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

Percentage Accurate: 47.2% → 84.2%

Time: 8.3s

Alternatives: 13

Speedup: 0.9×

• 26.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z2_, z0_, z1_] := 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]

Initial Program: 47.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z2_, z0_, z1_] := 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]

Alternative 1: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0}{\left|z1\right|}\\ t_1 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ t_2 := {\tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)}^{2} \cdot t\_0\\ \mathbf{if}\;z2 \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z2 \leq -8.5 \cdot 10^{-300}:\\ \;\;\;\;\left(t\_2 \cdot \frac{1}{\left|z1\right|}\right) \cdot z0 - -1\\ \mathbf{elif}\;z2 \leq 1.1 \cdot 10^{-227}:\\ \;\;\;\;t\_0 \cdot \frac{z0}{0} - -1\\ \mathbf{elif}\;z2 \leq 0.0066:\\ \;\;\;\;\left(t\_2 \cdot \left(-z0\right)\right) \cdot \frac{-1}{\left|z1\right|} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ z0 (fabs z1)))
       (t_1 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0))
       (t_2 (* (pow (tan (* PI (- (+ z2 z2) -0.5))) 2.0) t_0)))
  (if (<= z2 -6.5e+15)
    t_1
    (if (<= z2 -8.5e-300)
      (- (* (* t_2 (/ 1.0 (fabs z1))) z0) -1.0)
      (if (<= z2 1.1e-227)
        (- (* t_0 (/ z0 0.0)) -1.0)
        (if (<= z2 0.0066)
          (- (* (* t_2 (- z0)) (/ -1.0 (fabs z1))) -1.0)
          t_1))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = z0 / fabs(z1);
	double t_1 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double t_2 = pow(tan((((double) M_PI) * ((z2 + z2) - -0.5))), 2.0) * t_0;
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_1;
	} else if (z2 <= -8.5e-300) {
		tmp = ((t_2 * (1.0 / fabs(z1))) * z0) - -1.0;
	} else if (z2 <= 1.1e-227) {
		tmp = (t_0 * (z0 / 0.0)) - -1.0;
	} else if (z2 <= 0.0066) {
		tmp = ((t_2 * -z0) * (-1.0 / fabs(z1))) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = z0 / Math.abs(z1);
	double t_1 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double t_2 = Math.pow(Math.tan((Math.PI * ((z2 + z2) - -0.5))), 2.0) * t_0;
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_1;
	} else if (z2 <= -8.5e-300) {
		tmp = ((t_2 * (1.0 / Math.abs(z1))) * z0) - -1.0;
	} else if (z2 <= 1.1e-227) {
		tmp = (t_0 * (z0 / 0.0)) - -1.0;
	} else if (z2 <= 0.0066) {
		tmp = ((t_2 * -z0) * (-1.0 / Math.abs(z1))) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = z0 / math.fabs(z1)
	t_1 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	t_2 = math.pow(math.tan((math.pi * ((z2 + z2) - -0.5))), 2.0) * t_0
	tmp = 0
	if z2 <= -6.5e+15:
		tmp = t_1
	elif z2 <= -8.5e-300:
		tmp = ((t_2 * (1.0 / math.fabs(z1))) * z0) - -1.0
	elif z2 <= 1.1e-227:
		tmp = (t_0 * (z0 / 0.0)) - -1.0
	elif z2 <= 0.0066:
		tmp = ((t_2 * -z0) * (-1.0 / math.fabs(z1))) - -1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(z0 / abs(z1))
	t_1 = Float64(Float64(Float64(Float64(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	t_2 = Float64((tan(Float64(pi * Float64(Float64(z2 + z2) - -0.5))) ^ 2.0) * t_0)
	tmp = 0.0
	if (z2 <= -6.5e+15)
		tmp = t_1;
	elseif (z2 <= -8.5e-300)
		tmp = Float64(Float64(Float64(t_2 * Float64(1.0 / abs(z1))) * z0) - -1.0);
	elseif (z2 <= 1.1e-227)
		tmp = Float64(Float64(t_0 * Float64(z0 / 0.0)) - -1.0);
	elseif (z2 <= 0.0066)
		tmp = Float64(Float64(Float64(t_2 * Float64(-z0)) * Float64(-1.0 / abs(z1))) - -1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = z0 / abs(z1);
	t_1 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	t_2 = (tan((pi * ((z2 + z2) - -0.5))) ^ 2.0) * t_0;
	tmp = 0.0;
	if (z2 <= -6.5e+15)
		tmp = t_1;
	elseif (z2 <= -8.5e-300)
		tmp = ((t_2 * (1.0 / abs(z1))) * z0) - -1.0;
	elseif (z2 <= 1.1e-227)
		tmp = (t_0 * (z0 / 0.0)) - -1.0;
	elseif (z2 <= 0.0066)
		tmp = ((t_2 * -z0) * (-1.0 / abs(z1))) - -1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Tan[N[(Pi * N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[z2, -6.5e+15], t$95$1, If[LessEqual[z2, -8.5e-300], N[(N[(N[(t$95$2 * N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 1.1e-227], N[(N[(t$95$0 * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 0.0066], N[(N[(N[(t$95$2 * (-z0)), $MachinePrecision] * N[(-1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -6.5e15 or 0.0066 < z2

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

   if -6.5e15 < z2 < -8.4999999999999995e-300

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

   if -8.4999999999999995e-300 < z2 < 1.0999999999999999e-227

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

   10. Applied rewrites 34.5%

       \[\leadsto \color{blue}{\frac{z0}{z1} \cdot \frac{z0}{0}} - -1 \]

