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

Percentage Accurate: 40.7% → 71.0%

Time: 14.4s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z2_, z0_, z1_] := N[(1.0 / N[(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] * z1), $MachinePrecision]), $MachinePrecision]

Initial Program: 40.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z2_, z0_, z1_] := N[(1.0 / N[(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] * z1), $MachinePrecision]), $MachinePrecision]

Alternative 1: 71.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(0.5 \cdot \pi\right)\\ t_1 := t\_0 \cdot t\_0\\ t_2 := \tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)\\ t_3 := \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\\ t_4 := 2 \cdot \left(\pi + t\_1 \cdot \pi\right)\\ \mathbf{if}\;t\_2 \leq 100:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\frac{\left|z0\right| \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(t\_4 \cdot t\_0\right) + \left(t\_3 - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot t\_4 - t\_3 \cdot t\_1\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(t\_4 \cdot t\_1\right)\right)\right) \cdot z2\right) \cdot z2 + t\_4\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{z1}\right)}^{2} - -1} \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{e^{\log \left(t\_2 \cdot \left|z0\right|\right) \cdot 2}}{z1}}{z1} - -1} \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (tan (* 0.5 PI)))
       (t_1 (* t_0 t_0))
       (t_2 (tan (* (- (+ z2 z2) -0.5) PI)))
       (t_3 (* (* -1.3333333333333333 (* PI PI)) PI))
       (t_4 (* 2.0 (+ PI (* t_1 PI)))))
  (if (<= t_2 100.0)
    (/
     1.0
     (*
      (sqrt
       (-
        (pow
         (/
          (*
           (fabs z0)
           (-
            (*
             (+
              (*
               (+
                (* (+ PI PI) (* t_4 t_0))
                (*
                 (-
                  t_3
                  (+
                   (- (* (* -2.0 (* PI PI)) t_4) (* t_3 t_1))
                   (* (* -4.0 (* PI PI)) (* t_4 t_1))))
                 z2))
               z2)
              t_4)
             z2)
            (tan (* PI -0.5))))
          z1)
         2.0)
        -1.0))
      z1))
    (/
     1.0
     (*
      (sqrt
       (- (/ (/ (exp (* (log (* t_2 (fabs z0))) 2.0)) z1) z1) -1.0))
      z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = tan((0.5 * ((double) M_PI)));
	double t_1 = t_0 * t_0;
	double t_2 = tan((((z2 + z2) - -0.5) * ((double) M_PI)));
	double t_3 = (-1.3333333333333333 * (((double) M_PI) * ((double) M_PI))) * ((double) M_PI);
	double t_4 = 2.0 * (((double) M_PI) + (t_1 * ((double) M_PI)));
	double tmp;
	if (t_2 <= 100.0) {
		tmp = 1.0 / (sqrt((pow(((fabs(z0) * (((((((((double) M_PI) + ((double) M_PI)) * (t_4 * t_0)) + ((t_3 - ((((-2.0 * (((double) M_PI) * ((double) M_PI))) * t_4) - (t_3 * t_1)) + ((-4.0 * (((double) M_PI) * ((double) M_PI))) * (t_4 * t_1)))) * z2)) * z2) + t_4) * z2) - tan((((double) M_PI) * -0.5)))) / z1), 2.0) - -1.0)) * z1);
	} else {
		tmp = 1.0 / (sqrt((((exp((log((t_2 * fabs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1);
	}
	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 = t_0 * t_0;
	double t_2 = Math.tan((((z2 + z2) - -0.5) * Math.PI));
	double t_3 = (-1.3333333333333333 * (Math.PI * Math.PI)) * Math.PI;
	double t_4 = 2.0 * (Math.PI + (t_1 * Math.PI));
	double tmp;
	if (t_2 <= 100.0) {
		tmp = 1.0 / (Math.sqrt((Math.pow(((Math.abs(z0) * (((((((Math.PI + Math.PI) * (t_4 * t_0)) + ((t_3 - ((((-2.0 * (Math.PI * Math.PI)) * t_4) - (t_3 * t_1)) + ((-4.0 * (Math.PI * Math.PI)) * (t_4 * t_1)))) * z2)) * z2) + t_4) * z2) - Math.tan((Math.PI * -0.5)))) / z1), 2.0) - -1.0)) * z1);
	} else {
		tmp = 1.0 / (Math.sqrt((((Math.exp((Math.log((t_2 * Math.abs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.tan((0.5 * math.pi))
	t_1 = t_0 * t_0
	t_2 = math.tan((((z2 + z2) - -0.5) * math.pi))
	t_3 = (-1.3333333333333333 * (math.pi * math.pi)) * math.pi
	t_4 = 2.0 * (math.pi + (t_1 * math.pi))
	tmp = 0
	if t_2 <= 100.0:
		tmp = 1.0 / (math.sqrt((math.pow(((math.fabs(z0) * (((((((math.pi + math.pi) * (t_4 * t_0)) + ((t_3 - ((((-2.0 * (math.pi * math.pi)) * t_4) - (t_3 * t_1)) + ((-4.0 * (math.pi * math.pi)) * (t_4 * t_1)))) * z2)) * z2) + t_4) * z2) - math.tan((math.pi * -0.5)))) / z1), 2.0) - -1.0)) * z1)
	else:
		tmp = 1.0 / (math.sqrt((((math.exp((math.log((t_2 * math.fabs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = tan(Float64(0.5 * pi))
	t_1 = Float64(t_0 * t_0)
	t_2 = tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi))
	t_3 = Float64(Float64(-1.3333333333333333 * Float64(pi * pi)) * pi)
	t_4 = Float64(2.0 * Float64(pi + Float64(t_1 * pi)))
	tmp = 0.0
	if (t_2 <= 100.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64((Float64(Float64(abs(z0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(pi + pi) * Float64(t_4 * t_0)) + Float64(Float64(t_3 - Float64(Float64(Float64(Float64(-2.0 * Float64(pi * pi)) * t_4) - Float64(t_3 * t_1)) + Float64(Float64(-4.0 * Float64(pi * pi)) * Float64(t_4 * t_1)))) * z2)) * z2) + t_4) * z2) - tan(Float64(pi * -0.5)))) / z1) ^ 2.0) - -1.0)) * z1));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64(exp(Float64(log(Float64(t_2 * abs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = tan((0.5 * pi));
	t_1 = t_0 * t_0;
	t_2 = tan((((z2 + z2) - -0.5) * pi));
	t_3 = (-1.3333333333333333 * (pi * pi)) * pi;
	t_4 = 2.0 * (pi + (t_1 * pi));
	tmp = 0.0;
	if (t_2 <= 100.0)
		tmp = 1.0 / (sqrt(((((abs(z0) * (((((((pi + pi) * (t_4 * t_0)) + ((t_3 - ((((-2.0 * (pi * pi)) * t_4) - (t_3 * t_1)) + ((-4.0 * (pi * pi)) * (t_4 * t_1)))) * z2)) * z2) + t_4) * z2) - tan((pi * -0.5)))) / z1) ^ 2.0) - -1.0)) * z1);
	else
		tmp = 1.0 / (sqrt((((exp((log((t_2 * abs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1);
	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[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-1.3333333333333333 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(Pi + N[(t$95$1 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 100.0], N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[(N[Abs[z0], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(Pi + Pi), $MachinePrecision] * N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$3 - N[(N[(N[(N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] - N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision] * z2), $MachinePrecision] + t$95$4), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Exp[N[(N[Log[N[(t$95$2 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) < 100

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 64.2%

      \[\leadsto \frac{1}{\sqrt{{\left(\color{blue}{\left(z2 \cdot \left(\left(2 \cdot \pi + z2 \cdot \left(z2 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(0.5 \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right) - -2 \cdot \frac{\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right)\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1} \]

   6. Applied rewrites 68.6%

      \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\frac{z0 \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) + \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) - \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)\right)\right) \cdot z2\right) \cdot z2 + 2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{z1}\right)}}^{2} - -1} \cdot z1} \]

   if 100 < (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)))

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

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

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

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

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

      5. lower-unsound-log.f64 23.4%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 23.4%

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

   5. Applied rewrites 23.4%

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

Alternative 2: 64.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)\\ t_1 := \cos \left(0.5 \cdot \pi\right)\\ t_2 := \sin \left(0.5 \cdot \pi\right)\\ t_3 := -2 \cdot \frac{\pi \cdot {t\_2}^{2}}{{t\_1}^{2}}\\ \mathbf{if}\;t\_0 \leq 100:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\left(z2 \cdot \left(\left(2 \cdot \pi + 2 \cdot \frac{z2 \cdot \left(\pi \cdot \left(t\_2 \cdot \left(2 \cdot \pi - t\_3\right)\right)\right)}{t\_1}\right) - t\_3\right) + \frac{t\_2}{t\_1}\right) \cdot \frac{\left|z0\right|}{z1}\right)}^{2} - -1} \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{e^{\log \left(t\_0 \cdot \left|z0\right|\right) \cdot 2}}{z1}}{z1} - -1} \cdot z1}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (tan (* (- (+ z2 z2) -0.5) PI)))
       (t_1 (cos (* 0.5 PI)))
       (t_2 (sin (* 0.5 PI)))
       (t_3 (* -2.0 (/ (* PI (pow t_2 2.0)) (pow t_1 2.0)))))
  (if (<= t_0 100.0)
    (/
     1.0
     (*
      (sqrt
       (-
        (pow
         (*
          (+
           (*
            z2
            (-
             (+
              (* 2.0 PI)
              (*
               2.0
               (/ (* z2 (* PI (* t_2 (- (* 2.0 PI) t_3)))) t_1)))
             t_3))
           (/ t_2 t_1))
          (/ (fabs z0) z1))
         2.0)
        -1.0))
      z1))
    (/
     1.0
     (*
      (sqrt
       (- (/ (/ (exp (* (log (* t_0 (fabs z0))) 2.0)) z1) z1) -1.0))
      z1)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = tan((((z2 + z2) - -0.5) * ((double) M_PI)));
	double t_1 = cos((0.5 * ((double) M_PI)));
	double t_2 = sin((0.5 * ((double) M_PI)));
	double t_3 = -2.0 * ((((double) M_PI) * pow(t_2, 2.0)) / pow(t_1, 2.0));
	double tmp;
	if (t_0 <= 100.0) {
		tmp = 1.0 / (sqrt((pow((((z2 * (((2.0 * ((double) M_PI)) + (2.0 * ((z2 * (((double) M_PI) * (t_2 * ((2.0 * ((double) M_PI)) - t_3)))) / t_1))) - t_3)) + (t_2 / t_1)) * (fabs(z0) / z1)), 2.0) - -1.0)) * z1);
	} else {
		tmp = 1.0 / (sqrt((((exp((log((t_0 * fabs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1);
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.tan((((z2 + z2) - -0.5) * Math.PI));
	double t_1 = Math.cos((0.5 * Math.PI));
	double t_2 = Math.sin((0.5 * Math.PI));
	double t_3 = -2.0 * ((Math.PI * Math.pow(t_2, 2.0)) / Math.pow(t_1, 2.0));
	double tmp;
	if (t_0 <= 100.0) {
		tmp = 1.0 / (Math.sqrt((Math.pow((((z2 * (((2.0 * Math.PI) + (2.0 * ((z2 * (Math.PI * (t_2 * ((2.0 * Math.PI) - t_3)))) / t_1))) - t_3)) + (t_2 / t_1)) * (Math.abs(z0) / z1)), 2.0) - -1.0)) * z1);
	} else {
		tmp = 1.0 / (Math.sqrt((((Math.exp((Math.log((t_0 * Math.abs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1);
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.tan((((z2 + z2) - -0.5) * math.pi))
	t_1 = math.cos((0.5 * math.pi))
	t_2 = math.sin((0.5 * math.pi))
	t_3 = -2.0 * ((math.pi * math.pow(t_2, 2.0)) / math.pow(t_1, 2.0))
	tmp = 0
	if t_0 <= 100.0:
		tmp = 1.0 / (math.sqrt((math.pow((((z2 * (((2.0 * math.pi) + (2.0 * ((z2 * (math.pi * (t_2 * ((2.0 * math.pi) - t_3)))) / t_1))) - t_3)) + (t_2 / t_1)) * (math.fabs(z0) / z1)), 2.0) - -1.0)) * z1)
	else:
		tmp = 1.0 / (math.sqrt((((math.exp((math.log((t_0 * math.fabs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1)
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi))
	t_1 = cos(Float64(0.5 * pi))
	t_2 = sin(Float64(0.5 * pi))
	t_3 = Float64(-2.0 * Float64(Float64(pi * (t_2 ^ 2.0)) / (t_1 ^ 2.0)))
	tmp = 0.0
	if (t_0 <= 100.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64((Float64(Float64(Float64(z2 * Float64(Float64(Float64(2.0 * pi) + Float64(2.0 * Float64(Float64(z2 * Float64(pi * Float64(t_2 * Float64(Float64(2.0 * pi) - t_3)))) / t_1))) - t_3)) + Float64(t_2 / t_1)) * Float64(abs(z0) / z1)) ^ 2.0) - -1.0)) * z1));
	else
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64(exp(Float64(log(Float64(t_0 * abs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = tan((((z2 + z2) - -0.5) * pi));
	t_1 = cos((0.5 * pi));
	t_2 = sin((0.5 * pi));
	t_3 = -2.0 * ((pi * (t_2 ^ 2.0)) / (t_1 ^ 2.0));
	tmp = 0.0;
	if (t_0 <= 100.0)
		tmp = 1.0 / (sqrt((((((z2 * (((2.0 * pi) + (2.0 * ((z2 * (pi * (t_2 * ((2.0 * pi) - t_3)))) / t_1))) - t_3)) + (t_2 / t_1)) * (abs(z0) / z1)) ^ 2.0) - -1.0)) * z1);
	else
		tmp = 1.0 / (sqrt((((exp((log((t_0 * abs(z0))) * 2.0)) / z1) / z1) - -1.0)) * z1);
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[(N[(Pi * N[Power[t$95$2, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 100.0], N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[(N[(z2 * N[(N[(N[(2.0 * Pi), $MachinePrecision] + N[(2.0 * N[(N[(z2 * N[(Pi * N[(t$95$2 * N[(N[(2.0 * Pi), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[z0], $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Exp[N[(N[Log[N[(t$95$0 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) < 100

