(tan (- (* (+ PI PI) z0) (* -1/2 PI)))

Percentage Accurate: 8.8% → 97.5%

Time: 11.7s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. z0 = 1.5400569355970817e+39

Specification

Math

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

FPCore

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

C

double code(double z0) {
	return tan((((((double) M_PI) + ((double) M_PI)) * z0) - (-0.5 * ((double) M_PI))));
}

Java

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

Python

def code(z0):
	return math.tan((((math.pi + math.pi) * z0) - (-0.5 * math.pi)))

Julia

function code(z0)
	return tan(Float64(Float64(Float64(pi + pi) * z0) - Float64(-0.5 * pi)))
end

MATLAB

function tmp = code(z0)
	tmp = tan((((pi + pi) * z0) - (-0.5 * pi)));
end

Wolfram

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

Initial Program: 8.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double z0) {
	return tan((((((double) M_PI) + ((double) M_PI)) * z0) - (-0.5 * ((double) M_PI))));
}

Java

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

Python

def code(z0):
	return math.tan((((math.pi + math.pi) * z0) - (-0.5 * math.pi)))

Julia

function code(z0)
	return tan(Float64(Float64(Float64(pi + pi) * z0) - Float64(-0.5 * pi)))
end

MATLAB

function tmp = code(z0)
	tmp = tan((((pi + pi) * z0) - (-0.5 * pi)));
end

Wolfram

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

Alternative 1: 97.5% accurate, 0.5× speedup

Math

\[\frac{\cos \left(\left(z0 + z0\right) \cdot \pi + \pi\right)}{\sin \left(z0 \cdot \left(\pi + \pi\right)\right)} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

3. Applied rewrites 97.5%

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

   1. lift-neg.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-+PI-rev N/A

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

   4. lower-cos.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lower-+.f64 97.5%

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

   7. lift-*.f64 N/A

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

   8. lift-+.f64 N/A

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

   9. count-2 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lower-*.f64 97.5%

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

   14. lift-*.f64 N/A

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

   15. count-2-rev N/A

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

   16. lower-+.f64 97.5%

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

5. Applied rewrites 97.5%

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

Alternative 2: 97.5% accurate, 0.6× speedup

Math

\[{\tan \left(\left(-z0\right) \cdot \left(\pi + \pi\right)\right)}^{-1} \]

FPCore

(FPCore (z0)
  :precision binary64
  (pow (tan (* (- z0) (+ PI PI))) -1))

C

double code(double z0) {
	return pow(tan((-z0 * (((double) M_PI) + ((double) M_PI)))), -1.0);
}

Java

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

Python

def code(z0):
	return math.pow(math.tan((-z0 * (math.pi + math.pi))), -1.0)

Julia

function code(z0)
	return tan(Float64(Float64(-z0) * Float64(pi + pi))) ^ -1.0
end

MATLAB

function tmp = code(z0)
	tmp = tan((-z0 * (pi + pi))) ^ -1.0;
end

Wolfram

code[z0_] := N[Power[N[Tan[N[((-z0) * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1], $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

3. Applied rewrites 97.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

   5. lift-sin.f64 N/A

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

   6. sin-neg-rev N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. count-2 N/A

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

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

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

   14. metadata-eval N/A

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

   15. lift-*.f64 N/A

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

5. Applied rewrites 97.5%

   \[\leadsto \color{blue}{{\tan \left(\left(-z0\right) \cdot \left(\pi + \pi\right)\right)}^{-1}} \]
6. Add Preprocessing

Alternative 3: 97.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double z0) {
	return 1.0 / tan((-z0 * (((double) M_PI) + ((double) M_PI))));
}

Java

public static double code(double z0) {
	return 1.0 / Math.tan((-z0 * (Math.PI + Math.PI)));
}

Python

def code(z0):
	return 1.0 / math.tan((-z0 * (math.pi + math.pi)))

Julia

function code(z0)
	return Float64(1.0 / tan(Float64(Float64(-z0) * Float64(pi + pi))))
end