   if 1.0999999999999999e-227 < z2 < 0.0066

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

Alternative 2: 82.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0}{\left|z1\right|}\\ t_1 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ t_2 := \left(z2 + z2\right) - -0.5\\ t_3 := {\tan \left(\pi \cdot t\_2\right)}^{2} \cdot t\_0\\ \mathbf{if}\;z2 \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z2 \leq -8.5 \cdot 10^{-300}:\\ \;\;\;\;\left(t\_3 \cdot \frac{1}{\left|z1\right|}\right) \cdot z0 - -1\\ \mathbf{elif}\;z2 \leq 1.1 \cdot 10^{-227}:\\ \;\;\;\;\frac{\left(0.5 - \cos \left(\left(\left(z2 + z2\right) - -1.5\right) \cdot \left(\pi + \pi\right)\right) \cdot 0.5\right) \cdot z0}{\left(\left(\cos \left(t\_2 \cdot \left(\pi + \pi\right)\right) + 1\right) \cdot 0.5\right) \cdot \left|z1\right|} \cdot t\_0 - -1\\ \mathbf{elif}\;z2 \leq 0.0066:\\ \;\;\;\;\left(t\_3 \cdot \left(-z0\right)\right) \cdot \frac{-1}{\left|z1\right|} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ z0 (fabs z1)))
       (t_1 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0))
       (t_2 (- (+ z2 z2) -0.5))
       (t_3 (* (pow (tan (* PI t_2)) 2.0) t_0)))
  (if (<= z2 -6.5e+15)
    t_1
    (if (<= z2 -8.5e-300)
      (- (* (* t_3 (/ 1.0 (fabs z1))) z0) -1.0)
      (if (<= z2 1.1e-227)
        (-
         (*
          (/
           (*
            (- 0.5 (* (cos (* (- (+ z2 z2) -1.5) (+ PI PI))) 0.5))
            z0)
           (* (* (+ (cos (* t_2 (+ PI PI))) 1.0) 0.5) (fabs z1)))
          t_0)
         -1.0)
        (if (<= z2 0.0066)
          (- (* (* t_3 (- z0)) (/ -1.0 (fabs z1))) -1.0)
          t_1))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = z0 / fabs(z1);
	double t_1 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double t_2 = (z2 + z2) - -0.5;
	double t_3 = pow(tan((((double) M_PI) * t_2)), 2.0) * t_0;
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_1;
	} else if (z2 <= -8.5e-300) {
		tmp = ((t_3 * (1.0 / fabs(z1))) * z0) - -1.0;
	} else if (z2 <= 1.1e-227) {
		tmp = ((((0.5 - (cos((((z2 + z2) - -1.5) * (((double) M_PI) + ((double) M_PI)))) * 0.5)) * z0) / (((cos((t_2 * (((double) M_PI) + ((double) M_PI)))) + 1.0) * 0.5) * fabs(z1))) * t_0) - -1.0;
	} else if (z2 <= 0.0066) {
		tmp = ((t_3 * -z0) * (-1.0 / fabs(z1))) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = z0 / Math.abs(z1);
	double t_1 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double t_2 = (z2 + z2) - -0.5;
	double t_3 = Math.pow(Math.tan((Math.PI * t_2)), 2.0) * t_0;
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_1;
	} else if (z2 <= -8.5e-300) {
		tmp = ((t_3 * (1.0 / Math.abs(z1))) * z0) - -1.0;
	} else if (z2 <= 1.1e-227) {
		tmp = ((((0.5 - (Math.cos((((z2 + z2) - -1.5) * (Math.PI + Math.PI))) * 0.5)) * z0) / (((Math.cos((t_2 * (Math.PI + Math.PI))) + 1.0) * 0.5) * Math.abs(z1))) * t_0) - -1.0;
	} else if (z2 <= 0.0066) {
		tmp = ((t_3 * -z0) * (-1.0 / Math.abs(z1))) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = z0 / math.fabs(z1)
	t_1 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	t_2 = (z2 + z2) - -0.5
	t_3 = math.pow(math.tan((math.pi * t_2)), 2.0) * t_0
	tmp = 0
	if z2 <= -6.5e+15:
		tmp = t_1
	elif z2 <= -8.5e-300:
		tmp = ((t_3 * (1.0 / math.fabs(z1))) * z0) - -1.0
	elif z2 <= 1.1e-227:
		tmp = ((((0.5 - (math.cos((((z2 + z2) - -1.5) * (math.pi + math.pi))) * 0.5)) * z0) / (((math.cos((t_2 * (math.pi + math.pi))) + 1.0) * 0.5) * math.fabs(z1))) * t_0) - -1.0
	elif z2 <= 0.0066:
		tmp = ((t_3 * -z0) * (-1.0 / math.fabs(z1))) - -1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(z0 / abs(z1))
	t_1 = Float64(Float64(Float64(Float64(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	t_2 = Float64(Float64(z2 + z2) - -0.5)
	t_3 = Float64((tan(Float64(pi * t_2)) ^ 2.0) * t_0)
	tmp = 0.0
	if (z2 <= -6.5e+15)
		tmp = t_1;
	elseif (z2 <= -8.5e-300)
		tmp = Float64(Float64(Float64(t_3 * Float64(1.0 / abs(z1))) * z0) - -1.0);
	elseif (z2 <= 1.1e-227)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 - Float64(cos(Float64(Float64(Float64(z2 + z2) - -1.5) * Float64(pi + pi))) * 0.5)) * z0) / Float64(Float64(Float64(cos(Float64(t_2 * Float64(pi + pi))) + 1.0) * 0.5) * abs(z1))) * t_0) - -1.0);
	elseif (z2 <= 0.0066)
		tmp = Float64(Float64(Float64(t_3 * Float64(-z0)) * Float64(-1.0 / abs(z1))) - -1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = z0 / abs(z1);
	t_1 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	t_2 = (z2 + z2) - -0.5;
	t_3 = (tan((pi * t_2)) ^ 2.0) * t_0;
	tmp = 0.0;
	if (z2 <= -6.5e+15)
		tmp = t_1;
	elseif (z2 <= -8.5e-300)
		tmp = ((t_3 * (1.0 / abs(z1))) * z0) - -1.0;
	elseif (z2 <= 1.1e-227)
		tmp = ((((0.5 - (cos((((z2 + z2) - -1.5) * (pi + pi))) * 0.5)) * z0) / (((cos((t_2 * (pi + pi))) + 1.0) * 0.5) * abs(z1))) * t_0) - -1.0;
	elseif (z2 <= 0.0066)
		tmp = ((t_3 * -z0) * (-1.0 / abs(z1))) - -1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Tan[N[(Pi * t$95$2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[z2, -6.5e+15], t$95$1, If[LessEqual[z2, -8.5e-300], N[(N[(N[(t$95$3 * N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 1.1e-227], N[(N[(N[(N[(N[(0.5 - N[(N[Cos[N[(N[(N[(z2 + z2), $MachinePrecision] - -1.5), $MachinePrecision] * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] / N[(N[(N[(N[Cos[N[(t$95$2 * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + 1.0), $MachinePrecision] * 0.5), $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 0.0066], N[(N[(N[(t$95$3 * (-z0)), $MachinePrecision] * N[(-1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -6.5e15 or 0.0066 < z2