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 61.9%

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

   if 100 < (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)))

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

      1. lift-pow.f64 N/A

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

      2. pow-to-exp N/A

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

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

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

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

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

      5. lower-unsound-log.f64 23.4%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 23.4%

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

   5. Applied rewrites 23.4%

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

Alternative 3: 59.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(0.5 + 2 \cdot z2\right)\\ t_1 := {\sin t\_0}^{2}\\ t_2 := \sqrt{\frac{{z0}^{2} \cdot t\_1}{{\left(\cos \left(0.5 \cdot \pi\right) + -2 \cdot \left(z2 \cdot \left(\pi \cdot \sin \left(0.5 \cdot \pi\right)\right)\right)\right)}^{2}}}\\ t_3 := \frac{1}{\sqrt{{\left(\frac{\left(\left(\left(\left({\tan \left(\pi \cdot 0.5\right)}^{2} - -1\right) \cdot \pi\right) \cdot 2\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right)}^{2} - -1} \cdot z1}\\ t_4 := \sqrt{\frac{t\_1}{{z1}^{2} \cdot {\cos t\_0}^{2}}}\\ \mathbf{if}\;z2 \leq -3.6 \cdot 10^{+229}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z2 \leq -8.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\left(z0 \cdot \left(t\_4 + 0.5 \cdot \frac{1}{{z0}^{2} \cdot t\_4}\right)\right) \cdot z1}\\ \mathbf{elif}\;z2 \leq 5 \cdot 10^{+53}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{{\left(z0 \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)\right)}^{2}}{z1}}{z1} - -1} \cdot z1}\\ \mathbf{elif}\;z2 \leq 3 \cdot 10^{+175}:\\ \;\;\;\;\frac{1}{t\_2 + 0.5 \cdot \frac{{z1}^{2}}{t\_2}}\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* PI (+ 0.5 (* 2.0 z2))))
       (t_1 (pow (sin t_0) 2.0))
       (t_2
        (sqrt
         (/
          (* (pow z0 2.0) t_1)
          (pow
           (+
            (cos (* 0.5 PI))
            (* -2.0 (* z2 (* PI (sin (* 0.5 PI))))))
           2.0))))
       (t_3
        (/
         1.0
         (*
          (sqrt
           (-
            (pow
             (/
              (*
               (-
                (*
                 (* (* (- (pow (tan (* PI 0.5)) 2.0) -1.0) PI) 2.0)
                 z2)
                (tan (* PI -0.5)))
               z0)
              z1)
             2.0)
            -1.0))
          z1)))
       (t_4 (sqrt (/ t_1 (* (pow z1 2.0) (pow (cos t_0) 2.0))))))
  (if (<= z2 -3.6e+229)
    t_3
    (if (<= z2 -8.5e+25)
      (/
       1.0
       (* (* z0 (+ t_4 (* 0.5 (/ 1.0 (* (pow z0 2.0) t_4))))) z1))
      (if (<= z2 5e+53)
        (/
         1.0
         (*
          (sqrt
           (-
            (/
             (/ (pow (* z0 (tan (* PI (- (+ z2 z2) -0.5)))) 2.0) z1)
             z1)
            -1.0))
          z1))
        (if (<= z2 3e+175)
          (/ 1.0 (+ t_2 (* 0.5 (/ (pow z1 2.0) t_2))))
          t_3))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((double) M_PI) * (0.5 + (2.0 * z2));
	double t_1 = pow(sin(t_0), 2.0);
	double t_2 = sqrt(((pow(z0, 2.0) * t_1) / pow((cos((0.5 * ((double) M_PI))) + (-2.0 * (z2 * (((double) M_PI) * sin((0.5 * ((double) M_PI))))))), 2.0)));
	double t_3 = 1.0 / (sqrt((pow((((((((pow(tan((((double) M_PI) * 0.5)), 2.0) - -1.0) * ((double) M_PI)) * 2.0) * z2) - tan((((double) M_PI) * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_4 = sqrt((t_1 / (pow(z1, 2.0) * pow(cos(t_0), 2.0))));
	double tmp;
	if (z2 <= -3.6e+229) {
		tmp = t_3;
	} else if (z2 <= -8.5e+25) {
		tmp = 1.0 / ((z0 * (t_4 + (0.5 * (1.0 / (pow(z0, 2.0) * t_4))))) * z1);
	} else if (z2 <= 5e+53) {
		tmp = 1.0 / (sqrt((((pow((z0 * tan((((double) M_PI) * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 3e+175) {
		tmp = 1.0 / (t_2 + (0.5 * (pow(z1, 2.0) / t_2)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.PI * (0.5 + (2.0 * z2));
	double t_1 = Math.pow(Math.sin(t_0), 2.0);
	double t_2 = Math.sqrt(((Math.pow(z0, 2.0) * t_1) / Math.pow((Math.cos((0.5 * Math.PI)) + (-2.0 * (z2 * (Math.PI * Math.sin((0.5 * Math.PI)))))), 2.0)));
	double t_3 = 1.0 / (Math.sqrt((Math.pow((((((((Math.pow(Math.tan((Math.PI * 0.5)), 2.0) - -1.0) * Math.PI) * 2.0) * z2) - Math.tan((Math.PI * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_4 = Math.sqrt((t_1 / (Math.pow(z1, 2.0) * Math.pow(Math.cos(t_0), 2.0))));
	double tmp;
	if (z2 <= -3.6e+229) {
		tmp = t_3;
	} else if (z2 <= -8.5e+25) {
		tmp = 1.0 / ((z0 * (t_4 + (0.5 * (1.0 / (Math.pow(z0, 2.0) * t_4))))) * z1);
	} else if (z2 <= 5e+53) {
		tmp = 1.0 / (Math.sqrt((((Math.pow((z0 * Math.tan((Math.PI * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 3e+175) {
		tmp = 1.0 / (t_2 + (0.5 * (Math.pow(z1, 2.0) / t_2)));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.pi * (0.5 + (2.0 * z2))
	t_1 = math.pow(math.sin(t_0), 2.0)
	t_2 = math.sqrt(((math.pow(z0, 2.0) * t_1) / math.pow((math.cos((0.5 * math.pi)) + (-2.0 * (z2 * (math.pi * math.sin((0.5 * math.pi)))))), 2.0)))
	t_3 = 1.0 / (math.sqrt((math.pow((((((((math.pow(math.tan((math.pi * 0.5)), 2.0) - -1.0) * math.pi) * 2.0) * z2) - math.tan((math.pi * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1)
	t_4 = math.sqrt((t_1 / (math.pow(z1, 2.0) * math.pow(math.cos(t_0), 2.0))))
	tmp = 0
	if z2 <= -3.6e+229:
		tmp = t_3
	elif z2 <= -8.5e+25:
		tmp = 1.0 / ((z0 * (t_4 + (0.5 * (1.0 / (math.pow(z0, 2.0) * t_4))))) * z1)
	elif z2 <= 5e+53:
		tmp = 1.0 / (math.sqrt((((math.pow((z0 * math.tan((math.pi * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1)
	elif z2 <= 3e+175:
		tmp = 1.0 / (t_2 + (0.5 * (math.pow(z1, 2.0) / t_2)))
	else:
		tmp = t_3
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(pi * Float64(0.5 + Float64(2.0 * z2)))
	t_1 = sin(t_0) ^ 2.0
	t_2 = sqrt(Float64(Float64((z0 ^ 2.0) * t_1) / (Float64(cos(Float64(0.5 * pi)) + Float64(-2.0 * Float64(z2 * Float64(pi * sin(Float64(0.5 * pi)))))) ^ 2.0)))
	t_3 = Float64(1.0 / Float64(sqrt(Float64((Float64(Float64(Float64(Float64(Float64(Float64(Float64((tan(Float64(pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan(Float64(pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1))
	t_4 = sqrt(Float64(t_1 / Float64((z1 ^ 2.0) * (cos(t_0) ^ 2.0))))
	tmp = 0.0
	if (z2 <= -3.6e+229)
		tmp = t_3;
	elseif (z2 <= -8.5e+25)
		tmp = Float64(1.0 / Float64(Float64(z0 * Float64(t_4 + Float64(0.5 * Float64(1.0 / Float64((z0 ^ 2.0) * t_4))))) * z1));
	elseif (z2 <= 5e+53)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64((Float64(z0 * tan(Float64(pi * Float64(Float64(z2 + z2) - -0.5)))) ^ 2.0) / z1) / z1) - -1.0)) * z1));
	elseif (z2 <= 3e+175)
		tmp = Float64(1.0 / Float64(t_2 + Float64(0.5 * Float64((z1 ^ 2.0) / t_2))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = pi * (0.5 + (2.0 * z2));
	t_1 = sin(t_0) ^ 2.0;
	t_2 = sqrt((((z0 ^ 2.0) * t_1) / ((cos((0.5 * pi)) + (-2.0 * (z2 * (pi * sin((0.5 * pi)))))) ^ 2.0)));
	t_3 = 1.0 / (sqrt(((((((((((tan((pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan((pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1);
	t_4 = sqrt((t_1 / ((z1 ^ 2.0) * (cos(t_0) ^ 2.0))));
	tmp = 0.0;
	if (z2 <= -3.6e+229)
		tmp = t_3;
	elseif (z2 <= -8.5e+25)
		tmp = 1.0 / ((z0 * (t_4 + (0.5 * (1.0 / ((z0 ^ 2.0) * t_4))))) * z1);
	elseif (z2 <= 5e+53)
		tmp = 1.0 / (sqrt((((((z0 * tan((pi * ((z2 + z2) - -0.5)))) ^ 2.0) / z1) / z1) - -1.0)) * z1);
	elseif (z2 <= 3e+175)
		tmp = 1.0 / (t_2 + (0.5 * ((z1 ^ 2.0) / t_2)));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(Pi * N[(0.5 + N[(2.0 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[(N[Power[z0, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision] / N[Power[N[(N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[(z2 * N[(Pi * N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[Power[N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$1 / N[(N[Power[z1, 2.0], $MachinePrecision] * N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z2, -3.6e+229], t$95$3, If[LessEqual[z2, -8.5e+25], N[(1.0 / N[(N[(z0 * N[(t$95$4 + N[(0.5 * N[(1.0 / N[(N[Power[z0, 2.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 5e+53], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 3e+175], N[(1.0 / N[(t$95$2 + N[(0.5 * N[(N[Power[z1, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -3.5999999999999999e229 or 3.0000000000000002e175 < z2

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 64.2%

      \[\leadsto \frac{1}{\sqrt{{\left(\color{blue}{\left(z2 \cdot \left(\left(2 \cdot \pi + z2 \cdot \left(z2 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(0.5 \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right) - -2 \cdot \frac{\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right)\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1} \]

   6. Applied rewrites 68.6%

      \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\frac{z0 \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) + \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) - \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)\right)\right) \cdot z2\right) \cdot z2 + 2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{z1}\right)}}^{2} - -1} \cdot z1} \]