MATLAB

function tmp = code(z0)
	tmp = 1.0 / tan((-z0 * (pi + pi)));
end

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

3. Applied rewrites 97.5%

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

   1. lift-/.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. distribute-frac-neg N/A

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

   4. distribute-neg-frac2 N/A

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

   5. lift-sin.f64 N/A

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

   6. sin-neg-rev N/A

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

   7. lift-*.f64 N/A

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

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

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-+.f64 N/A

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

   12. count-2 N/A

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

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

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

   14. metadata-eval N/A

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

   15. lift-*.f64 N/A

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

5. Applied rewrites 97.5%

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

Alternative 4: 10.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq \frac{-290953239129259}{12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424}:\\ \;\;\;\;\tan \left(\left(\left(\left(z0 + z0\right) - \frac{-1}{2}\right) + 1\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(-\left(\left(\left(\frac{-1}{2} - z0\right) - z0\right) + 1\right) \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<=
     z0
     -290953239129259/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424)
  (tan (* (+ (- (+ z0 z0) -1/2) 1) PI))
  (tan (- (* (+ (- (- -1/2 z0) z0) 1) PI)))))

C

double code(double z0) {
	double tmp;
	if (z0 <= -2.3e-308) {
		tmp = tan(((((z0 + z0) - -0.5) + 1.0) * ((double) M_PI)));
	} else {
		tmp = tan(-((((-0.5 - z0) - z0) + 1.0) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (z0 <= -2.3e-308) {
		tmp = Math.tan(((((z0 + z0) - -0.5) + 1.0) * Math.PI));
	} else {
		tmp = Math.tan(-((((-0.5 - z0) - z0) + 1.0) * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if z0 <= -2.3e-308:
		tmp = math.tan(((((z0 + z0) - -0.5) + 1.0) * math.pi))
	else:
		tmp = math.tan(-((((-0.5 - z0) - z0) + 1.0) * math.pi))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (z0 <= -2.3e-308)
		tmp = tan(Float64(Float64(Float64(Float64(z0 + z0) - -0.5) + 1.0) * pi));
	else
		tmp = tan(Float64(-Float64(Float64(Float64(Float64(-0.5 - z0) - z0) + 1.0) * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (z0 <= -2.3e-308)
		tmp = tan(((((z0 + z0) - -0.5) + 1.0) * pi));
	else
		tmp = tan(-((((-0.5 - z0) - z0) + 1.0) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[z0, -290953239129259/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424], N[Tan[N[(N[(N[(N[(z0 + z0), $MachinePrecision] - -1/2), $MachinePrecision] + 1), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], N[Tan[(-N[(N[(N[(N[(-1/2 - z0), $MachinePrecision] - z0), $MachinePrecision] + 1), $MachinePrecision] * Pi), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < -2.2999999999999999e-308

   1. Initial program 8.8%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 97.5%

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

   5. Applied rewrites 8.6%

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

   if -2.2999999999999999e-308 < z0

   1. Initial program 8.8%

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

      1. lift-tan.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. tan-neg N/A

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

      5. tan-+PI-rev N/A

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

      6. tan-neg-rev N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-+.f64 N/A

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

   3. Applied rewrites 8.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. count-2-rev N/A

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

      4. associate--r+ N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 8.7%

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

   5. Applied rewrites 8.7%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 8.7%

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

   7. Applied rewrites 8.7%

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

Alternative 5: 10.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq \frac{-290953239129259}{12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424}:\\ \;\;\;\;\tan \left(\left(\left(\left(z0 + z0\right) - \frac{-1}{2}\right) + 1\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(-\left(\left(\frac{-1}{2} - \left(z0 + z0\right)\right) + 1\right) \cdot \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<=
     z0
     -290953239129259/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424)
  (tan (* (+ (- (+ z0 z0) -1/2) 1) PI))
  (tan (- (* (+ (- -1/2 (+ z0 z0)) 1) PI)))))