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

   if -6.5e15 < z2 < -8.4999999999999995e-300

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

   if -8.4999999999999995e-300 < z2 < 1.0999999999999999e-227

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-sin-a-rev N/A

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

      6. sqr-neg-rev N/A

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

      7. sin-+PI N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. sin-+PI N/A

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

      11. lift-PI.f64 N/A

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

      12. lift-+.f64 N/A

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

   7. Applied rewrites 44.6%

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

   9. Applied rewrites 43.3%

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

   if 1.0999999999999999e-227 < z2 < 0.0066

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

Alternative 3: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{z0}{\left|z1\right|}\\ t_1 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ t_2 := \tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)\\ \mathbf{if}\;z2 \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z2 \leq -8.5 \cdot 10^{-300}:\\ \;\;\;\;\left(\left({t\_2}^{2} \cdot t\_0\right) \cdot \frac{1}{\left|z1\right|}\right) \cdot z0 - -1\\ \mathbf{elif}\;z2 \leq 1.1 \cdot 10^{-227}:\\ \;\;\;\;t\_0 \cdot \frac{z0}{0} - -1\\ \mathbf{elif}\;z2 \leq 0.0066:\\ \;\;\;\;\frac{\frac{{\left(z0 \cdot t\_2\right)}^{2}}{\left|z1\right|}}{\left|z1\right|} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ z0 (fabs z1)))
       (t_1 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0))
       (t_2 (tan (* PI (- (+ z2 z2) -0.5)))))
  (if (<= z2 -6.5e+15)
    t_1
    (if (<= z2 -8.5e-300)
      (- (* (* (* (pow t_2 2.0) t_0) (/ 1.0 (fabs z1))) z0) -1.0)
      (if (<= z2 1.1e-227)
        (- (* t_0 (/ z0 0.0)) -1.0)
        (if (<= z2 0.0066)
          (- (/ (/ (pow (* z0 t_2) 2.0) (fabs z1)) (fabs z1)) -1.0)
          t_1))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = z0 / fabs(z1);
	double t_1 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double t_2 = tan((((double) M_PI) * ((z2 + z2) - -0.5)));
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_1;
	} else if (z2 <= -8.5e-300) {
		tmp = (((pow(t_2, 2.0) * t_0) * (1.0 / fabs(z1))) * z0) - -1.0;
	} else if (z2 <= 1.1e-227) {
		tmp = (t_0 * (z0 / 0.0)) - -1.0;
	} else if (z2 <= 0.0066) {
		tmp = ((pow((z0 * t_2), 2.0) / fabs(z1)) / fabs(z1)) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = z0 / Math.abs(z1);
	double t_1 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double t_2 = Math.tan((Math.PI * ((z2 + z2) - -0.5)));
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_1;
	} else if (z2 <= -8.5e-300) {
		tmp = (((Math.pow(t_2, 2.0) * t_0) * (1.0 / Math.abs(z1))) * z0) - -1.0;
	} else if (z2 <= 1.1e-227) {
		tmp = (t_0 * (z0 / 0.0)) - -1.0;
	} else if (z2 <= 0.0066) {
		tmp = ((Math.pow((z0 * t_2), 2.0) / Math.abs(z1)) / Math.abs(z1)) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = z0 / math.fabs(z1)
	t_1 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	t_2 = math.tan((math.pi * ((z2 + z2) - -0.5)))
	tmp = 0
	if z2 <= -6.5e+15:
		tmp = t_1
	elif z2 <= -8.5e-300:
		tmp = (((math.pow(t_2, 2.0) * t_0) * (1.0 / math.fabs(z1))) * z0) - -1.0
	elif z2 <= 1.1e-227:
		tmp = (t_0 * (z0 / 0.0)) - -1.0
	elif z2 <= 0.0066:
		tmp = ((math.pow((z0 * t_2), 2.0) / math.fabs(z1)) / math.fabs(z1)) - -1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(z0 / abs(z1))
	t_1 = Float64(Float64(Float64(Float64(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	t_2 = tan(Float64(pi * Float64(Float64(z2 + z2) - -0.5)))
	tmp = 0.0
	if (z2 <= -6.5e+15)
		tmp = t_1;
	elseif (z2 <= -8.5e-300)
		tmp = Float64(Float64(Float64(Float64((t_2 ^ 2.0) * t_0) * Float64(1.0 / abs(z1))) * z0) - -1.0);
	elseif (z2 <= 1.1e-227)
		tmp = Float64(Float64(t_0 * Float64(z0 / 0.0)) - -1.0);
	elseif (z2 <= 0.0066)
		tmp = Float64(Float64(Float64((Float64(z0 * t_2) ^ 2.0) / abs(z1)) / abs(z1)) - -1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = z0 / abs(z1);
	t_1 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	t_2 = tan((pi * ((z2 + z2) - -0.5)));
	tmp = 0.0;
	if (z2 <= -6.5e+15)
		tmp = t_1;
	elseif (z2 <= -8.5e-300)
		tmp = ((((t_2 ^ 2.0) * t_0) * (1.0 / abs(z1))) * z0) - -1.0;
	elseif (z2 <= 1.1e-227)
		tmp = (t_0 * (z0 / 0.0)) - -1.0;
	elseif (z2 <= 0.0066)
		tmp = ((((z0 * t_2) ^ 2.0) / abs(z1)) / abs(z1)) - -1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[(Pi * N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[z2, -6.5e+15], t$95$1, If[LessEqual[z2, -8.5e-300], N[(N[(N[(N[(N[Power[t$95$2, 2.0], $MachinePrecision] * t$95$0), $MachinePrecision] * N[(1.0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * z0), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 1.1e-227], N[(N[(t$95$0 * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 0.0066], N[(N[(N[(N[Power[N[(z0 * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z2 < -6.5e15 or 0.0066 < z2

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

   if -6.5e15 < z2 < -8.4999999999999995e-300

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. *-commutative N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

   if -8.4999999999999995e-300 < z2 < 1.0999999999999999e-227

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

   10. Applied rewrites 34.5%

       \[\leadsto \color{blue}{\frac{z0}{z1} \cdot \frac{z0}{0}} - -1 \]

   if 1.0999999999999999e-227 < z2 < 0.0066

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. associate-*r/ N/A

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

      7. lower-/.f64 N/A

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

   3. Applied rewrites 53.4%

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

Alternative 4: 82.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} t_0 := \left(-1.3333333333333333 \cdot \left(\pi \cdot \pi\right)\right) \cdot \pi\\ t_1 := \frac{\left|z0\right|}{\left|z1\right|}\\ t_2 := \tan \left(0.5 \cdot \pi\right)\\ t_3 := t\_2 \cdot t\_2\\ t_4 := 2 \cdot \left(\pi + t\_3 \cdot \pi\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-127}:\\ \;\;\;\;{\left(\frac{\left|z0\right| \cdot \left(\left(\left(\left(\pi + \pi\right) \cdot \left(t\_4 \cdot t\_2\right) + \left(t\_0 - \left(\left(\left(-2 \cdot \left(\pi \cdot \pi\right)\right) \cdot t\_4 - t\_0 \cdot t\_3\right) + \left(-4 \cdot \left(\pi \cdot \pi\right)\right) \cdot \left(t\_4 \cdot t\_3\right)\right)\right) \cdot z2\right) \cdot z2 + t\_4\right) \cdot z2 - \tan \left(\pi \cdot -0.5\right)\right)}{\left|z1\right|}\right)}^{2} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{\left|z0\right|}{0} - -1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (* -1.3333333333333333 (* PI PI)) PI))
       (t_1 (/ (fabs z0) (fabs z1)))
       (t_2 (tan (* 0.5 PI)))
       (t_3 (* t_2 t_2))
       (t_4 (* 2.0 (+ PI (* t_3 PI)))))
  (if (<= t_1 2e-127)
    (-
     (pow
      (/
       (*
        (fabs z0)
        (-
         (*
          (+
           (*
            (+
             (* (+ PI PI) (* t_4 t_2))
             (*
              (-
               t_0
               (+
                (- (* (* -2.0 (* PI PI)) t_4) (* t_0 t_3))
                (* (* -4.0 (* PI PI)) (* t_4 t_3))))
              z2))
            z2)
           t_4)
          z2)
         (tan (* PI -0.5))))
       (fabs z1))
      2.0)
     -1.0)
    (- (* t_1 (/ (fabs z0) 0.0)) -1.0))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (-1.3333333333333333 * (((double) M_PI) * ((double) M_PI))) * ((double) M_PI);
	double t_1 = fabs(z0) / fabs(z1);
	double t_2 = tan((0.5 * ((double) M_PI)));
	double t_3 = t_2 * t_2;
	double t_4 = 2.0 * (((double) M_PI) + (t_3 * ((double) M_PI)));
	double tmp;
	if (t_1 <= 2e-127) {
		tmp = pow(((fabs(z0) * (((((((((double) M_PI) + ((double) M_PI)) * (t_4 * t_2)) + ((t_0 - ((((-2.0 * (((double) M_PI) * ((double) M_PI))) * t_4) - (t_0 * t_3)) + ((-4.0 * (((double) M_PI) * ((double) M_PI))) * (t_4 * t_3)))) * z2)) * z2) + t_4) * z2) - tan((((double) M_PI) * -0.5)))) / fabs(z1)), 2.0) - -1.0;
	} else {
		tmp = (t_1 * (fabs(z0) / 0.0)) - -1.0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (-1.3333333333333333 * (Math.PI * Math.PI)) * Math.PI;
	double t_1 = Math.abs(z0) / Math.abs(z1);
	double t_2 = Math.tan((0.5 * Math.PI));
	double t_3 = t_2 * t_2;
	double t_4 = 2.0 * (Math.PI + (t_3 * Math.PI));
	double tmp;
	if (t_1 <= 2e-127) {
		tmp = Math.pow(((Math.abs(z0) * (((((((Math.PI + Math.PI) * (t_4 * t_2)) + ((t_0 - ((((-2.0 * (Math.PI * Math.PI)) * t_4) - (t_0 * t_3)) + ((-4.0 * (Math.PI * Math.PI)) * (t_4 * t_3)))) * z2)) * z2) + t_4) * z2) - Math.tan((Math.PI * -0.5)))) / Math.abs(z1)), 2.0) - -1.0;
	} else {
		tmp = (t_1 * (Math.abs(z0) / 0.0)) - -1.0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (-1.3333333333333333 * (math.pi * math.pi)) * math.pi
	t_1 = math.fabs(z0) / math.fabs(z1)
	t_2 = math.tan((0.5 * math.pi))
	t_3 = t_2 * t_2
	t_4 = 2.0 * (math.pi + (t_3 * math.pi))
	tmp = 0
	if t_1 <= 2e-127:
		tmp = math.pow(((math.fabs(z0) * (((((((math.pi + math.pi) * (t_4 * t_2)) + ((t_0 - ((((-2.0 * (math.pi * math.pi)) * t_4) - (t_0 * t_3)) + ((-4.0 * (math.pi * math.pi)) * (t_4 * t_3)))) * z2)) * z2) + t_4) * z2) - math.tan((math.pi * -0.5)))) / math.fabs(z1)), 2.0) - -1.0
	else:
		tmp = (t_1 * (math.fabs(z0) / 0.0)) - -1.0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(-1.3333333333333333 * Float64(pi * pi)) * pi)
	t_1 = Float64(abs(z0) / abs(z1))
	t_2 = tan(Float64(0.5 * pi))
	t_3 = Float64(t_2 * t_2)
	t_4 = Float64(2.0 * Float64(pi + Float64(t_3 * pi)))
	tmp = 0.0
	if (t_1 <= 2e-127)
		tmp = Float64((Float64(Float64(abs(z0) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(pi + pi) * Float64(t_4 * t_2)) + Float64(Float64(t_0 - Float64(Float64(Float64(Float64(-2.0 * Float64(pi * pi)) * t_4) - Float64(t_0 * t_3)) + Float64(Float64(-4.0 * Float64(pi * pi)) * Float64(t_4 * t_3)))) * z2)) * z2) + t_4) * z2) - tan(Float64(pi * -0.5)))) / abs(z1)) ^ 2.0) - -1.0);
	else
		tmp = Float64(Float64(t_1 * Float64(abs(z0) / 0.0)) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (-1.3333333333333333 * (pi * pi)) * pi;
	t_1 = abs(z0) / abs(z1);
	t_2 = tan((0.5 * pi));
	t_3 = t_2 * t_2;
	t_4 = 2.0 * (pi + (t_3 * pi));
	tmp = 0.0;
	if (t_1 <= 2e-127)
		tmp = (((abs(z0) * (((((((pi + pi) * (t_4 * t_2)) + ((t_0 - ((((-2.0 * (pi * pi)) * t_4) - (t_0 * t_3)) + ((-4.0 * (pi * pi)) * (t_4 * t_3)))) * z2)) * z2) + t_4) * z2) - tan((pi * -0.5)))) / abs(z1)) ^ 2.0) - -1.0;
	else
		tmp = (t_1 * (abs(z0) / 0.0)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(-1.3333333333333333 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Tan[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(2.0 * N[(Pi + N[(t$95$3 * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-127], 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$2), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 - N[(N[(N[(N[(-2.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision] - N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(N[(-4.0 * N[(Pi * Pi), $MachinePrecision]), $MachinePrecision] * N[(t$95$4 * t$95$3), $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] / N[Abs[z1], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision], N[(N[(t$95$1 * N[(N[Abs[z0], $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 z0 z1) < 2.0000000000000001e-127