   7. Taylor expanded in z2 around 0

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

   9. Applied rewrites 55.0%

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

   11. Applied rewrites 55.0%

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

   if -3.5999999999999999e229 < z2 < -8.5000000000000007e25

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 35.2%

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

   if -8.5000000000000007e25 < z2 < 5.0000000000000004e53

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   if 5.0000000000000004e53 < z2 < 3.0000000000000002e175

   1. Initial program 40.7%

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

   2. Taylor expanded in z1 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 30.4%

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

   7. Applied rewrites 30.4%

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

   8. Taylor expanded in z2 around 0

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

      1. lower-+.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-PI.f64 38.0%

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

   10. Applied rewrites 38.0%

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

Alternative 4: 57.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \pi \cdot \left(0.5 + 2 \cdot z2\right)\\ t_1 := \left(\frac{1}{z1} + z1\right) \cdot 0.5\\ t_2 := \frac{1}{\sqrt{{\left(\frac{\left(\left(\left(\left({\tan \left(\pi \cdot 0.5\right)}^{2} - -1\right) \cdot \pi\right) \cdot 2\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right)}^{2} - -1} \cdot z1}\\ t_3 := \sqrt{\frac{{\sin t\_0}^{2}}{{z1}^{2} \cdot {\cos t\_0}^{2}}}\\ \mathbf{if}\;z2 \leq -3.6 \cdot 10^{+229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z2 \leq -8.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\left(z0 \cdot \left(t\_3 + 0.5 \cdot \frac{1}{{z0}^{2} \cdot t\_3}\right)\right) \cdot z1}\\ \mathbf{elif}\;z2 \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{{\left(z0 \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)\right)}^{2}}{z1}}{z1} - -1} \cdot z1}\\ \mathbf{elif}\;z2 \leq 1.55 \cdot 10^{+175}:\\ \;\;\;\;\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{t\_1}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* PI (+ 0.5 (* 2.0 z2))))
       (t_1 (* (+ (/ 1.0 z1) z1) 0.5))
       (t_2
        (/
         1.0
         (*
          (sqrt
           (-
            (pow
             (/
              (*
               (-
                (*
                 (* (* (- (pow (tan (* PI 0.5)) 2.0) -1.0) PI) 2.0)
                 z2)
                (tan (* PI -0.5)))
               z0)
              z1)
             2.0)
            -1.0))
          z1)))
       (t_3
        (sqrt
         (/
          (pow (sin t_0) 2.0)
          (* (pow z1 2.0) (pow (cos t_0) 2.0))))))
  (if (<= z2 -3.6e+229)
    t_2
    (if (<= z2 -8.5e+25)
      (/
       1.0
       (* (* z0 (+ t_3 (* 0.5 (/ 1.0 (* (pow z0 2.0) t_3))))) z1))
      (if (<= z2 2.1e+51)
        (/
         1.0
         (*
          (sqrt
           (-
            (/
             (/ (pow (* z0 (tan (* PI (- (+ z2 z2) -0.5)))) 2.0) z1)
             z1)
            -1.0))
          z1))
        (if (<= z2 1.55e+175)
          (* (+ 1.0 (/ (* (- (/ 1.0 z1) z1) 0.5) t_1)) t_1)
          t_2))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((double) M_PI) * (0.5 + (2.0 * z2));
	double t_1 = ((1.0 / z1) + z1) * 0.5;
	double t_2 = 1.0 / (sqrt((pow((((((((pow(tan((((double) M_PI) * 0.5)), 2.0) - -1.0) * ((double) M_PI)) * 2.0) * z2) - tan((((double) M_PI) * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_3 = sqrt((pow(sin(t_0), 2.0) / (pow(z1, 2.0) * pow(cos(t_0), 2.0))));
	double tmp;
	if (z2 <= -3.6e+229) {
		tmp = t_2;
	} else if (z2 <= -8.5e+25) {
		tmp = 1.0 / ((z0 * (t_3 + (0.5 * (1.0 / (pow(z0, 2.0) * t_3))))) * z1);
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (sqrt((((pow((z0 * tan((((double) M_PI) * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_1)) * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.PI * (0.5 + (2.0 * z2));
	double t_1 = ((1.0 / z1) + z1) * 0.5;
	double t_2 = 1.0 / (Math.sqrt((Math.pow((((((((Math.pow(Math.tan((Math.PI * 0.5)), 2.0) - -1.0) * Math.PI) * 2.0) * z2) - Math.tan((Math.PI * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_3 = Math.sqrt((Math.pow(Math.sin(t_0), 2.0) / (Math.pow(z1, 2.0) * Math.pow(Math.cos(t_0), 2.0))));
	double tmp;
	if (z2 <= -3.6e+229) {
		tmp = t_2;
	} else if (z2 <= -8.5e+25) {
		tmp = 1.0 / ((z0 * (t_3 + (0.5 * (1.0 / (Math.pow(z0, 2.0) * t_3))))) * z1);
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (Math.sqrt((((Math.pow((z0 * Math.tan((Math.PI * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_1)) * t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.pi * (0.5 + (2.0 * z2))
	t_1 = ((1.0 / z1) + z1) * 0.5
	t_2 = 1.0 / (math.sqrt((math.pow((((((((math.pow(math.tan((math.pi * 0.5)), 2.0) - -1.0) * math.pi) * 2.0) * z2) - math.tan((math.pi * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1)
	t_3 = math.sqrt((math.pow(math.sin(t_0), 2.0) / (math.pow(z1, 2.0) * math.pow(math.cos(t_0), 2.0))))
	tmp = 0
	if z2 <= -3.6e+229:
		tmp = t_2
	elif z2 <= -8.5e+25:
		tmp = 1.0 / ((z0 * (t_3 + (0.5 * (1.0 / (math.pow(z0, 2.0) * t_3))))) * z1)
	elif z2 <= 2.1e+51:
		tmp = 1.0 / (math.sqrt((((math.pow((z0 * math.tan((math.pi * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1)
	elif z2 <= 1.55e+175:
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_1)) * t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(pi * Float64(0.5 + Float64(2.0 * z2)))
	t_1 = Float64(Float64(Float64(1.0 / z1) + z1) * 0.5)
	t_2 = Float64(1.0 / Float64(sqrt(Float64((Float64(Float64(Float64(Float64(Float64(Float64(Float64((tan(Float64(pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan(Float64(pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1))
	t_3 = sqrt(Float64((sin(t_0) ^ 2.0) / Float64((z1 ^ 2.0) * (cos(t_0) ^ 2.0))))
	tmp = 0.0
	if (z2 <= -3.6e+229)
		tmp = t_2;
	elseif (z2 <= -8.5e+25)
		tmp = Float64(1.0 / Float64(Float64(z0 * Float64(t_3 + Float64(0.5 * Float64(1.0 / Float64((z0 ^ 2.0) * t_3))))) * z1));
	elseif (z2 <= 2.1e+51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64((Float64(z0 * tan(Float64(pi * Float64(Float64(z2 + z2) - -0.5)))) ^ 2.0) / z1) / z1) - -1.0)) * z1));
	elseif (z2 <= 1.55e+175)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(1.0 / z1) - z1) * 0.5) / t_1)) * t_1);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = pi * (0.5 + (2.0 * z2));
	t_1 = ((1.0 / z1) + z1) * 0.5;
	t_2 = 1.0 / (sqrt(((((((((((tan((pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan((pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1);
	t_3 = sqrt(((sin(t_0) ^ 2.0) / ((z1 ^ 2.0) * (cos(t_0) ^ 2.0))));
	tmp = 0.0;
	if (z2 <= -3.6e+229)
		tmp = t_2;
	elseif (z2 <= -8.5e+25)
		tmp = 1.0 / ((z0 * (t_3 + (0.5 * (1.0 / ((z0 ^ 2.0) * t_3))))) * z1);
	elseif (z2 <= 2.1e+51)
		tmp = 1.0 / (sqrt((((((z0 * tan((pi * ((z2 + z2) - -0.5)))) ^ 2.0) / z1) / z1) - -1.0)) * z1);
	elseif (z2 <= 1.55e+175)
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_1)) * t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(Pi * N[(0.5 + N[(2.0 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[Power[N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[z1, 2.0], $MachinePrecision] * N[Power[N[Cos[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z2, -3.6e+229], t$95$2, If[LessEqual[z2, -8.5e+25], N[(1.0 / N[(N[(z0 * N[(t$95$3 + N[(0.5 * N[(1.0 / N[(N[Power[z0, 2.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 2.1e+51], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 1.55e+175], N[(N[(1.0 + N[(N[(N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision] * 0.5), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -3.5999999999999999e229 or 1.5499999999999999e175 < z2

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 64.2%

      \[\leadsto \frac{1}{\sqrt{{\left(\color{blue}{\left(z2 \cdot \left(\left(2 \cdot \pi + z2 \cdot \left(z2 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(0.5 \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right) - -2 \cdot \frac{\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right)\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1} \]

   6. Applied rewrites 68.6%

      \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\frac{z0 \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) + \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) - \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)\right)\right) \cdot z2\right) \cdot z2 + 2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{z1}\right)}}^{2} - -1} \cdot z1} \]

   7. Taylor expanded in z2 around 0

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

   9. Applied rewrites 55.0%

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

   11. Applied rewrites 55.0%

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

   if -3.5999999999999999e229 < z2 < -8.5000000000000007e25

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 35.2%

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

   if -8.5000000000000007e25 < z2 < 2.1000000000000001e51

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   if 2.1000000000000001e51 < z2 < 1.5499999999999999e175