C

double code(double z0) {
	double tmp;
	if (z0 <= -2.3e-308) {
		tmp = tan(((((z0 + z0) - -0.5) + 1.0) * ((double) M_PI)));
	} else {
		tmp = tan(-(((-0.5 - (z0 + z0)) + 1.0) * ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (z0 <= -2.3e-308) {
		tmp = Math.tan(((((z0 + z0) - -0.5) + 1.0) * Math.PI));
	} else {
		tmp = Math.tan(-(((-0.5 - (z0 + z0)) + 1.0) * Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if z0 <= -2.3e-308:
		tmp = math.tan(((((z0 + z0) - -0.5) + 1.0) * math.pi))
	else:
		tmp = math.tan(-(((-0.5 - (z0 + z0)) + 1.0) * math.pi))
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (z0 <= -2.3e-308)
		tmp = tan(Float64(Float64(Float64(Float64(z0 + z0) - -0.5) + 1.0) * pi));
	else
		tmp = tan(Float64(-Float64(Float64(Float64(-0.5 - Float64(z0 + z0)) + 1.0) * pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (z0 <= -2.3e-308)
		tmp = tan(((((z0 + z0) - -0.5) + 1.0) * pi));
	else
		tmp = tan(-(((-0.5 - (z0 + z0)) + 1.0) * pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[z0, -290953239129259/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424], N[Tan[N[(N[(N[(N[(z0 + z0), $MachinePrecision] - -1/2), $MachinePrecision] + 1), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], N[Tan[(-N[(N[(N[(-1/2 - N[(z0 + z0), $MachinePrecision]), $MachinePrecision] + 1), $MachinePrecision] * Pi), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z0 < -2.2999999999999999e-308

   1. Initial program 8.8%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 97.5%

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

   5. Applied rewrites 8.6%

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

   if -2.2999999999999999e-308 < z0

   1. Initial program 8.8%

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

      1. lift-tan.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. tan-neg N/A

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

      5. tan-+PI-rev N/A

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

      6. tan-neg-rev N/A

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

      7. lower-tan.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. lower-+.f64 N/A

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

   3. Applied rewrites 8.8%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft1-in N/A

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

      5. lower-*.f64 N/A

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

      6. lower-+.f64 8.7%

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

      7. lift-*.f64 N/A

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

      8. count-2-rev N/A

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

      9. lower-+.f64 8.7%

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

   5. Applied rewrites 8.7%

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

Alternative 6: 10.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \left(z0 + z0\right) - \frac{-1}{2}\\ \mathbf{if}\;z0 \leq \frac{-290953239129259}{12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424}:\\ \;\;\;\;\tan \left(\left(t\_0 + 1\right) \cdot \pi\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(t\_0 \cdot \pi - \pi\right)\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (let* ((t_0 (- (+ z0 z0) -1/2)))
  (if (<=
       z0
       -290953239129259/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424)
    (tan (* (+ t_0 1) PI))
    (tan (- (* t_0 PI) PI)))))

C

double code(double z0) {
	double t_0 = (z0 + z0) - -0.5;
	double tmp;
	if (z0 <= -2.3e-308) {
		tmp = tan(((t_0 + 1.0) * ((double) M_PI)));
	} else {
		tmp = tan(((t_0 * ((double) M_PI)) - ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double z0) {
	double t_0 = (z0 + z0) - -0.5;
	double tmp;
	if (z0 <= -2.3e-308) {
		tmp = Math.tan(((t_0 + 1.0) * Math.PI));
	} else {
		tmp = Math.tan(((t_0 * Math.PI) - Math.PI));
	}
	return tmp;
}

Python

def code(z0):
	t_0 = (z0 + z0) - -0.5
	tmp = 0
	if z0 <= -2.3e-308:
		tmp = math.tan(((t_0 + 1.0) * math.pi))
	else:
		tmp = math.tan(((t_0 * math.pi) - math.pi))
	return tmp

