - -0.5) * Math.PI)), 2.0) * (Math.abs(z0) * Math.abs(z0)))) / Math.abs(z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	tmp = 0
	if math.fabs(z0) <= 6.8e+156:
		tmp = math.sqrt((math.pow((math.tan(((z2 - (-0.5 - z2)) * math.pi)) * (math.fabs(z0) / math.fabs(z1))), 2.0) - -1.0))
	else:
		tmp = math.sqrt((math.pow(math.tan((((z2 + z2) - -0.5) * math.pi)), 2.0) * (math.fabs(z0) * math.fabs(z0)))) / math.fabs(z1)
	return tmp

Julia

function code(z2, z0, z1)
	tmp = 0.0
	if (abs(z0) <= 6.8e+156)
		tmp = sqrt(Float64((Float64(tan(Float64(Float64(z2 - Float64(-0.5 - z2)) * pi)) * Float64(abs(z0) / abs(z1))) ^ 2.0) - -1.0));
	else
		tmp = Float64(sqrt(Float64((tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) ^ 2.0) * Float64(abs(z0) * abs(z0)))) / abs(z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	tmp = 0.0;
	if (abs(z0) <= 6.8e+156)
		tmp = sqrt((((tan(((z2 - (-0.5 - z2)) * pi)) * (abs(z0) / abs(z1))) ^ 2.0) - -1.0));
	else
		tmp = sqrt(((tan((((z2 + z2) - -0.5) * pi)) ^ 2.0) * (abs(z0) * abs(z0)))) / abs(z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := If[LessEqual[N[Abs[z0], $MachinePrecision], 6.8e+156], N[Sqrt[N[(N[Power[N[(N[Tan[N[(N[(z2 - N[(-0.5 - z2), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(N[Power[N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < 6.8000000000000002e156

   1. Initial program 44.8%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 44.8%

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

   3. Applied rewrites 44.8%

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

   if 6.8000000000000002e156 < z0

   1. Initial program 44.8%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 45.2%

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

   5. Taylor expanded in z1 around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 14.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. lift-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. frac-times N/A

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

   9. Applied rewrites 14.3%

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

Alternative 13: 45.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (sqrt (- (pow (/ (* (tan (* 0.5 PI)) z0) z1) 2.0) -1.0)))

C

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

Java

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

Python

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

Julia

function code(z2, z0, z1)
	return sqrt(Float64((Float64(Float64(tan(Float64(0.5 * pi)) * z0) / z1) ^ 2.0) - -1.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

2. Taylor expanded in z2 around 0

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

4. Applied rewrites 45.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 45.6%

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

6. Applied rewrites 45.6%

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

Alternative 14: 45.6% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (sqrt (- (pow (* z0 (/ (tan (* 0.5 PI)) z1)) 2.0) -1.0)))

C

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

Java

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

Python

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

Julia

function code(z2, z0, z1)
	return sqrt(Float64((Float64(z0 * Float64(tan(Float64(0.5 * pi)) / z1)) ^ 2.0) - -1.0))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

2. Taylor expanded in z2 around 0

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

4. Applied rewrites 45.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 45.6%

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

6. Applied rewrites 45.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 45.6%

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

8. Applied rewrites 45.6%

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

Alternative 15: 18.9% accurate, 22.0× speedup

Math

\[\sqrt{1} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (sqrt 1.0))

C

double code(double z2, double z0, double z1) {
	return sqrt(1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z2, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    code = sqrt(1.0d0)
end function

Java

public static double code(double z2, double z0, double z1) {
	return Math.sqrt(1.0);
}

Python

def code(z2, z0, z1):
	return math.sqrt(1.0)

Julia

function code(z2, z0, z1)
	return sqrt(1.0)
end

MATLAB

function tmp = code(z2, z0, z1)
	tmp = sqrt(1.0);
end

Wolfram

code[z2_, z0_, z1_] := N[Sqrt[1.0], $MachinePrecision]

Derivation

1. Initial program 44.8%

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

2. Taylor expanded in z0 around 0

   \[\leadsto \sqrt{\color{blue}{1}} \]
3. Step-by-step derivation

4. Applied rewrites 18.9%

   \[\leadsto \sqrt{\color{blue}{1}} \]
5. Add Preprocessing

Reproduce

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