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{\left(\frac{1}{z1} + z1\right) \cdot 0.5}\right) \cdot \left(\left(\frac{1}{z1} + z1\right) \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 57.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{{\left(\frac{\left(\left(\left(\left({\tan \left(\pi \cdot 0.5\right)}^{2} - -1\right) \cdot \pi\right) \cdot 2\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right)}^{2} - -1} \cdot z1}\\ t_1 := \pi \cdot \left(0.5 + 2 \cdot z2\right)\\ t_2 := \sqrt{\frac{{\sin t\_1}^{2}}{{z1}^{2} \cdot {\cos t\_1}^{2}}}\\ t_3 := \left(\frac{1}{z1} + z1\right) \cdot 0.5\\ \mathbf{if}\;z2 \leq -3.6 \cdot 10^{+229}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq -1.05 \cdot 10^{+26}:\\ \;\;\;\;\frac{1}{z0 \cdot \left(0.5 \cdot \frac{z1}{{z0}^{2} \cdot t\_2} + z1 \cdot t\_2\right)}\\ \mathbf{elif}\;z2 \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{{\left(z0 \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)\right)}^{2}}{z1}}{z1} - -1} \cdot z1}\\ \mathbf{elif}\;z2 \leq 1.55 \cdot 10^{+175}:\\ \;\;\;\;\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{t\_3}\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (*
          (sqrt
           (-
            (pow
             (/
              (*
               (-
                (*
                 (* (* (- (pow (tan (* PI 0.5)) 2.0) -1.0) PI) 2.0)
                 z2)
                (tan (* PI -0.5)))
               z0)
              z1)
             2.0)
            -1.0))
          z1)))
       (t_1 (* PI (+ 0.5 (* 2.0 z2))))
       (t_2
        (sqrt
         (/
          (pow (sin t_1) 2.0)
          (* (pow z1 2.0) (pow (cos t_1) 2.0)))))
       (t_3 (* (+ (/ 1.0 z1) z1) 0.5)))
  (if (<= z2 -3.6e+229)
    t_0
    (if (<= z2 -1.05e+26)
      (/
       1.0
       (* z0 (+ (* 0.5 (/ z1 (* (pow z0 2.0) t_2))) (* z1 t_2))))
      (if (<= z2 2.1e+51)
        (/
         1.0
         (*
          (sqrt
           (-
            (/
             (/ (pow (* z0 (tan (* PI (- (+ z2 z2) -0.5)))) 2.0) z1)
             z1)
            -1.0))
          z1))
        (if (<= z2 1.55e+175)
          (* (+ 1.0 (/ (* (- (/ 1.0 z1) z1) 0.5) t_3)) t_3)
          t_0))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / (sqrt((pow((((((((pow(tan((((double) M_PI) * 0.5)), 2.0) - -1.0) * ((double) M_PI)) * 2.0) * z2) - tan((((double) M_PI) * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_1 = ((double) M_PI) * (0.5 + (2.0 * z2));
	double t_2 = sqrt((pow(sin(t_1), 2.0) / (pow(z1, 2.0) * pow(cos(t_1), 2.0))));
	double t_3 = ((1.0 / z1) + z1) * 0.5;
	double tmp;
	if (z2 <= -3.6e+229) {
		tmp = t_0;
	} else if (z2 <= -1.05e+26) {
		tmp = 1.0 / (z0 * ((0.5 * (z1 / (pow(z0, 2.0) * t_2))) + (z1 * t_2)));
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (sqrt((((pow((z0 * tan((((double) M_PI) * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_3)) * t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / (Math.sqrt((Math.pow((((((((Math.pow(Math.tan((Math.PI * 0.5)), 2.0) - -1.0) * Math.PI) * 2.0) * z2) - Math.tan((Math.PI * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_1 = Math.PI * (0.5 + (2.0 * z2));
	double t_2 = Math.sqrt((Math.pow(Math.sin(t_1), 2.0) / (Math.pow(z1, 2.0) * Math.pow(Math.cos(t_1), 2.0))));
	double t_3 = ((1.0 / z1) + z1) * 0.5;
	double tmp;
	if (z2 <= -3.6e+229) {
		tmp = t_0;
	} else if (z2 <= -1.05e+26) {
		tmp = 1.0 / (z0 * ((0.5 * (z1 / (Math.pow(z0, 2.0) * t_2))) + (z1 * t_2)));
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (Math.sqrt((((Math.pow((z0 * Math.tan((Math.PI * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_3)) * t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = 1.0 / (math.sqrt((math.pow((((((((math.pow(math.tan((math.pi * 0.5)), 2.0) - -1.0) * math.pi) * 2.0) * z2) - math.tan((math.pi * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1)
	t_1 = math.pi * (0.5 + (2.0 * z2))
	t_2 = math.sqrt((math.pow(math.sin(t_1), 2.0) / (math.pow(z1, 2.0) * math.pow(math.cos(t_1), 2.0))))
	t_3 = ((1.0 / z1) + z1) * 0.5
	tmp = 0
	if z2 <= -3.6e+229:
		tmp = t_0
	elif z2 <= -1.05e+26:
		tmp = 1.0 / (z0 * ((0.5 * (z1 / (math.pow(z0, 2.0) * t_2))) + (z1 * t_2)))
	elif z2 <= 2.1e+51:
		tmp = 1.0 / (math.sqrt((((math.pow((z0 * math.tan((math.pi * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1)
	elif z2 <= 1.55e+175:
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_3)) * t_3
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(1.0 / Float64(sqrt(Float64((Float64(Float64(Float64(Float64(Float64(Float64(Float64((tan(Float64(pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan(Float64(pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1))
	t_1 = Float64(pi * Float64(0.5 + Float64(2.0 * z2)))
	t_2 = sqrt(Float64((sin(t_1) ^ 2.0) / Float64((z1 ^ 2.0) * (cos(t_1) ^ 2.0))))
	t_3 = Float64(Float64(Float64(1.0 / z1) + z1) * 0.5)
	tmp = 0.0
	if (z2 <= -3.6e+229)
		tmp = t_0;
	elseif (z2 <= -1.05e+26)
		tmp = Float64(1.0 / Float64(z0 * Float64(Float64(0.5 * Float64(z1 / Float64((z0 ^ 2.0) * t_2))) + Float64(z1 * t_2))));
	elseif (z2 <= 2.1e+51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64((Float64(z0 * tan(Float64(pi * Float64(Float64(z2 + z2) - -0.5)))) ^ 2.0) / z1) / z1) - -1.0)) * z1));
	elseif (z2 <= 1.55e+175)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(1.0 / z1) - z1) * 0.5) / t_3)) * t_3);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = 1.0 / (sqrt(((((((((((tan((pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan((pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1);
	t_1 = pi * (0.5 + (2.0 * z2));
	t_2 = sqrt(((sin(t_1) ^ 2.0) / ((z1 ^ 2.0) * (cos(t_1) ^ 2.0))));
	t_3 = ((1.0 / z1) + z1) * 0.5;
	tmp = 0.0;
	if (z2 <= -3.6e+229)
		tmp = t_0;
	elseif (z2 <= -1.05e+26)
		tmp = 1.0 / (z0 * ((0.5 * (z1 / ((z0 ^ 2.0) * t_2))) + (z1 * t_2)));
	elseif (z2 <= 2.1e+51)
		tmp = 1.0 / (sqrt((((((z0 * tan((pi * ((z2 + z2) - -0.5)))) ^ 2.0) / z1) / z1) - -1.0)) * z1);
	elseif (z2 <= 1.55e+175)
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_3)) * t_3;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[Power[N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(Pi * N[(0.5 + N[(2.0 * z2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[z1, 2.0], $MachinePrecision] * N[Power[N[Cos[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[z2, -3.6e+229], t$95$0, If[LessEqual[z2, -1.05e+26], N[(1.0 / N[(z0 * N[(N[(0.5 * N[(z1 / N[(N[Power[z0, 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 2.1e+51], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 1.55e+175], N[(N[(1.0 + N[(N[(N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision] * 0.5), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -3.5999999999999999e229 or 1.5499999999999999e175 < z2

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 64.2%

      \[\leadsto \frac{1}{\sqrt{{\left(\color{blue}{\left(z2 \cdot \left(\left(2 \cdot \pi + z2 \cdot \left(z2 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(0.5 \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right) - -2 \cdot \frac{\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right)\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1} \]

   6. Applied rewrites 68.6%

      \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\frac{z0 \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) + \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) - \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)\right)\right) \cdot z2\right) \cdot z2 + 2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{z1}\right)}}^{2} - -1} \cdot z1} \]

   7. Taylor expanded in z2 around 0

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

   9. Applied rewrites 55.0%

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

   11. Applied rewrites 55.0%

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

   if -3.5999999999999999e229 < z2 < -1.05e26

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 33.4%

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

   if -1.05e26 < z2 < 2.1000000000000001e51

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   if 2.1000000000000001e51 < z2 < 1.5499999999999999e175

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{\left(\frac{1}{z1} + z1\right) \cdot 0.5}\right) \cdot \left(\left(\frac{1}{z1} + z1\right) \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 56.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{{\left(\frac{\left(\left(\left(\left({\tan \left(\pi \cdot 0.5\right)}^{2} - -1\right) \cdot \pi\right) \cdot 2\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right)}^{2} - -1} \cdot z1}\\ t_1 := \left(\frac{1}{z1} + z1\right) \cdot 0.5\\ t_2 := \left(z2 + z2\right) - -0.5\\ t_3 := \tan \left(t\_2 \cdot \pi\right)\\ \mathbf{if}\;z2 \leq -2.7 \cdot 10^{+271}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq -1.25 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{e^{\log \left(\left(t\_3 \cdot t\_3\right) \cdot \left(z0 \cdot z0\right)\right) \cdot 0.5} + 0.5 \cdot \left(z1 \cdot \frac{z1}{\sqrt{{\left(t\_3 \cdot z0\right)}^{2}}}\right)}\\ \mathbf{elif}\;z2 \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{{\left(z0 \cdot \tan \left(\pi \cdot t\_2\right)\right)}^{2}}{z1}}{z1} - -1} \cdot z1}\\ \mathbf{elif}\;z2 \leq 1.55 \cdot 10^{+175}:\\ \;\;\;\;\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{t\_1}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (*
          (sqrt
           (-
            (pow
             (/
              (*
               (-
                (*
                 (* (* (- (pow (tan (* PI 0.5)) 2.0) -1.0) PI) 2.0)
                 z2)
                (tan (* PI -0.5)))
               z0)
              z1)
             2.0)
            -1.0))
          z1)))
       (t_1 (* (+ (/ 1.0 z1) z1) 0.5))
       (t_2 (- (+ z2 z2) -0.5))
       (t_3 (tan (* t_2 PI))))
  (if (<= z2 -2.7e+271)
    t_0
    (if (<= z2 -1.25e+138)
      (/
       1.0
       (+
        (exp (* (log (* (* t_3 t_3) (* z0 z0))) 0.5))
        (* 0.5 (* z1 (/ z1 (sqrt (pow (* t_3 z0) 2.0)))))))
      (if (<= z2 2.1e+51)
        (/
         1.0
         (*
          (sqrt
           (- (/ (/ (pow (* z0 (tan (* PI t_2))) 2.0) z1) z1) -1.0))
          z1))
        (if (<= z2 1.55e+175)
          (* (+ 1.0 (/ (* (- (/ 1.0 z1) z1) 0.5) t_1)) t_1)
          t_0))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / (sqrt((pow((((((((pow(tan((((double) M_PI) * 0.5)), 2.0) - -1.0) * ((double) M_PI)) * 2.0) * z2) - tan((((double) M_PI) * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_1 = ((1.0 / z1) + z1) * 0.5;
	double t_2 = (z2 + z2) - -0.5;
	double t_3 = tan((t_2 * ((double) M_PI)));
	double tmp;
	if (z2 <= -2.7e+271) {
		tmp = t_0;
	} else if (z2 <= -1.25e+138) {
		tmp = 1.0 / (exp((log(((t_3 * t_3) * (z0 * z0))) * 0.5)) + (0.5 * (z1 * (z1 / sqrt(pow((t_3 * z0), 2.0))))));
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (sqrt((((pow((z0 * tan((((double) M_PI) * t_2))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_1)) * t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / (Math.sqrt((Math.pow((((((((Math.pow(Math.tan((Math.PI * 0.5)), 2.0) - -1.0) * Math.PI) * 2.0) * z2) - Math.tan((Math.PI * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_1 = ((1.0 / z1) + z1) * 0.5;
	double t_2 = (z2 + z2) - -0.5;
	double t_3 = Math.tan((t_2 * Math.PI));
	double tmp;
	if (z2 <= -2.7e+271) {
		tmp = t_0;
	} else if (z2 <= -1.25e+138) {
		tmp = 1.0 / (Math.exp((Math.log(((t_3 * t_3) * (z0 * z0))) * 0.5)) + (0.5 * (z1 * (z1 / Math.sqrt(Math.pow((t_3 * z0), 2.0))))));
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (Math.sqrt((((Math.pow((z0 * Math.tan((Math.PI * t_2))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_1)) * t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = 1.0 / (math.sqrt((math.pow((((((((math.pow(math.tan((math.pi * 0.5)), 2.0) - -1.0) * math.pi) * 2.0) * z2) - math.tan((math.pi * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1)
	t_1 = ((1.0 / z1) + z1) * 0.5
	t_2 = (z2 + z2) - -0.5
	t_3 = math.tan((t_2 * math.pi))
	tmp = 0
	if z2 <= -2.7e+271:
		tmp = t_0
	elif z2 <= -1.25e+138:
		tmp = 1.0 / (math.exp((math.log(((t_3 * t_3) * (z0 * z0))) * 0.5)) + (0.5 * (z1 * (z1 / math.sqrt(math.pow((t_3 * z0), 2.0))))))
	elif z2 <= 2.1e+51:
		tmp = 1.0 / (math.sqrt((((math.pow((z0 * math.tan((math.pi * t_2))), 2.0) / z1) / z1) - -1.0)) * z1)
	elif z2 <= 1.55e+175:
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_1)) * t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(1.0 / Float64(sqrt(Float64((Float64(Float64(Float64(Float64(Float64(Float64(Float64((tan(Float64(pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan(Float64(pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1))
	t_1 = Float64(Float64(Float64(1.0 / z1) + z1) * 0.5)
	t_2 = Float64(Float64(z2 + z2) - -0.5)
	t_3 = tan(Float64(t_2 * pi))
	tmp = 0.0
	if (z2 <= -2.7e+271)
		tmp = t_0;
	elseif (z2 <= -1.25e+138)
		tmp = Float64(1.0 / Float64(exp(Float64(log(Float64(Float64(t_3 * t_3) * Float64(z0 * z0))) * 0.5)) + Float64(0.5 * Float64(z1 * Float64(z1 / sqrt((Float64(t_3 * z0) ^ 2.0)))))));
	elseif (z2 <= 2.1e+51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64((Float64(z0 * tan(Float64(pi * t_2))) ^ 2.0) / z1) / z1) - -1.0)) * z1));
	elseif (z2 <= 1.55e+175)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(1.0 / z1) - z1) * 0.5) / t_1)) * t_1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = 1.0 / (sqrt(((((((((((tan((pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan((pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1);
	t_1 = ((1.0 / z1) + z1) * 0.5;
	t_2 = (z2 + z2) - -0.5;
	t_3 = tan((t_2 * pi));
	tmp = 0.0;
	if (z2 <= -2.7e+271)
		tmp = t_0;
	elseif (z2 <= -1.25e+138)
		tmp = 1.0 / (exp((log(((t_3 * t_3) * (z0 * z0))) * 0.5)) + (0.5 * (z1 * (z1 / sqrt(((t_3 * z0) ^ 2.0))))));
	elseif (z2 <= 2.1e+51)
		tmp = 1.0 / (sqrt((((((z0 * tan((pi * t_2))) ^ 2.0) / z1) / z1) - -1.0)) * z1);
	elseif (z2 <= 1.55e+175)
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_1)) * t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[Power[N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$3 = N[Tan[N[(t$95$2 * Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z2, -2.7e+271], t$95$0, If[LessEqual[z2, -1.25e+138], N[(1.0 / N[(N[Exp[N[(N[Log[N[(N[(t$95$3 * t$95$3), $MachinePrecision] * N[(z0 * z0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(z1 * N[(z1 / N[Sqrt[N[Power[N[(t$95$3 * z0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 2.1e+51], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 1.55e+175], N[(N[(1.0 + N[(N[(N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision] * 0.5), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -2.6999999999999999e271 or 1.5499999999999999e175 < z2