   1. Initial program 47.2%

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

   2. Taylor expanded in z2 around 0

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

   4. Applied rewrites 70.7%

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

   6. Applied rewrites 75.6%

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

   if 2.0000000000000001e-127 < (/.f64 z0 z1)

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

   10. Applied rewrites 34.5%

       \[\leadsto \color{blue}{\frac{z0}{z1} \cdot \frac{z0}{0}} - -1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ \mathbf{if}\;z2 \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;{\left(\tan \left(\left(z2 - \left(-0.5 - z2\right)\right) \cdot \pi\right) \cdot \frac{z0}{\left|z1\right|}\right)}^{2} - -1\\ \mathbf{elif}\;z2 \leq 0.007:\\ \;\;\;\;\frac{{\left(z0 \cdot \tan \left(\pi \cdot 0.5\right)\right)}^{2}}{\left|z1\right| \cdot \left|z1\right|} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0)))
  (if (<= z2 -6.5e+15)
    t_0
    (if (<= z2 -3.6e-97)
      (-
       (pow (* (tan (* (- z2 (- -0.5 z2)) PI)) (/ z0 (fabs z1))) 2.0)
       -1.0)
      (if (<= z2 0.007)
        (-
         (/ (pow (* z0 (tan (* PI 0.5))) 2.0) (* (fabs z1) (fabs z1)))
         -1.0)
        t_0)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_0;
	} else if (z2 <= -3.6e-97) {
		tmp = pow((tan(((z2 - (-0.5 - z2)) * ((double) M_PI))) * (z0 / fabs(z1))), 2.0) - -1.0;
	} else if (z2 <= 0.007) {
		tmp = (pow((z0 * tan((((double) M_PI) * 0.5))), 2.0) / (fabs(z1) * fabs(z1))) - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_0;
	} else if (z2 <= -3.6e-97) {
		tmp = Math.pow((Math.tan(((z2 - (-0.5 - z2)) * Math.PI)) * (z0 / Math.abs(z1))), 2.0) - -1.0;
	} else if (z2 <= 0.007) {
		tmp = (Math.pow((z0 * Math.tan((Math.PI * 0.5))), 2.0) / (Math.abs(z1) * Math.abs(z1))) - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	tmp = 0
	if z2 <= -6.5e+15:
		tmp = t_0
	elif z2 <= -3.6e-97:
		tmp = math.pow((math.tan(((z2 - (-0.5 - z2)) * math.pi)) * (z0 / math.fabs(z1))), 2.0) - -1.0
	elif z2 <= 0.007:
		tmp = (math.pow((z0 * math.tan((math.pi * 0.5))), 2.0) / (math.fabs(z1) * math.fabs(z1))) - -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(Float64(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	tmp = 0.0
	if (z2 <= -6.5e+15)
		tmp = t_0;
	elseif (z2 <= -3.6e-97)
		tmp = Float64((Float64(tan(Float64(Float64(z2 - Float64(-0.5 - z2)) * pi)) * Float64(z0 / abs(z1))) ^ 2.0) - -1.0);
	elseif (z2 <= 0.007)
		tmp = Float64(Float64((Float64(z0 * tan(Float64(pi * 0.5))) ^ 2.0) / Float64(abs(z1) * abs(z1))) - -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	tmp = 0.0;
	if (z2 <= -6.5e+15)
		tmp = t_0;
	elseif (z2 <= -3.6e-97)
		tmp = ((tan(((z2 - (-0.5 - z2)) * pi)) * (z0 / abs(z1))) ^ 2.0) - -1.0;
	elseif (z2 <= 0.007)
		tmp = (((z0 * tan((pi * 0.5))) ^ 2.0) / (abs(z1) * abs(z1))) - -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[z2, -6.5e+15], t$95$0, If[LessEqual[z2, -3.6e-97], N[(N[Power[N[(N[Tan[N[(N[(z2 - N[(-0.5 - z2), $MachinePrecision]), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 0.007], N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -6.5e15 or 0.0070000000000000001 < z2