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 64.2%

      \[\leadsto \frac{1}{\sqrt{{\left(\color{blue}{\left(z2 \cdot \left(\left(2 \cdot \pi + z2 \cdot \left(z2 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(0.5 \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right) - -2 \cdot \frac{\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right)\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1} \]

   6. Applied rewrites 68.6%

      \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\frac{z0 \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) + \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) - \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)\right)\right) \cdot z2\right) \cdot z2 + 2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{z1}\right)}}^{2} - -1} \cdot z1} \]

   7. Taylor expanded in z2 around 0

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

   9. Applied rewrites 55.0%

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

   11. Applied rewrites 55.0%

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

   if -2.6999999999999999e271 < z2 < -1.25e138

   1. Initial program 40.7%

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

   2. Taylor expanded in z1 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 35.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

   6. Applied rewrites 35.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

   8. Applied rewrites 35.8%

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

      1. lift-/.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. unpow2 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 43.1%

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

      7. lift-exp.f64 N/A

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

      8. lift-*.f64 N/A

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

   10. Applied rewrites 43.1%

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

   if -1.25e138 < z2 < 2.1000000000000001e51

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   if 2.1000000000000001e51 < z2 < 1.5499999999999999e175

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{\left(\frac{1}{z1} + z1\right) \cdot 0.5}\right) \cdot \left(\left(\frac{1}{z1} + z1\right) \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sqrt{{\left(\frac{\left(\left(\left(\left({\tan \left(\pi \cdot 0.5\right)}^{2} - -1\right) \cdot \pi\right) \cdot 2\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right)}^{2} - -1} \cdot z1}\\ t_1 := \left(z2 + z2\right) - -0.5\\ t_2 := \sqrt{{\left(\tan \left(t\_1 \cdot \pi\right) \cdot z0\right)}^{2}}\\ t_3 := \left(\frac{1}{z1} + z1\right) \cdot 0.5\\ \mathbf{if}\;z2 \leq -2.7 \cdot 10^{+271}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq -1.25 \cdot 10^{+138}:\\ \;\;\;\;\frac{1}{t\_2 - \frac{-0.5 \cdot \left(z1 \cdot z1\right)}{t\_2}}\\ \mathbf{elif}\;z2 \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{{\left(z0 \cdot \tan \left(\pi \cdot t\_1\right)\right)}^{2}}{z1}}{z1} - -1} \cdot z1}\\ \mathbf{elif}\;z2 \leq 1.55 \cdot 10^{+175}:\\ \;\;\;\;\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{t\_3}\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0
        (/
         1.0
         (*
          (sqrt
           (-
            (pow
             (/
              (*
               (-
                (*
                 (* (* (- (pow (tan (* PI 0.5)) 2.0) -1.0) PI) 2.0)
                 z2)
                (tan (* PI -0.5)))
               z0)
              z1)
             2.0)
            -1.0))
          z1)))
       (t_1 (- (+ z2 z2) -0.5))
       (t_2 (sqrt (pow (* (tan (* t_1 PI)) z0) 2.0)))
       (t_3 (* (+ (/ 1.0 z1) z1) 0.5)))
  (if (<= z2 -2.7e+271)
    t_0
    (if (<= z2 -1.25e+138)
      (/ 1.0 (- t_2 (/ (* -0.5 (* z1 z1)) t_2)))
      (if (<= z2 2.1e+51)
        (/
         1.0
         (*
          (sqrt
           (- (/ (/ (pow (* z0 (tan (* PI t_1))) 2.0) z1) z1) -1.0))
          z1))
        (if (<= z2 1.55e+175)
          (* (+ 1.0 (/ (* (- (/ 1.0 z1) z1) 0.5) t_3)) t_3)
          t_0))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / (sqrt((pow((((((((pow(tan((((double) M_PI) * 0.5)), 2.0) - -1.0) * ((double) M_PI)) * 2.0) * z2) - tan((((double) M_PI) * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_1 = (z2 + z2) - -0.5;
	double t_2 = sqrt(pow((tan((t_1 * ((double) M_PI))) * z0), 2.0));
	double t_3 = ((1.0 / z1) + z1) * 0.5;
	double tmp;
	if (z2 <= -2.7e+271) {
		tmp = t_0;
	} else if (z2 <= -1.25e+138) {
		tmp = 1.0 / (t_2 - ((-0.5 * (z1 * z1)) / t_2));
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (sqrt((((pow((z0 * tan((((double) M_PI) * t_1))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_3)) * t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / (Math.sqrt((Math.pow((((((((Math.pow(Math.tan((Math.PI * 0.5)), 2.0) - -1.0) * Math.PI) * 2.0) * z2) - Math.tan((Math.PI * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_1 = (z2 + z2) - -0.5;
	double t_2 = Math.sqrt(Math.pow((Math.tan((t_1 * Math.PI)) * z0), 2.0));
	double t_3 = ((1.0 / z1) + z1) * 0.5;
	double tmp;
	if (z2 <= -2.7e+271) {
		tmp = t_0;
	} else if (z2 <= -1.25e+138) {
		tmp = 1.0 / (t_2 - ((-0.5 * (z1 * z1)) / t_2));
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (Math.sqrt((((Math.pow((z0 * Math.tan((Math.PI * t_1))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_3)) * t_3;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = 1.0 / (math.sqrt((math.pow((((((((math.pow(math.tan((math.pi * 0.5)), 2.0) - -1.0) * math.pi) * 2.0) * z2) - math.tan((math.pi * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1)
	t_1 = (z2 + z2) - -0.5
	t_2 = math.sqrt(math.pow((math.tan((t_1 * math.pi)) * z0), 2.0))
	t_3 = ((1.0 / z1) + z1) * 0.5
	tmp = 0
	if z2 <= -2.7e+271:
		tmp = t_0
	elif z2 <= -1.25e+138:
		tmp = 1.0 / (t_2 - ((-0.5 * (z1 * z1)) / t_2))
	elif z2 <= 2.1e+51:
		tmp = 1.0 / (math.sqrt((((math.pow((z0 * math.tan((math.pi * t_1))), 2.0) / z1) / z1) - -1.0)) * z1)
	elif z2 <= 1.55e+175:
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_3)) * t_3
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(1.0 / Float64(sqrt(Float64((Float64(Float64(Float64(Float64(Float64(Float64(Float64((tan(Float64(pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan(Float64(pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1))
	t_1 = Float64(Float64(z2 + z2) - -0.5)
	t_2 = sqrt((Float64(tan(Float64(t_1 * pi)) * z0) ^ 2.0))
	t_3 = Float64(Float64(Float64(1.0 / z1) + z1) * 0.5)
	tmp = 0.0
	if (z2 <= -2.7e+271)
		tmp = t_0;
	elseif (z2 <= -1.25e+138)
		tmp = Float64(1.0 / Float64(t_2 - Float64(Float64(-0.5 * Float64(z1 * z1)) / t_2)));
	elseif (z2 <= 2.1e+51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64((Float64(z0 * tan(Float64(pi * t_1))) ^ 2.0) / z1) / z1) - -1.0)) * z1));
	elseif (z2 <= 1.55e+175)
		tmp = Float64(Float64(1.0 + Float64(Float64(Float64(Float64(1.0 / z1) - z1) * 0.5) / t_3)) * t_3);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = 1.0 / (sqrt(((((((((((tan((pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan((pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1);
	t_1 = (z2 + z2) - -0.5;
	t_2 = sqrt(((tan((t_1 * pi)) * z0) ^ 2.0));
	t_3 = ((1.0 / z1) + z1) * 0.5;
	tmp = 0.0;
	if (z2 <= -2.7e+271)
		tmp = t_0;
	elseif (z2 <= -1.25e+138)
		tmp = 1.0 / (t_2 - ((-0.5 * (z1 * z1)) / t_2));
	elseif (z2 <= 2.1e+51)
		tmp = 1.0 / (sqrt((((((z0 * tan((pi * t_1))) ^ 2.0) / z1) / z1) - -1.0)) * z1);
	elseif (z2 <= 1.55e+175)
		tmp = (1.0 + ((((1.0 / z1) - z1) * 0.5) / t_3)) * t_3;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[Power[N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[Power[N[(N[Tan[N[(t$95$1 * Pi), $MachinePrecision]], $MachinePrecision] * z0), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[z2, -2.7e+271], t$95$0, If[LessEqual[z2, -1.25e+138], N[(1.0 / N[(t$95$2 - N[(N[(-0.5 * N[(z1 * z1), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 2.1e+51], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 1.55e+175], N[(N[(1.0 + N[(N[(N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision] * 0.5), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], t$95$0]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -2.6999999999999999e271 or 1.5499999999999999e175 < z2

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 64.2%

      \[\leadsto \frac{1}{\sqrt{{\left(\color{blue}{\left(z2 \cdot \left(\left(2 \cdot \pi + z2 \cdot \left(z2 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(0.5 \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right) - -2 \cdot \frac{\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right)\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1} \]

   6. Applied rewrites 68.6%

      \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\frac{z0 \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) + \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) - \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)\right)\right) \cdot z2\right) \cdot z2 + 2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{z1}\right)}}^{2} - -1} \cdot z1} \]

   7. Taylor expanded in z2 around 0

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

   9. Applied rewrites 55.0%

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

   11. Applied rewrites 55.0%

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

   if -2.6999999999999999e271 < z2 < -1.25e138

   1. Initial program 40.7%

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

   2. Taylor expanded in z1 around 0

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

      1. lower-+.f64 N/A

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

   4. Applied rewrites 35.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

   6. Applied rewrites 35.8%

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

      1. lift-sqrt.f64 N/A

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

      2. pow1/2 N/A

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

      3. pow-to-exp N/A

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

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

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

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

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

   8. Applied rewrites 35.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

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

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

   10. Applied rewrites 35.8%

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

   if -1.25e138 < z2 < 2.1000000000000001e51

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   if 2.1000000000000001e51 < z2 < 1.5499999999999999e175

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{\left(\frac{1}{z1} + z1\right) \cdot 0.5}\right) \cdot \left(\left(\frac{1}{z1} + z1\right) \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 55.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{1}{z1} + z1\right) \cdot 0.5\\ t_1 := \frac{1}{\sqrt{{\left(\frac{\left(\left(\left(\left({\tan \left(\pi \cdot 0.5\right)}^{2} - -1\right) \cdot \pi\right) \cdot 2\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right) \cdot z0}{z1}\right)}^{2} - -1} \cdot z1}\\ t_2 := \frac{1}{z1} - z1\\ \mathbf{if}\;z2 \leq -1.12 \cdot 10^{+235}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z2 \leq -9 \cdot 10^{+148}:\\ \;\;\;\;t\_0 - t\_2 \cdot -0.5\\ \mathbf{elif}\;z2 \leq 2.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{{\left(z0 \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)\right)}^{2}}{z1}}{z1} - -1} \cdot z1}\\ \mathbf{elif}\;z2 \leq 1.55 \cdot 10^{+175}:\\ \;\;\;\;\left(1 + \frac{t\_2 \cdot 0.5}{t\_0}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (+ (/ 1.0 z1) z1) 0.5))
       (t_1
        (/
         1.0
         (*
          (sqrt
           (-
            (pow
             (/
              (*
               (-
                (*
                 (* (* (- (pow (tan (* PI 0.5)) 2.0) -1.0) PI) 2.0)
                 z2)
                (tan (* PI -0.5)))
               z0)
              z1)
             2.0)
            -1.0))
          z1)))
       (t_2 (- (/ 1.0 z1) z1)))
  (if (<= z2 -1.12e+235)
    t_1
    (if (<= z2 -9e+148)
      (- t_0 (* t_2 -0.5))
      (if (<= z2 2.1e+51)
        (/
         1.0
         (*
          (sqrt
           (-
            (/
             (/ (pow (* z0 (tan (* PI (- (+ z2 z2) -0.5)))) 2.0) z1)
             z1)
            -1.0))
          z1))
        (if (<= z2 1.55e+175)
          (* (+ 1.0 (/ (* t_2 0.5) t_0)) t_0)
          t_1))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((1.0 / z1) + z1) * 0.5;
	double t_1 = 1.0 / (sqrt((pow((((((((pow(tan((((double) M_PI) * 0.5)), 2.0) - -1.0) * ((double) M_PI)) * 2.0) * z2) - tan((((double) M_PI) * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_2 = (1.0 / z1) - z1;
	double tmp;
	if (z2 <= -1.12e+235) {
		tmp = t_1;
	} else if (z2 <= -9e+148) {
		tmp = t_0 - (t_2 * -0.5);
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (sqrt((((pow((z0 * tan((((double) M_PI) * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((t_2 * 0.5) / t_0)) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = ((1.0 / z1) + z1) * 0.5;
	double t_1 = 1.0 / (Math.sqrt((Math.pow((((((((Math.pow(Math.tan((Math.PI * 0.5)), 2.0) - -1.0) * Math.PI) * 2.0) * z2) - Math.tan((Math.PI * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1);
	double t_2 = (1.0 / z1) - z1;
	double tmp;
	if (z2 <= -1.12e+235) {
		tmp = t_1;
	} else if (z2 <= -9e+148) {
		tmp = t_0 - (t_2 * -0.5);
	} else if (z2 <= 2.1e+51) {
		tmp = 1.0 / (Math.sqrt((((Math.pow((z0 * Math.tan((Math.PI * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else if (z2 <= 1.55e+175) {
		tmp = (1.0 + ((t_2 * 0.5) / t_0)) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((1.0 / z1) + z1) * 0.5
	t_1 = 1.0 / (math.sqrt((math.pow((((((((math.pow(math.tan((math.pi * 0.5)), 2.0) - -1.0) * math.pi) * 2.0) * z2) - math.tan((math.pi * -0.5))) * z0) / z1), 2.0) - -1.0)) * z1)
	t_2 = (1.0 / z1) - z1
	tmp = 0
	if z2 <= -1.12e+235:
		tmp = t_1
	elif z2 <= -9e+148:
		tmp = t_0 - (t_2 * -0.5)
	elif z2 <= 2.1e+51:
		tmp = 1.0 / (math.sqrt((((math.pow((z0 * math.tan((math.pi * ((z2 + z2) - -0.5)))), 2.0) / z1) / z1) - -1.0)) * z1)
	elif z2 <= 1.55e+175:
		tmp = (1.0 + ((t_2 * 0.5) / t_0)) * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(1.0 / z1) + z1) * 0.5)
	t_1 = Float64(1.0 / Float64(sqrt(Float64((Float64(Float64(Float64(Float64(Float64(Float64(Float64((tan(Float64(pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan(Float64(pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1))
	t_2 = Float64(Float64(1.0 / z1) - z1)
	tmp = 0.0
	if (z2 <= -1.12e+235)
		tmp = t_1;
	elseif (z2 <= -9e+148)
		tmp = Float64(t_0 - Float64(t_2 * -0.5));
	elseif (z2 <= 2.1e+51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64((Float64(z0 * tan(Float64(pi * Float64(Float64(z2 + z2) - -0.5)))) ^ 2.0) / z1) / z1) - -1.0)) * z1));
	elseif (z2 <= 1.55e+175)
		tmp = Float64(Float64(1.0 + Float64(Float64(t_2 * 0.5) / t_0)) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = ((1.0 / z1) + z1) * 0.5;
	t_1 = 1.0 / (sqrt(((((((((((tan((pi * 0.5)) ^ 2.0) - -1.0) * pi) * 2.0) * z2) - tan((pi * -0.5))) * z0) / z1) ^ 2.0) - -1.0)) * z1);
	t_2 = (1.0 / z1) - z1;
	tmp = 0.0;
	if (z2 <= -1.12e+235)
		tmp = t_1;
	elseif (z2 <= -9e+148)
		tmp = t_0 - (t_2 * -0.5);
	elseif (z2 <= 2.1e+51)
		tmp = 1.0 / (sqrt((((((z0 * tan((pi * ((z2 + z2) - -0.5)))) ^ 2.0) / z1) / z1) - -1.0)) * z1);
	elseif (z2 <= 1.55e+175)
		tmp = (1.0 + ((t_2 * 0.5) / t_0)) * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[(N[(N[(N[(N[(N[(N[Power[N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision] * Pi), $MachinePrecision] * 2.0), $MachinePrecision] * z2), $MachinePrecision] - N[Tan[N[(Pi * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / z1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision]}, If[LessEqual[z2, -1.12e+235], t$95$1, If[LessEqual[z2, -9e+148], N[(t$95$0 - N[(t$95$2 * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 2.1e+51], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], If[LessEqual[z2, 1.55e+175], N[(N[(1.0 + N[(N[(t$95$2 * 0.5), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -1.1199999999999999e235 or 1.5499999999999999e175 < z2

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 64.2%

      \[\leadsto \frac{1}{\sqrt{{\left(\color{blue}{\left(z2 \cdot \left(\left(2 \cdot \pi + z2 \cdot \left(z2 \cdot \left(-1.3333333333333333 \cdot {\pi}^{3} - \left(-4 \cdot \frac{{\pi}^{2} \cdot \left({\sin \left(0.5 \cdot \pi\right)}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{{\cos \left(0.5 \cdot \pi\right)}^{2}} + \left(-2 \cdot \left({\pi}^{2} \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right) + 1.3333333333333333 \cdot \frac{{\pi}^{3} \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)\right) - -2 \cdot \frac{\pi \cdot \left(\sin \left(0.5 \cdot \pi\right) \cdot \left(2 \cdot \pi - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right)\right)}{\cos \left(0.5 \cdot \pi\right)}\right)\right) - -2 \cdot \frac{\pi \cdot {\sin \left(0.5 \cdot \pi\right)}^{2}}{{\cos \left(0.5 \cdot \pi\right)}^{2}}\right) + \frac{\sin \left(0.5 \cdot \pi\right)}{\cos \left(0.5 \cdot \pi\right)}\right)} \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1} \]

   6. Applied rewrites 68.6%

      \[\leadsto \frac{1}{\sqrt{{\color{blue}{\left(\frac{z0 \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) + \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) - \left(\left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(\left(2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right)\right)\right)\right) \cdot z2\right) \cdot z2 + 2 \cdot \left(\pi + \left(\tan \left(0.5 \cdot \pi\right) \cdot \tan \left(0.5 \cdot \pi\right)\right) \cdot \pi\right)\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{z1}\right)}}^{2} - -1} \cdot z1} \]

   7. Taylor expanded in z2 around 0

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

   9. Applied rewrites 55.0%

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

   11. Applied rewrites 55.0%

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

   if -1.1199999999999999e235 < z2 < -8.9999999999999999e148

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5} \]

   if -8.9999999999999999e148 < z2 < 2.1000000000000001e51

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   if 2.1000000000000001e51 < z2 < 1.5499999999999999e175

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{\left(\frac{1}{z1} + z1\right) \cdot 0.5}\right) \cdot \left(\left(\frac{1}{z1} + z1\right) \cdot 0.5\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{z1} - z1\\ t_1 := \left(z2 + z2\right) - -0.5\\ t_2 := t\_1 \cdot \pi\\ t_3 := \left(\frac{1}{z1} + z1\right) \cdot 0.5\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+150}:\\ \;\;\;\;t\_3 - t\_0 \cdot -0.5\\ \mathbf{elif}\;t\_2 \leq 10^{+51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{{\left(z0 \cdot \tan \left(\pi \cdot t\_1\right)\right)}^{2}}{z1}}{z1} - -1} \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_0 \cdot 0.5}{t\_3}\right) \cdot t\_3\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (/ 1.0 z1) z1))
       (t_1 (- (+ z2 z2) -0.5))
       (t_2 (* t_1 PI))
       (t_3 (* (+ (/ 1.0 z1) z1) 0.5)))
  (if (<= t_2 -4e+150)
    (- t_3 (* t_0 -0.5))
    (if (<= t_2 1e+51)
      (/
       1.0
       (*
        (sqrt
         (- (/ (/ (pow (* z0 (tan (* PI t_1))) 2.0) z1) z1) -1.0))
        z1))
      (* (+ 1.0 (/ (* t_0 0.5) t_3)) t_3)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (1.0 / z1) - z1;
	double t_1 = (z2 + z2) - -0.5;
	double t_2 = t_1 * ((double) M_PI);
	double t_3 = ((1.0 / z1) + z1) * 0.5;
	double tmp;
	if (t_2 <= -4e+150) {
		tmp = t_3 - (t_0 * -0.5);
	} else if (t_2 <= 1e+51) {
		tmp = 1.0 / (sqrt((((pow((z0 * tan((((double) M_PI) * t_1))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else {
		tmp = (1.0 + ((t_0 * 0.5) / t_3)) * t_3;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (1.0 / z1) - z1;
	double t_1 = (z2 + z2) - -0.5;
	double t_2 = t_1 * Math.PI;
	double t_3 = ((1.0 / z1) + z1) * 0.5;
	double tmp;
	if (t_2 <= -4e+150) {
		tmp = t_3 - (t_0 * -0.5);
	} else if (t_2 <= 1e+51) {
		tmp = 1.0 / (Math.sqrt((((Math.pow((z0 * Math.tan((Math.PI * t_1))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else {
		tmp = (1.0 + ((t_0 * 0.5) / t_3)) * t_3;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (1.0 / z1) - z1
	t_1 = (z2 + z2) - -0.5
	t_2 = t_1 * math.pi
	t_3 = ((1.0 / z1) + z1) * 0.5
	tmp = 0
	if t_2 <= -4e+150:
		tmp = t_3 - (t_0 * -0.5)
	elif t_2 <= 1e+51:
		tmp = 1.0 / (math.sqrt((((math.pow((z0 * math.tan((math.pi * t_1))), 2.0) / z1) / z1) - -1.0)) * z1)
	else:
		tmp = (1.0 + ((t_0 * 0.5) / t_3)) * t_3
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(1.0 / z1) - z1)
	t_1 = Float64(Float64(z2 + z2) - -0.5)
	t_2 = Float64(t_1 * pi)
	t_3 = Float64(Float64(Float64(1.0 / z1) + z1) * 0.5)
	tmp = 0.0
	if (t_2 <= -4e+150)
		tmp = Float64(t_3 - Float64(t_0 * -0.5));
	elseif (t_2 <= 1e+51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64((Float64(z0 * tan(Float64(pi * t_1))) ^ 2.0) / z1) / z1) - -1.0)) * z1));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t_0 * 0.5) / t_3)) * t_3);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (1.0 / z1) - z1;
	t_1 = (z2 + z2) - -0.5;
	t_2 = t_1 * pi;
	t_3 = ((1.0 / z1) + z1) * 0.5;
	tmp = 0.0;
	if (t_2 <= -4e+150)
		tmp = t_3 - (t_0 * -0.5);
	elseif (t_2 <= 1e+51)
		tmp = 1.0 / (sqrt((((((z0 * tan((pi * t_1))) ^ 2.0) / z1) / z1) - -1.0)) * z1);
	else
		tmp = (1.0 + ((t_0 * 0.5) / t_3)) * t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * Pi), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+150], N[(t$95$3 - N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+51], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t$95$0 * 0.5), $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)) < -3.9999999999999999e150

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5} \]

   if -3.9999999999999999e150 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)) < 9.9999999999999999e50