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

   if -6.5e15 < z2 < -3.6e-97

   1. Initial program 47.2%

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

      1. lift--.f64 N/A

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

      2. sub-negate-rev N/A

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

      3. lift-+.f64 N/A

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

      4. associate--r+ N/A

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

      5. sub-negate N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 47.2%

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

   3. Applied rewrites 47.2%

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

   if -3.6e-97 < z2 < 0.0070000000000000001

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 50.7%

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

   4. Taylor expanded in z2 around 0

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

   6. Applied rewrites 51.5%

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

Alternative 6: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ \mathbf{if}\;z2 \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq -3.6 \cdot 10^{-97}:\\ \;\;\;\;{\left(z0 \cdot \frac{\tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)}{\left|z1\right|}\right)}^{2} - -1\\ \mathbf{elif}\;z2 \leq 0.007:\\ \;\;\;\;\frac{{\left(z0 \cdot \tan \left(\pi \cdot 0.5\right)\right)}^{2}}{\left|z1\right| \cdot \left|z1\right|} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0)))
  (if (<= z2 -6.5e+15)
    t_0
    (if (<= z2 -3.6e-97)
      (-
       (pow (* z0 (/ (tan (* PI (- (+ z2 z2) -0.5))) (fabs z1))) 2.0)
       -1.0)
      (if (<= z2 0.007)
        (-
         (/ (pow (* z0 (tan (* PI 0.5))) 2.0) (* (fabs z1) (fabs z1)))
         -1.0)
        t_0)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_0;
	} else if (z2 <= -3.6e-97) {
		tmp = pow((z0 * (tan((((double) M_PI) * ((z2 + z2) - -0.5))) / fabs(z1))), 2.0) - -1.0;
	} else if (z2 <= 0.007) {
		tmp = (pow((z0 * tan((((double) M_PI) * 0.5))), 2.0) / (fabs(z1) * fabs(z1))) - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double tmp;
	if (z2 <= -6.5e+15) {
		tmp = t_0;
	} else if (z2 <= -3.6e-97) {
		tmp = Math.pow((z0 * (Math.tan((Math.PI * ((z2 + z2) - -0.5))) / Math.abs(z1))), 2.0) - -1.0;
	} else if (z2 <= 0.007) {
		tmp = (Math.pow((z0 * Math.tan((Math.PI * 0.5))), 2.0) / (Math.abs(z1) * Math.abs(z1))) - -1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	tmp = 0
	if z2 <= -6.5e+15:
		tmp = t_0
	elif z2 <= -3.6e-97:
		tmp = math.pow((z0 * (math.tan((math.pi * ((z2 + z2) - -0.5))) / math.fabs(z1))), 2.0) - -1.0
	elif z2 <= 0.007:
		tmp = (math.pow((z0 * math.tan((math.pi * 0.5))), 2.0) / (math.fabs(z1) * math.fabs(z1))) - -1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(Float64(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	tmp = 0.0
	if (z2 <= -6.5e+15)
		tmp = t_0;
	elseif (z2 <= -3.6e-97)
		tmp = Float64((Float64(z0 * Float64(tan(Float64(pi * Float64(Float64(z2 + z2) - -0.5))) / abs(z1))) ^ 2.0) - -1.0);
	elseif (z2 <= 0.007)
		tmp = Float64(Float64((Float64(z0 * tan(Float64(pi * 0.5))) ^ 2.0) / Float64(abs(z1) * abs(z1))) - -1.0);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	tmp = 0.0;
	if (z2 <= -6.5e+15)
		tmp = t_0;
	elseif (z2 <= -3.6e-97)
		tmp = ((z0 * (tan((pi * ((z2 + z2) - -0.5))) / abs(z1))) ^ 2.0) - -1.0;
	elseif (z2 <= 0.007)
		tmp = (((z0 * tan((pi * 0.5))) ^ 2.0) / (abs(z1) * abs(z1))) - -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[z2, -6.5e+15], t$95$0, If[LessEqual[z2, -3.6e-97], N[(N[Power[N[(z0 * N[(N[Tan[N[(Pi * N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 0.007], N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -6.5e15 or 0.0070000000000000001 < z2

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

   if -6.5e15 < z2 < -3.6e-97

   1. Initial program 47.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 47.5%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 47.5%

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

   3. Applied rewrites 47.5%

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

   if -3.6e-97 < z2 < 0.0070000000000000001

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 50.7%

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

   4. Taylor expanded in z2 around 0

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

   6. Applied rewrites 51.5%

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

Alternative 7: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ t_1 := \frac{z0}{\left|z1\right|}\\ t_2 := {\left(\tan \left(2.5 \cdot \pi\right) \cdot t\_1\right)}^{2} - -1\\ \mathbf{if}\;z2 \leq -64000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z2 \leq -8.5 \cdot 10^{-300}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z2 \leq 1.2 \cdot 10^{-227}:\\ \;\;\;\;t\_1 \cdot \frac{z0}{0} - -1\\ \mathbf{elif}\;z2 \leq 0.0066:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0))
       (t_1 (/ z0 (fabs z1)))
       (t_2 (- (pow (* (tan (* 2.5 PI)) t_1) 2.0) -1.0)))
  (if (<= z2 -64000000.0)
    t_0
    (if (<= z2 -8.5e-300)
      t_2
      (if (<= z2 1.2e-227)
        (- (* t_1 (/ z0 0.0)) -1.0)
        (if (<= z2 0.0066) t_2 t_0))))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double t_1 = z0 / fabs(z1);
	double t_2 = pow((tan((2.5 * ((double) M_PI))) * t_1), 2.0) - -1.0;
	double tmp;
	if (z2 <= -64000000.0) {
		tmp = t_0;
	} else if (z2 <= -8.5e-300) {
		tmp = t_2;
	} else if (z2 <= 1.2e-227) {
		tmp = (t_1 * (z0 / 0.0)) - -1.0;
	} else if (z2 <= 0.0066) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double t_1 = z0 / Math.abs(z1);
	double t_2 = Math.pow((Math.tan((2.5 * Math.PI)) * t_1), 2.0) - -1.0;
	double tmp;
	if (z2 <= -64000000.0) {
		tmp = t_0;
	} else if (z2 <= -8.5e-300) {
		tmp = t_2;
	} else if (z2 <= 1.2e-227) {
		tmp = (t_1 * (z0 / 0.0)) - -1.0;
	} else if (z2 <= 0.0066) {
		tmp = t_2;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	t_1 = z0 / math.fabs(z1)
	t_2 = math.pow((math.tan((2.5 * math.pi)) * t_1), 2.0) - -1.0
	tmp = 0
	if z2 <= -64000000.0:
		tmp = t_0
	elif z2 <= -8.5e-300:
		tmp = t_2
	elif z2 <= 1.2e-227:
		tmp = (t_1 * (z0 / 0.0)) - -1.0
	elif z2 <= 0.0066:
		tmp = t_2
	else:
		tmp = t_0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(Float64(Float64(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	t_1 = Float64(z0 / abs(z1))
	t_2 = Float64((Float64(tan(Float64(2.5 * pi)) * t_1) ^ 2.0) - -1.0)
	tmp = 0.0
	if (z2 <= -64000000.0)
		tmp = t_0;
	elseif (z2 <= -8.5e-300)
		tmp = t_2;
	elseif (z2 <= 1.2e-227)
		tmp = Float64(Float64(t_1 * Float64(z0 / 0.0)) - -1.0);
	elseif (z2 <= 0.0066)
		tmp = t_2;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	t_1 = z0 / abs(z1);
	t_2 = ((tan((2.5 * pi)) * t_1) ^ 2.0) - -1.0;
	tmp = 0.0;
	if (z2 <= -64000000.0)
		tmp = t_0;
	elseif (z2 <= -8.5e-300)
		tmp = t_2;
	elseif (z2 <= 1.2e-227)
		tmp = (t_1 * (z0 / 0.0)) - -1.0;
	elseif (z2 <= 0.0066)
		tmp = t_2;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(N[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(z0 / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(N[Tan[N[(2.5 * Pi), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[z2, -64000000.0], t$95$0, If[LessEqual[z2, -8.5e-300], t$95$2, If[LessEqual[z2, 1.2e-227], N[(N[(t$95$1 * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[z2, 0.0066], t$95$2, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z2 < -6.4e7 or 0.0066 < z2

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

   if -6.4e7 < z2 < -8.4999999999999995e-300 or 1.2e-227 < z2 < 0.0066

   1. Initial program 47.2%

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

      1. lift-tan.f64 N/A

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

      2. tan-+PI-rev N/A

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

      3. tan-+PI-rev N/A

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

      4. lower-tan.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-PI.f64 N/A

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

      7. distribute-lft1-in N/A

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

      8. lift-PI.f64 N/A

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

      9. distribute-lft1-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. lift--.f64 N/A

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

      13. associate-+l- N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval 47.2%