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   if 9.9999999999999999e50 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{\left(\frac{1}{z1} + z1\right) \cdot 0.5}\right) \cdot \left(\left(\frac{1}{z1} + z1\right) \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 52.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{z1} - z1\\ t_1 := \left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\\ t_2 := \left(\frac{1}{z1} + z1\right) \cdot 0.5\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+150}:\\ \;\;\;\;t\_2 - t\_0 \cdot -0.5\\ \mathbf{elif}\;t\_1 \leq 10^{+51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{\frac{{\left(z0 \cdot \tan \left(\pi \cdot 0.5\right)\right)}^{2}}{z1}}{z1} - -1} \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \frac{t\_0 \cdot 0.5}{t\_2}\right) \cdot t\_2\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (/ 1.0 z1) z1))
       (t_1 (* (- (+ z2 z2) -0.5) PI))
       (t_2 (* (+ (/ 1.0 z1) z1) 0.5)))
  (if (<= t_1 -4e+150)
    (- t_2 (* t_0 -0.5))
    (if (<= t_1 1e+51)
      (/
       1.0
       (*
        (sqrt
         (- (/ (/ (pow (* z0 (tan (* PI 0.5))) 2.0) z1) z1) -1.0))
        z1))
      (* (+ 1.0 (/ (* t_0 0.5) t_2)) t_2)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (1.0 / z1) - z1;
	double t_1 = ((z2 + z2) - -0.5) * ((double) M_PI);
	double t_2 = ((1.0 / z1) + z1) * 0.5;
	double tmp;
	if (t_1 <= -4e+150) {
		tmp = t_2 - (t_0 * -0.5);
	} else if (t_1 <= 1e+51) {
		tmp = 1.0 / (sqrt((((pow((z0 * tan((((double) M_PI) * 0.5))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else {
		tmp = (1.0 + ((t_0 * 0.5) / t_2)) * t_2;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (1.0 / z1) - z1;
	double t_1 = ((z2 + z2) - -0.5) * Math.PI;
	double t_2 = ((1.0 / z1) + z1) * 0.5;
	double tmp;
	if (t_1 <= -4e+150) {
		tmp = t_2 - (t_0 * -0.5);
	} else if (t_1 <= 1e+51) {
		tmp = 1.0 / (Math.sqrt((((Math.pow((z0 * Math.tan((Math.PI * 0.5))), 2.0) / z1) / z1) - -1.0)) * z1);
	} else {
		tmp = (1.0 + ((t_0 * 0.5) / t_2)) * t_2;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (1.0 / z1) - z1
	t_1 = ((z2 + z2) - -0.5) * math.pi
	t_2 = ((1.0 / z1) + z1) * 0.5
	tmp = 0
	if t_1 <= -4e+150:
		tmp = t_2 - (t_0 * -0.5)
	elif t_1 <= 1e+51:
		tmp = 1.0 / (math.sqrt((((math.pow((z0 * math.tan((math.pi * 0.5))), 2.0) / z1) / z1) - -1.0)) * z1)
	else:
		tmp = (1.0 + ((t_0 * 0.5) / t_2)) * t_2
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(1.0 / z1) - z1)
	t_1 = Float64(Float64(Float64(z2 + z2) - -0.5) * pi)
	t_2 = Float64(Float64(Float64(1.0 / z1) + z1) * 0.5)
	tmp = 0.0
	if (t_1 <= -4e+150)
		tmp = Float64(t_2 - Float64(t_0 * -0.5));
	elseif (t_1 <= 1e+51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(Float64(Float64((Float64(z0 * tan(Float64(pi * 0.5))) ^ 2.0) / z1) / z1) - -1.0)) * z1));
	else
		tmp = Float64(Float64(1.0 + Float64(Float64(t_0 * 0.5) / t_2)) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (1.0 / z1) - z1;
	t_1 = ((z2 + z2) - -0.5) * pi;
	t_2 = ((1.0 / z1) + z1) * 0.5;
	tmp = 0.0;
	if (t_1 <= -4e+150)
		tmp = t_2 - (t_0 * -0.5);
	elseif (t_1 <= 1e+51)
		tmp = 1.0 / (sqrt((((((z0 * tan((pi * 0.5))) ^ 2.0) / z1) / z1) - -1.0)) * z1);
	else
		tmp = (1.0 + ((t_0 * 0.5) / t_2)) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+150], N[(t$95$2 - N[(t$95$0 * -0.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+51], N[(1.0 / N[(N[Sqrt[N[(N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / z1), $MachinePrecision] / z1), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[(N[(t$95$0 * 0.5), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)) < -3.9999999999999999e150

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5} \]

   if -3.9999999999999999e150 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)) < 9.9999999999999999e50

   1. Initial program 40.7%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 46.8%

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

   4. Taylor expanded in z2 around 0

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

   6. Applied rewrites 47.7%

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

   if 9.9999999999999999e50 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(1 + \frac{\left(\frac{1}{z1} - z1\right) \cdot 0.5}{\left(\frac{1}{z1} + z1\right) \cdot 0.5}\right) \cdot \left(\left(\frac{1}{z1} + z1\right) \cdot 0.5\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 50.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\\ t_1 := \left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\tan \left(\left(z2 - \left(-0.5 - z2\right)\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (- (+ z2 z2) -0.5) PI))
       (t_1 (- (* (+ (/ 1.0 z1) z1) 0.5) (* (- (/ 1.0 z1) z1) -0.5))))
  (if (<= t_0 -5e+16)
    t_1
    (if (<= t_0 2.0)
      (/
       1.0
       (*
        (sqrt
         (-
          (pow (* (tan (* (- z2 (- -0.5 z2)) PI)) (/ z0 z1)) 2.0)
          -1.0))
        z1))
      t_1))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((z2 + z2) - -0.5) * ((double) M_PI);
	double t_1 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	double tmp;
	if (t_0 <= -5e+16) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 / (sqrt((pow((tan(((z2 - (-0.5 - z2)) * ((double) M_PI))) * (z0 / z1)), 2.0) - -1.0)) * z1);
	} else {
		tmp = t_1;
	}
	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 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	double tmp;
	if (t_0 <= -5e+16) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = 1.0 / (Math.sqrt((Math.pow((Math.tan(((z2 - (-0.5 - z2)) * Math.PI)) * (z0 / z1)), 2.0) - -1.0)) * z1);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((z2 + z2) - -0.5) * math.pi
	t_1 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5)
	tmp = 0
	if t_0 <= -5e+16:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = 1.0 / (math.sqrt((math.pow((math.tan(((z2 - (-0.5 - z2)) * math.pi)) * (z0 / z1)), 2.0) - -1.0)) * z1)
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(z2 + z2) - -0.5) * pi)
	t_1 = Float64(Float64(Float64(Float64(1.0 / z1) + z1) * 0.5) - Float64(Float64(Float64(1.0 / z1) - z1) * -0.5))
	tmp = 0.0
	if (t_0 <= -5e+16)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64((Float64(tan(Float64(Float64(z2 - Float64(-0.5 - z2)) * pi)) * Float64(z0 / z1)) ^ 2.0) - -1.0)) * z1));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = ((z2 + z2) - -0.5) * pi;
	t_1 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	tmp = 0.0;
	if (t_0 <= -5e+16)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = 1.0 / (sqrt((((tan(((z2 - (-0.5 - z2)) * pi)) * (z0 / z1)) ^ 2.0) - -1.0)) * z1);
	else
		tmp = t_1;
	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[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e+16], t$95$1, If[LessEqual[t$95$0, 2.0], N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[Tan[N[(N[(z2 - N[(-0.5 - z2), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(z0 / z1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)) < -5e16 or 2 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5} \]

   if -5e16 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)) < 2

   1. Initial program 40.7%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate-rev N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 40.7%

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

   3. Applied rewrites 40.7%

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

Alternative 12: 50.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5\\ \mathbf{if}\;z2 \leq -3.8 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq 19000000000:\\ \;\;\;\;\frac{1}{\sqrt{{\left(z0 \cdot \frac{\tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)}{z1}\right)}^{2} - -1} \cdot z1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (* (+ (/ 1.0 z1) z1) 0.5) (* (- (/ 1.0 z1) z1) -0.5))))
  (if (<= z2 -3.8e+15)
    t_0
    (if (<= z2 19000000000.0)
      (/
       1.0
       (*
        (sqrt
         (-
          (pow (* z0 (/ (tan (* PI (- (+ z2 z2) -0.5))) z1)) 2.0)
          -1.0))
        z1))
      t_0))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	double tmp;
	if (z2 <= -3.8e+15) {
		tmp = t_0;
	} else if (z2 <= 19000000000.0) {
		tmp = 1.0 / (sqrt((pow((z0 * (tan((((double) M_PI) * ((z2 + z2) - -0.5))) / z1)), 2.0) - -1.0)) * z1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	double tmp;
	if (z2 <= -3.8e+15) {
		tmp = t_0;
	} else if (z2 <= 19000000000.0) {
		tmp = 1.0 / (Math.sqrt((Math.pow((z0 * (Math.tan((Math.PI * ((z2 + z2) - -0.5))) / z1)), 2.0) - -1.0)) * z1);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5)
	tmp = 0
	if z2 <= -3.8e+15:
		tmp = t_0
	elif z2 <= 19000000000.0:
		tmp = 1.0 / (math.sqrt((math.pow((z0 * (math.tan((math.pi * ((z2 + z2) - -0.5))) / z1)), 2.0) - -1.0)) * z1)
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(Float64(1.0 / z1) + z1) * 0.5) - Float64(Float64(Float64(1.0 / z1) - z1) * -0.5))
	tmp = 0.0
	if (z2 <= -3.8e+15)
		tmp = t_0;
	elseif (z2 <= 19000000000.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64((Float64(z0 * Float64(tan(Float64(pi * Float64(Float64(z2 + z2) - -0.5))) / z1)) ^ 2.0) - -1.0)) * z1));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	tmp = 0.0;
	if (z2 <= -3.8e+15)
		tmp = t_0;
	elseif (z2 <= 19000000000.0)
		tmp = 1.0 / (sqrt((((z0 * (tan((pi * ((z2 + z2) - -0.5))) / z1)) ^ 2.0) - -1.0)) * z1);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z2, -3.8e+15], t$95$0, If[LessEqual[z2, 19000000000.0], N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(z0 * N[(N[Tan[N[(Pi * N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / z1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * z1), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z2 < -3.8e15 or 1.9e10 < z2

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5} \]

   if -3.8e15 < z2 < 1.9e10

   1. Initial program 40.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 40.7%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 40.7%

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

   3. Applied rewrites 40.7%

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

Alternative 13: 49.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|z1\right|}\\ \mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z1\right| \leq 1800:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\tan \left(2.5 \cdot \pi + \pi\right) \cdot \frac{z0}{\left|z1\right|}\right)}^{2} - -1} \cdot \left|z1\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + \left|z1\right|\right) \cdot 0.5 - \left(t\_0 - \left|z1\right|\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ 1.0 (fabs z1))))
  (*
   (copysign 1.0 z1)
   (if (<= (fabs z1) 1800.0)
     (/
      1.0
      (*
       (sqrt
        (-
         (pow (* (tan (+ (* 2.5 PI) PI)) (/ z0 (fabs z1))) 2.0)
         -1.0))
       (fabs z1)))
     (- (* (+ t_0 (fabs z1)) 0.5) (* (- t_0 (fabs z1)) -0.5))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / fabs(z1);
	double tmp;
	if (fabs(z1) <= 1800.0) {
		tmp = 1.0 / (sqrt((pow((tan(((2.5 * ((double) M_PI)) + ((double) M_PI))) * (z0 / fabs(z1))), 2.0) - -1.0)) * fabs(z1));
	} else {
		tmp = ((t_0 + fabs(z1)) * 0.5) - ((t_0 - fabs(z1)) * -0.5);
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / Math.abs(z1);
	double tmp;
	if (Math.abs(z1) <= 1800.0) {
		tmp = 1.0 / (Math.sqrt((Math.pow((Math.tan(((2.5 * Math.PI) + Math.PI)) * (z0 / Math.abs(z1))), 2.0) - -1.0)) * Math.abs(z1));
	} else {
		tmp = ((t_0 + Math.abs(z1)) * 0.5) - ((t_0 - Math.abs(z1)) * -0.5);
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z2, z0, z1):
	t_0 = 1.0 / math.fabs(z1)
	tmp = 0
	if math.fabs(z1) <= 1800.0:
		tmp = 1.0 / (math.sqrt((math.pow((math.tan(((2.5 * math.pi) + math.pi)) * (z0 / math.fabs(z1))), 2.0) - -1.0)) * math.fabs(z1))
	else:
		tmp = ((t_0 + math.fabs(z1)) * 0.5) - ((t_0 - math.fabs(z1)) * -0.5)
	return math.copysign(1.0, z1) * tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(1.0 / abs(z1))
	tmp = 0.0
	if (abs(z1) <= 1800.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64((Float64(tan(Float64(Float64(2.5 * pi) + pi)) * Float64(z0 / abs(z1))) ^ 2.0) - -1.0)) * abs(z1)));
	else
		tmp = Float64(Float64(Float64(t_0 + abs(z1)) * 0.5) - Float64(Float64(t_0 - abs(z1)) * -0.5));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = 1.0 / abs(z1);
	tmp = 0.0;
	if (abs(z1) <= 1800.0)
		tmp = 1.0 / (sqrt((((tan(((2.5 * pi) + pi)) * (z0 / abs(z1))) ^ 2.0) - -1.0)) * abs(z1));
	else
		tmp = ((t_0 + abs(z1)) * 0.5) - ((t_0 - abs(z1)) * -0.5);
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z1], $MachinePrecision], 1800.0], N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[Tan[N[(N[(2.5 * Pi), $MachinePrecision] + Pi), $MachinePrecision]], $MachinePrecision] * N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + N[Abs[z1], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[(t$95$0 - N[Abs[z1], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z1 < 1800

   1. Initial program 40.7%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-+PI-rev N/A

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

      4. tan-+PI-rev N/A

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

      5. lower-tan.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. lower-+.f64 N/A

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

   3. Applied rewrites 40.7%

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

   4. Taylor expanded in z2 around 0

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

   6. Applied rewrites 41.3%

      \[\leadsto \frac{1}{\sqrt{{\left(\tan \left(\color{blue}{2.5} \cdot \pi + \pi\right) \cdot \frac{z0}{z1}\right)}^{2} - -1} \cdot z1} \]

   if 1800 < z1

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

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

Alternative 14: 48.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left|z1\right|}\\ \mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\left|z1\right| \leq 1800:\\ \;\;\;\;\frac{1}{\sqrt{{\left(\tan \left(0.5 \cdot \pi\right) \cdot \frac{z0}{\left|z1\right|}\right)}^{2} - -1} \cdot \left|z1\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(t\_0 + \left|z1\right|\right) \cdot 0.5 - \left(t\_0 - \left|z1\right|\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ 1.0 (fabs z1))))
  (*
   (copysign 1.0 z1)
   (if (<= (fabs z1) 1800.0)
     (/
      1.0
      (*
       (sqrt (- (pow (* (tan (* 0.5 PI)) (/ z0 (fabs z1))) 2.0) -1.0))
       (fabs z1)))
     (- (* (+ t_0 (fabs z1)) 0.5) (* (- t_0 (fabs z1)) -0.5))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / fabs(z1);
	double tmp;
	if (fabs(z1) <= 1800.0) {
		tmp = 1.0 / (sqrt((pow((tan((0.5 * ((double) M_PI))) * (z0 / fabs(z1))), 2.0) - -1.0)) * fabs(z1));
	} else {
		tmp = ((t_0 + fabs(z1)) * 0.5) - ((t_0 - fabs(z1)) * -0.5);
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = 1.0 / Math.abs(z1);
	double tmp;
	if (Math.abs(z1) <= 1800.0) {
		tmp = 1.0 / (Math.sqrt((Math.pow((Math.tan((0.5 * Math.PI)) * (z0 / Math.abs(z1))), 2.0) - -1.0)) * Math.abs(z1));
	} else {
		tmp = ((t_0 + Math.abs(z1)) * 0.5) - ((t_0 - Math.abs(z1)) * -0.5);
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z2, z0, z1):
	t_0 = 1.0 / math.fabs(z1)
	tmp = 0
	if math.fabs(z1) <= 1800.0:
		tmp = 1.0 / (math.sqrt((math.pow((math.tan((0.5 * math.pi)) * (z0 / math.fabs(z1))), 2.0) - -1.0)) * math.fabs(z1))
	else:
		tmp = ((t_0 + math.fabs(z1)) * 0.5) - ((t_0 - math.fabs(z1)) * -0.5)
	return math.copysign(1.0, z1) * tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(1.0 / abs(z1))
	tmp = 0.0
	if (abs(z1) <= 1800.0)
		tmp = Float64(1.0 / Float64(sqrt(Float64((Float64(tan(Float64(0.5 * pi)) * Float64(z0 / abs(z1))) ^ 2.0) - -1.0)) * abs(z1)));
	else
		tmp = Float64(Float64(Float64(t_0 + abs(z1)) * 0.5) - Float64(Float64(t_0 - abs(z1)) * -0.5));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = 1.0 / abs(z1);
	tmp = 0.0;
	if (abs(z1) <= 1800.0)
		tmp = 1.0 / (sqrt((((tan((0.5 * pi)) * (z0 / abs(z1))) ^ 2.0) - -1.0)) * abs(z1));
	else
		tmp = ((t_0 + abs(z1)) * 0.5) - ((t_0 - abs(z1)) * -0.5);
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[z1], $MachinePrecision], 1800.0], N[(1.0 / N[(N[Sqrt[N[(N[Power[N[(N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision] * N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t$95$0 + N[Abs[z1], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[(t$95$0 - N[Abs[z1], $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z1 < 1800

   1. Initial program 40.7%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 41.4%

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

   if 1800 < z1

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

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

Alternative 15: 40.4% accurate, 3.4× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\\ t_1 := \left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5\\ \mathbf{if}\;t\_0 \leq -390000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 400:\\ \;\;\;\;\frac{1}{z1}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (- (+ z2 z2) -0.5) PI))
       (t_1 (- (* (+ (/ 1.0 z1) z1) 0.5) (* (- (/ 1.0 z1) z1) -0.5))))
  (if (<= t_0 -390000000.0) t_1 (if (<= t_0 400.0) (/ 1.0 z1) t_1))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((z2 + z2) - -0.5) * ((double) M_PI);
	double t_1 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	double tmp;
	if (t_0 <= -390000000.0) {
		tmp = t_1;
	} else if (t_0 <= 400.0) {
		tmp = 1.0 / z1;
	} else {
		tmp = t_1;
	}
	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 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	double tmp;
	if (t_0 <= -390000000.0) {
		tmp = t_1;
	} else if (t_0 <= 400.0) {
		tmp = 1.0 / z1;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((z2 + z2) - -0.5) * math.pi
	t_1 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5)
	tmp = 0
	if t_0 <= -390000000.0:
		tmp = t_1
	elif t_0 <= 400.0:
		tmp = 1.0 / z1
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(z2 + z2) - -0.5) * pi)
	t_1 = Float64(Float64(Float64(Float64(1.0 / z1) + z1) * 0.5) - Float64(Float64(Float64(1.0 / z1) - z1) * -0.5))
	tmp = 0.0
	if (t_0 <= -390000000.0)
		tmp = t_1;
	elseif (t_0 <= 400.0)
		tmp = Float64(1.0 / z1);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = ((z2 + z2) - -0.5) * pi;
	t_1 = (((1.0 / z1) + z1) * 0.5) - (((1.0 / z1) - z1) * -0.5);
	tmp = 0.0;
	if (t_0 <= -390000000.0)
		tmp = t_1;
	elseif (t_0 <= 400.0)
		tmp = 1.0 / z1;
	else
		tmp = t_1;
	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[(N[(N[(1.0 / z1), $MachinePrecision] + z1), $MachinePrecision] * 0.5), $MachinePrecision] - N[(N[(N[(1.0 / z1), $MachinePrecision] - z1), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -390000000.0], t$95$1, If[LessEqual[t$95$0, 400.0], N[(1.0 / z1), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)) < -3.9e8 or 400 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

   8. Applied rewrites 11.6%

      \[\leadsto \color{blue}{\frac{{\left(\frac{z1 + \frac{1}{z1}}{2}\right)}^{3} + {\left(-\frac{z1 - \frac{1}{z1}}{2}\right)}^{3}}{\frac{z1 + \frac{1}{z1}}{2} \cdot \frac{z1 + \frac{1}{z1}}{2} + \left(\left(-\frac{z1 - \frac{1}{z1}}{2}\right) \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right) - \frac{z1 + \frac{1}{z1}}{2} \cdot \left(-\frac{z1 - \frac{1}{z1}}{2}\right)\right)}} \]

   10. Applied rewrites 33.9%

       \[\leadsto \color{blue}{\left(\frac{1}{z1} + z1\right) \cdot 0.5 - \left(\frac{1}{z1} - z1\right) \cdot -0.5} \]

   if -3.9e8 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)) < 400

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 28.9% accurate, 1.0× speedup

Math

\[\mathsf{copysign}\left(1, z1\right) \cdot \begin{array}{l} \mathbf{if}\;\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \leq 9.6:\\ \;\;\;\;\sqrt{\frac{1}{\left|z1\right| \cdot \left|z1\right|}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\left|z1\right|}\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (*
 (copysign 1.0 z1)
 (if (<= (tan (* (- (+ z2 z2) -0.5) PI)) 9.6)
   (sqrt (/ 1.0 (* (fabs z1) (fabs z1))))
   (/ 1.0 (fabs z1)))))

C

double code(double z2, double z0, double z1) {
	double tmp;
	if (tan((((z2 + z2) - -0.5) * ((double) M_PI))) <= 9.6) {
		tmp = sqrt((1.0 / (fabs(z1) * fabs(z1))));
	} else {
		tmp = 1.0 / fabs(z1);
	}
	return copysign(1.0, z1) * tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double tmp;
	if (Math.tan((((z2 + z2) - -0.5) * Math.PI)) <= 9.6) {
		tmp = Math.sqrt((1.0 / (Math.abs(z1) * Math.abs(z1))));
	} else {
		tmp = 1.0 / Math.abs(z1);
	}
	return Math.copySign(1.0, z1) * tmp;
}

Python

def code(z2, z0, z1):
	tmp = 0
	if math.tan((((z2 + z2) - -0.5) * math.pi)) <= 9.6:
		tmp = math.sqrt((1.0 / (math.fabs(z1) * math.fabs(z1))))
	else:
		tmp = 1.0 / math.fabs(z1)
	return math.copysign(1.0, z1) * tmp

Julia

function code(z2, z0, z1)
	tmp = 0.0
	if (tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) <= 9.6)
		tmp = sqrt(Float64(1.0 / Float64(abs(z1) * abs(z1))));
	else
		tmp = Float64(1.0 / abs(z1));
	end
	return Float64(copysign(1.0, z1) * tmp)
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	tmp = 0.0;
	if (tan((((z2 + z2) - -0.5) * pi)) <= 9.6)
		tmp = sqrt((1.0 / (abs(z1) * abs(z1))));
	else
		tmp = 1.0 / abs(z1);
	end
	tmp_2 = (sign(z1) * abs(1.0)) * tmp;
end

Wolfram

code[z2_, z0_, z1_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z1]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], 9.6], N[Sqrt[N[(1.0 / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) < 9.5999999999999996

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{z1}} \]

      2. inv-pow N/A

         \[\leadsto \color{blue}{{z1}^{-1}} \]

      3. pow-to-exp N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

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

         \[\leadsto e^{\color{blue}{\log z1 \cdot -1}} \]

      6. lower-unsound-log.f64 8.9%

         \[\leadsto e^{\color{blue}{\log z1} \cdot -1} \]

   6. Applied rewrites 8.9%

      \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]
   7. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \color{blue}{e^{\log z1 \cdot -1}} \]

      2. exp-fabs N/A

         \[\leadsto \color{blue}{\left|e^{\log z1 \cdot -1}\right|} \]

      3. lift-exp.f64 N/A

         \[\leadsto \left|\color{blue}{e^{\log z1 \cdot -1}}\right| \]

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

         \[\leadsto \color{blue}{\sqrt{e^{\log z1 \cdot -1} \cdot e^{\log z1 \cdot -1}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{e^{\log z1 \cdot -1} \cdot e^{\log z1 \cdot -1}}} \]

   8. Applied rewrites 20.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{z1} \cdot \frac{1}{z1}}} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{z1} \cdot \frac{1}{z1}}} \]

      2. lift-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{z1}} \cdot \frac{1}{z1}} \]

      3. lift-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{z1} \cdot \color{blue}{\frac{1}{z1}}} \]

      4. frac-times N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1 \cdot 1}{z1 \cdot z1}}} \]

      5. metadata-eval N/A

         \[\leadsto \sqrt{\frac{\color{blue}{1}}{z1 \cdot z1}} \]

      6. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{z1 \cdot z1}}} \]

   10. Applied rewrites 20.9%

       \[\leadsto \sqrt{\color{blue}{\frac{1}{z1 \cdot z1}}} \]

   if 9.5999999999999996 < (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64)))

   1. Initial program 40.7%

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

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
   3. Step-by-step derivation

   4. Applied rewrites 19.2%

      \[\leadsto \frac{1}{\color{blue}{z1}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 19.2% accurate, 21.5× speedup

Math

\[\frac{1}{z1} \]

FPCore

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

C

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

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 = 1.0d0 / z1
end function

Java

public static double code(double z2, double z0, double z1) {
	return 1.0 / z1;
}

Python

def code(z2, z0, z1):
	return 1.0 / z1

Julia

function code(z2, z0, z1)
	return Float64(1.0 / z1)
end

MATLAB

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

Wolfram

code[z2_, z0_, z1_] := N[(1.0 / z1), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

2. Taylor expanded in z0 around 0

   \[\leadsto \frac{1}{\color{blue}{z1}} \]
3. Step-by-step derivation

4. Applied rewrites 19.2%

   \[\leadsto \frac{1}{\color{blue}{z1}} \]
5. Add Preprocessing

Reproduce

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