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

   3. Applied rewrites 47.2%

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

   4. Taylor expanded in z2 around 0

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

   6. Applied rewrites 47.6%

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

   if -8.4999999999999995e-300 < z2 < 1.2e-227

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

   10. Applied rewrites 34.5%

       \[\leadsto \color{blue}{\frac{z0}{z1} \cdot \frac{z0}{0}} - -1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(z2 + z2\right) - -0.5\\ t_1 := t\_0 \cdot \pi\\ t_2 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;{\left(z0 \cdot \tan \left(\pi \cdot t\_0\right)\right)}^{2} \cdot \frac{1}{\left|z1\right| \cdot \left|z1\right|} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (+ z2 z2) -0.5))
       (t_1 (* t_0 PI))
       (t_2 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0)))
  (if (<= t_1 -5e+16)
    t_2
    (if (<= t_1 2.0)
      (-
       (*
        (pow (* z0 (tan (* PI t_0))) 2.0)
        (/ 1.0 (* (fabs z1) (fabs z1))))
       -1.0)
      t_2))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double t_1 = t_0 * ((double) M_PI);
	double t_2 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double tmp;
	if (t_1 <= -5e+16) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = (pow((z0 * tan((((double) M_PI) * t_0))), 2.0) * (1.0 / (fabs(z1) * fabs(z1)))) - -1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double t_1 = t_0 * Math.PI;
	double t_2 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double tmp;
	if (t_1 <= -5e+16) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = (Math.pow((z0 * Math.tan((Math.PI * t_0))), 2.0) * (1.0 / (Math.abs(z1) * Math.abs(z1)))) - -1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (z2 + z2) - -0.5
	t_1 = t_0 * math.pi
	t_2 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	tmp = 0
	if t_1 <= -5e+16:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = (math.pow((z0 * math.tan((math.pi * t_0))), 2.0) * (1.0 / (math.fabs(z1) * math.fabs(z1)))) - -1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(z2 + z2) - -0.5)
	t_1 = Float64(t_0 * pi)
	t_2 = Float64(Float64(Float64(Float64(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	tmp = 0.0
	if (t_1 <= -5e+16)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64((Float64(z0 * tan(Float64(pi * t_0))) ^ 2.0) * Float64(1.0 / Float64(abs(z1) * abs(z1)))) - -1.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (z2 + z2) - -0.5;
	t_1 = t_0 * pi;
	t_2 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	tmp = 0.0;
	if (t_1 <= -5e+16)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = (((z0 * tan((pi * t_0))) ^ 2.0) * (1.0 / (abs(z1) * abs(z1)))) - -1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+16], t$95$2, If[LessEqual[t$95$1, 2.0], N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], t$95$2]]]]]

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 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

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

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. frac-times N/A

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

      10. mult-flip N/A

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

      11. lower-*.f64 N/A

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

   3. Applied rewrites 51.0%

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

Alternative 9: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(z2 + z2\right) - -0.5\\ t_1 := t\_0 \cdot \pi\\ t_2 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+16}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{{\left(z0 \cdot \tan \left(\pi \cdot t\_0\right)\right)}^{2}}{\left|z1\right| \cdot \left|z1\right|} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (- (+ z2 z2) -0.5))
       (t_1 (* t_0 PI))
       (t_2 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0)))
  (if (<= t_1 -5e+16)
    t_2
    (if (<= t_1 2.0)
      (-
       (/ (pow (* z0 (tan (* PI t_0))) 2.0) (* (fabs z1) (fabs z1)))
       -1.0)
      t_2))))

C

double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double t_1 = t_0 * ((double) M_PI);
	double t_2 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double tmp;
	if (t_1 <= -5e+16) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = (pow((z0 * tan((((double) M_PI) * t_0))), 2.0) / (fabs(z1) * fabs(z1))) - -1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = (z2 + z2) - -0.5;
	double t_1 = t_0 * Math.PI;
	double t_2 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double tmp;
	if (t_1 <= -5e+16) {
		tmp = t_2;
	} else if (t_1 <= 2.0) {
		tmp = (Math.pow((z0 * Math.tan((Math.PI * t_0))), 2.0) / (Math.abs(z1) * Math.abs(z1))) - -1.0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = (z2 + z2) - -0.5
	t_1 = t_0 * math.pi
	t_2 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	tmp = 0
	if t_1 <= -5e+16:
		tmp = t_2
	elif t_1 <= 2.0:
		tmp = (math.pow((z0 * math.tan((math.pi * t_0))), 2.0) / (math.fabs(z1) * math.fabs(z1))) - -1.0
	else:
		tmp = t_2
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(Float64(z2 + z2) - -0.5)
	t_1 = Float64(t_0 * pi)
	t_2 = Float64(Float64(Float64(Float64(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	tmp = 0.0
	if (t_1 <= -5e+16)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = Float64(Float64((Float64(z0 * tan(Float64(pi * t_0))) ^ 2.0) / Float64(abs(z1) * abs(z1))) - -1.0);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = (z2 + z2) - -0.5;
	t_1 = t_0 * pi;
	t_2 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	tmp = 0.0;
	if (t_1 <= -5e+16)
		tmp = t_2;
	elseif (t_1 <= 2.0)
		tmp = (((z0 * tan((pi * t_0))) ^ 2.0) / (abs(z1) * abs(z1))) - -1.0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * Pi), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+16], t$95$2, If[LessEqual[t$95$1, 2.0], N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision], t$95$2]]]]]

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 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

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

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 50.7%

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

Alternative 10: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\\ t_1 := \frac{\left(-z0\right) \cdot \frac{z0}{0}}{-\left|z1\right|} - -1\\ \mathbf{if}\;t\_0 \leq -200000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\frac{{\left(z0 \cdot \tan \left(\pi \cdot 0.5\right)\right)}^{2}}{\left|z1\right| \cdot \left|z1\right|} - -1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (* (- (+ z2 z2) -0.5) PI))
       (t_1 (- (/ (* (- z0) (/ z0 0.0)) (- (fabs z1))) -1.0)))
  (if (<= t_0 -200000000.0)
    t_1
    (if (<= t_0 2.0)
      (-
       (/ (pow (* z0 (tan (* PI 0.5))) 2.0) (* (fabs z1) (fabs z1)))
       -1.0)
      t_1))))

C

double code(double z2, double z0, double z1) {
	double t_0 = ((z2 + z2) - -0.5) * ((double) M_PI);
	double t_1 = ((-z0 * (z0 / 0.0)) / -fabs(z1)) - -1.0;
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = (pow((z0 * tan((((double) M_PI) * 0.5))), 2.0) / (fabs(z1) * fabs(z1))) - -1.0;
	} 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 = ((-z0 * (z0 / 0.0)) / -Math.abs(z1)) - -1.0;
	double tmp;
	if (t_0 <= -200000000.0) {
		tmp = t_1;
	} else if (t_0 <= 2.0) {
		tmp = (Math.pow((z0 * Math.tan((Math.PI * 0.5))), 2.0) / (Math.abs(z1) * Math.abs(z1))) - -1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = ((z2 + z2) - -0.5) * math.pi
	t_1 = ((-z0 * (z0 / 0.0)) / -math.fabs(z1)) - -1.0
	tmp = 0
	if t_0 <= -200000000.0:
		tmp = t_1
	elif t_0 <= 2.0:
		tmp = (math.pow((z0 * math.tan((math.pi * 0.5))), 2.0) / (math.fabs(z1) * math.fabs(z1))) - -1.0
	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(-z0) * Float64(z0 / 0.0)) / Float64(-abs(z1))) - -1.0)
	tmp = 0.0
	if (t_0 <= -200000000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = Float64(Float64((Float64(z0 * tan(Float64(pi * 0.5))) ^ 2.0) / Float64(abs(z1) * abs(z1))) - -1.0);
	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 = ((-z0 * (z0 / 0.0)) / -abs(z1)) - -1.0;
	tmp = 0.0;
	if (t_0 <= -200000000.0)
		tmp = t_1;
	elseif (t_0 <= 2.0)
		tmp = (((z0 * tan((pi * 0.5))) ^ 2.0) / (abs(z1) * abs(z1))) - -1.0;
	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[((-z0) * N[(z0 / 0.0), $MachinePrecision]), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -200000000.0], t$95$1, If[LessEqual[t$95$0, 2.0], N[(N[(N[Power[N[(z0 * N[Tan[N[(Pi * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Abs[z1], $MachinePrecision] * N[Abs[z1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $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)) < -2e8 or 2 < (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]

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

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*r/ N/A

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

      9. frac-times N/A

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

      10. lower-/.f64 N/A

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

   3. Applied rewrites 50.7%

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

   4. Taylor expanded in z2 around 0

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

   6. Applied rewrites 51.5%

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

Alternative 11: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|z0\right|}{\left|z1\right|}\\ t_1 := t\_0 \cdot \frac{\left|z0\right|}{0} - -1\\ t_2 := \tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot t\_0\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-305}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ (fabs z0) (fabs z1)))
       (t_1 (- (* t_0 (/ (fabs z0) 0.0)) -1.0))
       (t_2 (* (tan (* (- (+ z2 z2) -0.5) PI)) t_0)))
  (if (<= t_2 -1e-305) t_1 (if (<= t_2 2e-171) 1.0 t_1))))

C

double code(double z2, double z0, double z1) {
	double t_0 = fabs(z0) / fabs(z1);
	double t_1 = (t_0 * (fabs(z0) / 0.0)) - -1.0;
	double t_2 = tan((((z2 + z2) - -0.5) * ((double) M_PI))) * t_0;
	double tmp;
	if (t_2 <= -1e-305) {
		tmp = t_1;
	} else if (t_2 <= 2e-171) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.abs(z0) / Math.abs(z1);
	double t_1 = (t_0 * (Math.abs(z0) / 0.0)) - -1.0;
	double t_2 = Math.tan((((z2 + z2) - -0.5) * Math.PI)) * t_0;
	double tmp;
	if (t_2 <= -1e-305) {
		tmp = t_1;
	} else if (t_2 <= 2e-171) {
		tmp = 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.fabs(z0) / math.fabs(z1)
	t_1 = (t_0 * (math.fabs(z0) / 0.0)) - -1.0
	t_2 = math.tan((((z2 + z2) - -0.5) * math.pi)) * t_0
	tmp = 0
	if t_2 <= -1e-305:
		tmp = t_1
	elif t_2 <= 2e-171:
		tmp = 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(abs(z0) / abs(z1))
	t_1 = Float64(Float64(t_0 * Float64(abs(z0) / 0.0)) - -1.0)
	t_2 = Float64(tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) * t_0)
	tmp = 0.0
	if (t_2 <= -1e-305)
		tmp = t_1;
	elseif (t_2 <= 2e-171)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = abs(z0) / abs(z1);
	t_1 = (t_0 * (abs(z0) / 0.0)) - -1.0;
	t_2 = tan((((z2 + z2) - -0.5) * pi)) * t_0;
	tmp = 0.0;
	if (t_2 <= -1e-305)
		tmp = t_1;
	elseif (t_2 <= 2e-171)
		tmp = 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[(N[Abs[z0], $MachinePrecision] / 0.0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-305], t$95$1, If[LessEqual[t$95$2, 2e-171], 1.0, t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < -1e-305 or 2e-171 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1))

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in z2 around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-PI.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-PI.f64 74.3%

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

   8. Applied rewrites 74.3%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

   10. Applied rewrites 34.5%

       \[\leadsto \color{blue}{\frac{z0}{z1} \cdot \frac{z0}{0}} - -1 \]

   if -1e-305 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < 2e-171

   1. Initial program 47.2%

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

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 18.6%

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

Alternative 12: 75.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{\left|z0\right|}{\left|z1\right|}\\ t_1 := \tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot t\_0\\ t_2 := \frac{\left|z0\right|}{0}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-305}:\\ \;\;\;\;t\_0 \cdot t\_2 - -1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-171}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-\left|z0\right|\right) \cdot t\_2}{-\left|z1\right|} - -1\\ \end{array} \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  (let* ((t_0 (/ (fabs z0) (fabs z1)))
       (t_1 (* (tan (* (- (+ z2 z2) -0.5) PI)) t_0))
       (t_2 (/ (fabs z0) 0.0)))
  (if (<= t_1 -1e-305)
    (- (* t_0 t_2) -1.0)
    (if (<= t_1 2e-171)
      1.0
      (- (/ (* (- (fabs z0)) t_2) (- (fabs z1))) -1.0)))))

C

double code(double z2, double z0, double z1) {
	double t_0 = fabs(z0) / fabs(z1);
	double t_1 = tan((((z2 + z2) - -0.5) * ((double) M_PI))) * t_0;
	double t_2 = fabs(z0) / 0.0;
	double tmp;
	if (t_1 <= -1e-305) {
		tmp = (t_0 * t_2) - -1.0;
	} else if (t_1 <= 2e-171) {
		tmp = 1.0;
	} else {
		tmp = ((-fabs(z0) * t_2) / -fabs(z1)) - -1.0;
	}
	return tmp;
}

Java

public static double code(double z2, double z0, double z1) {
	double t_0 = Math.abs(z0) / Math.abs(z1);
	double t_1 = Math.tan((((z2 + z2) - -0.5) * Math.PI)) * t_0;
	double t_2 = Math.abs(z0) / 0.0;
	double tmp;
	if (t_1 <= -1e-305) {
		tmp = (t_0 * t_2) - -1.0;
	} else if (t_1 <= 2e-171) {
		tmp = 1.0;
	} else {
		tmp = ((-Math.abs(z0) * t_2) / -Math.abs(z1)) - -1.0;
	}
	return tmp;
}

Python

def code(z2, z0, z1):
	t_0 = math.fabs(z0) / math.fabs(z1)
	t_1 = math.tan((((z2 + z2) - -0.5) * math.pi)) * t_0
	t_2 = math.fabs(z0) / 0.0
	tmp = 0
	if t_1 <= -1e-305:
		tmp = (t_0 * t_2) - -1.0
	elif t_1 <= 2e-171:
		tmp = 1.0
	else:
		tmp = ((-math.fabs(z0) * t_2) / -math.fabs(z1)) - -1.0
	return tmp

Julia

function code(z2, z0, z1)
	t_0 = Float64(abs(z0) / abs(z1))
	t_1 = Float64(tan(Float64(Float64(Float64(z2 + z2) - -0.5) * pi)) * t_0)
	t_2 = Float64(abs(z0) / 0.0)
	tmp = 0.0
	if (t_1 <= -1e-305)
		tmp = Float64(Float64(t_0 * t_2) - -1.0);
	elseif (t_1 <= 2e-171)
		tmp = 1.0;
	else
		tmp = Float64(Float64(Float64(Float64(-abs(z0)) * t_2) / Float64(-abs(z1))) - -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z2, z0, z1)
	t_0 = abs(z0) / abs(z1);
	t_1 = tan((((z2 + z2) - -0.5) * pi)) * t_0;
	t_2 = abs(z0) / 0.0;
	tmp = 0.0;
	if (t_1 <= -1e-305)
		tmp = (t_0 * t_2) - -1.0;
	elseif (t_1 <= 2e-171)
		tmp = 1.0;
	else
		tmp = ((-abs(z0) * t_2) / -abs(z1)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z2_, z0_, z1_] := Block[{t$95$0 = N[(N[Abs[z0], $MachinePrecision] / N[Abs[z1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Tan[N[(N[(N[(z2 + z2), $MachinePrecision] - -0.5), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[z0], $MachinePrecision] / 0.0), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-305], N[(N[(t$95$0 * t$95$2), $MachinePrecision] - -1.0), $MachinePrecision], If[LessEqual[t$95$1, 2e-171], 1.0, N[(N[(N[((-N[Abs[z0], $MachinePrecision]) * t$95$2), $MachinePrecision] / (-N[Abs[z1], $MachinePrecision])), $MachinePrecision] - -1.0), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < -1e-305

   1. Initial program 47.2%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. mult-flip N/A

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

      9. associate-*r* N/A

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

      10. lower-*.f64 N/A

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

   3. Applied rewrites 50.0%

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

      1. lift-pow.f64 N/A

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

      2. unpow2 N/A

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

      3. lift-tan.f64 N/A

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

      4. tan-quot N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. frac-times N/A

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

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right) \cdot \sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}{\cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right) \cdot \cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}} \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

   5. Applied rewrites 44.6%

      \[\leadsto \left(\left(\color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)\right)}} \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

   6. Taylor expanded in z2 around 0

      \[\leadsto \left(\color{blue}{\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)}} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\color{blue}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\color{blue}{z1} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      6. lower-PI.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)}\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \mathsf{PI}\left(\right)}\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      10. lower-cos.f64 N/A

          \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      11. lower-PI.f64 74.3%

          \[\leadsto \left(\frac{z0 \cdot \left(0.5 - 0.5 \cdot \cos \pi\right)}{z1 \cdot \left(0.5 + 0.5 \cdot \cos \pi\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

   8. Applied rewrites 74.3%

      \[\leadsto \left(\color{blue}{\frac{z0 \cdot \left(0.5 - 0.5 \cdot \cos \pi\right)}{z1 \cdot \left(0.5 + 0.5 \cdot \cos \pi\right)}} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1}} - -1 \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(-z0\right)\right)} \cdot \frac{-1}{z1} - -1 \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(\left(-z0\right) \cdot \frac{-1}{z1}\right)} - -1 \]

   10. Applied rewrites 34.5%

       \[\leadsto \color{blue}{\frac{z0}{z1} \cdot \frac{z0}{0}} - -1 \]

   if -1e-305 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1)) < 2e-171

   1. Initial program 47.2%

      \[{\left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)}^{2} - -1 \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \color{blue}{1} \]
   3. Step-by-step derivation

   4. Applied rewrites 18.6%

      \[\leadsto \color{blue}{1} \]

   if 2e-171 < (*.f64 (tan.f64 (*.f64 (-.f64 (+.f64 z2 z2) #s(literal -1/2 binary64)) (PI.f64))) (/.f64 z0 z1))

   1. Initial program 47.2%

      \[{\left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)}^{2} - -1 \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)}^{2}} - -1 \]

      2. unpow2 N/A

         \[\leadsto \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} - -1 \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)} - -1 \]

      4. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right) \cdot \tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right)\right) \cdot \frac{z0}{z1}} - -1 \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)\right)} \cdot \frac{z0}{z1} - -1 \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)\right) \cdot \color{blue}{\frac{z0}{z1}} - -1 \]

      7. frac-2neg N/A

         \[\leadsto \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(z0\right)}{\mathsf{neg}\left(z1\right)}} - -1 \]

      8. mult-flip N/A

         \[\leadsto \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(z0\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z1\right)}\right)} - -1 \]

      9. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)\right) \cdot \left(\mathsf{neg}\left(z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z1\right)}} - -1 \]

      10. lower-*.f64 N/A

          \[\leadsto \color{blue}{\left(\left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \left(\tan \left(\left(\left(z2 + z2\right) - \frac{-1}{2}\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)\right) \cdot \left(\mathsf{neg}\left(z0\right)\right)\right) \cdot \frac{1}{\mathsf{neg}\left(z1\right)}} - -1 \]

   3. Applied rewrites 50.0%

      \[\leadsto \color{blue}{\left(\left({\tan \left(\pi \cdot \left(\left(z2 + z2\right) - -0.5\right)\right)}^{2} \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1}} - -1 \]
   4. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(\left(\color{blue}{{\tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}^{2}} \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      2. unpow2 N/A

         \[\leadsto \left(\left(\color{blue}{\left(\tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right) \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)\right)} \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(\left(\left(\color{blue}{\tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)} \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)\right) \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      4. tan-quot N/A

         \[\leadsto \left(\left(\left(\color{blue}{\frac{\sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}{\cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}} \cdot \tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)\right) \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      5. lift-tan.f64 N/A

         \[\leadsto \left(\left(\left(\frac{\sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}{\cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)} \cdot \color{blue}{\tan \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}\right) \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      6. tan-quot N/A

         \[\leadsto \left(\left(\left(\frac{\sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}{\cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)} \cdot \color{blue}{\frac{\sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}{\cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}}\right) \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      7. frac-times N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right) \cdot \sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}{\cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right) \cdot \cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}} \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\color{blue}{\frac{\sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right) \cdot \sin \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}{\cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right) \cdot \cos \left(\pi \cdot \left(\left(z2 + z2\right) - \frac{-1}{2}\right)\right)}} \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

   5. Applied rewrites 44.6%

      \[\leadsto \left(\left(\color{blue}{\frac{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)\right)}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right)\right)}} \cdot \frac{z0}{z1}\right) \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

   6. Taylor expanded in z2 around 0

      \[\leadsto \left(\color{blue}{\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)}} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\color{blue}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{\color{blue}{z1} \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      3. lower--.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      4. lower-*.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      5. lower-cos.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      6. lower-PI.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)}} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      8. lower-+.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \color{blue}{\frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)}\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \color{blue}{\cos \mathsf{PI}\left(\right)}\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      10. lower-cos.f64 N/A

          \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \mathsf{PI}\left(\right)\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

      11. lower-PI.f64 74.3%

          \[\leadsto \left(\frac{z0 \cdot \left(0.5 - 0.5 \cdot \cos \pi\right)}{z1 \cdot \left(0.5 + 0.5 \cdot \cos \pi\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]

   8. Applied rewrites 74.3%

      \[\leadsto \left(\color{blue}{\frac{z0 \cdot \left(0.5 - 0.5 \cdot \cos \pi\right)}{z1 \cdot \left(0.5 + 0.5 \cdot \cos \pi\right)}} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1} - -1 \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(-z0\right)\right) \cdot \frac{-1}{z1}} - -1 \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(-z0\right)\right) \cdot \color{blue}{\frac{-1}{z1}} - -1 \]

      3. frac-2neg N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(-z0\right)\right) \cdot \color{blue}{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(z1\right)}} - -1 \]

      4. metadata-eval N/A

         \[\leadsto \left(\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(-z0\right)\right) \cdot \frac{\color{blue}{1}}{\mathsf{neg}\left(z1\right)} - -1 \]

      5. mult-flip-rev N/A

         \[\leadsto \color{blue}{\frac{\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(-z0\right)}{\mathsf{neg}\left(z1\right)}} - -1 \]

      6. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{z0 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \pi\right)}{z1 \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos \pi\right)} \cdot \left(-z0\right)}{\mathsf{neg}\left(z1\right)}} - -1 \]

   10. Applied rewrites 37.3%

       \[\leadsto \color{blue}{\frac{\left(-z0\right) \cdot \frac{z0}{0}}{-z1}} - -1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 18.6% accurate, 232.0× speedup

Math

\[1 \]

FPCore

(FPCore (z2 z0 z1)
  :precision binary64
  1.0)

C

double code(double z2, double z0, double z1) {
	return 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z2, z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z2
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    code = 1.0d0
end function

Java

public static double code(double z2, double z0, double z1) {
	return 1.0;
}

Python

def code(z2, z0, z1):
	return 1.0

Julia

function code(z2, z0, z1)
	return 1.0
end

MATLAB

function tmp = code(z2, z0, z1)
	tmp = 1.0;
end

Wolfram

code[z2_, z0_, z1_] := 1.0

Derivation

1. Initial program 47.2%

   \[{\left(\tan \left(\left(\left(z2 + z2\right) - -0.5\right) \cdot \pi\right) \cdot \frac{z0}{z1}\right)}^{2} - -1 \]

2. Taylor expanded in z0 around 0

   \[\leadsto \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 18.6%

   \[\leadsto \color{blue}{1} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z2 z0 z1)
  :name "(- (pow (* (tan (* (- (+ z2 z2) -1/2) PI)) (/ z0 z1)) 2) -1)"
  :precision binary64
  (- (pow (* (tan (* (- (+ z2 z2) -0.5) PI)) (/ z0 z1)) 2.0) -1.0))