Julia

function code(z0)
	t_0 = Float64(Float64(z0 + z0) - -0.5)
	tmp = 0.0
	if (z0 <= -2.3e-308)
		tmp = tan(Float64(Float64(t_0 + 1.0) * pi));
	else
		tmp = tan(Float64(Float64(t_0 * pi) - pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	t_0 = (z0 + z0) - -0.5;
	tmp = 0.0;
	if (z0 <= -2.3e-308)
		tmp = tan(((t_0 + 1.0) * pi));
	else
		tmp = tan(((t_0 * pi) - pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := Block[{t$95$0 = N[(N[(z0 + z0), $MachinePrecision] - -1/2), $MachinePrecision]}, If[LessEqual[z0, -290953239129259/12650140831706913647030959169932331690597290610258882397306334876714396222999709180747523981339820280949192366519800744461863046086612092304188337496296156870094839017285397585279181733880826021327485479904546566785125467714043293663631459728072472271300628532022423097020838413451906408261645469290375391456731733818343424], N[Tan[N[(N[(t$95$0 + 1), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], N[Tan[N[(N[(t$95$0 * Pi), $MachinePrecision] - Pi), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -2.2999999999999999e-308

   1. Initial program 8.8%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 97.5%

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

   5. Applied rewrites 8.6%

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

   if -2.2999999999999999e-308 < z0

   1. Initial program 8.8%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. frac-2neg N/A

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

      4. lower-/.f64 N/A

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

   3. Applied rewrites 97.5%

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

   5. Applied rewrites 8.8%

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

Alternative 7: 8.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double z0) {
	return tan((((z0 + z0) - -0.5) * ((double) M_PI)));
}

Java

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

Python

def code(z0):
	return math.tan((((z0 + z0) - -0.5) * math.pi))

Julia

function code(z0)
	return tan(Float64(Float64(Float64(z0 + z0) - -0.5) * pi))
end

MATLAB

function tmp = code(z0)
	tmp = tan((((z0 + z0) - -0.5) * pi));
end

Wolfram

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

Derivation

1. Initial program 8.8%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

3. Applied rewrites 97.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-neg.f64 N/A

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

   4. remove-double-neg N/A

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

5. Applied rewrites 8.8%

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

Alternative 8: 8.6% accurate, 1.0× speedup

Math

\[\tan \left(\left(\left(z0 + z0\right) - \frac{-5}{2}\right) \cdot \pi\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (tan (* (- (+ z0 z0) -5/2) PI)))

C

double code(double z0) {
	return tan((((z0 + z0) - -2.5) * ((double) M_PI)));
}

Java

public static double code(double z0) {
	return Math.tan((((z0 + z0) - -2.5) * Math.PI));
}

Python

def code(z0):
	return math.tan((((z0 + z0) - -2.5) * math.pi))

Julia

function code(z0)
	return tan(Float64(Float64(Float64(z0 + z0) - -2.5) * pi))
end

MATLAB

function tmp = code(z0)
	tmp = tan((((z0 + z0) - -2.5) * pi));
end

Wolfram

code[z0_] := N[Tan[N[(N[(N[(z0 + z0), $MachinePrecision] - -5/2), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 8.8%

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

   1. lift-tan.f64 N/A

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

   2. tan-quot N/A

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

   3. frac-2neg N/A

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

   4. lower-/.f64 N/A

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

3. Applied rewrites 97.5%

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

5. Applied rewrites 8.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 8.6%

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

   4. lift-+.f64 N/A

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

   5. lift--.f64 N/A

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

   6. associate-+l- N/A

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

   7. lower--.f64 N/A

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

   8. metadata-eval 8.6%

      \[\leadsto \tan \left(\left(\left(z0 + z0\right) - \color{blue}{\frac{-5}{2}}\right) \cdot \pi\right) \]

7. Applied rewrites 8.6%

   \[\leadsto \tan \color{blue}{\left(\left(\left(z0 + z0\right) - \frac{-5}{2}\right) \cdot \pi\right)} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0)
  :name "(tan (- (* (+ PI PI) z0) (* -1/2 PI)))"
  :precision binary64
  (tan (- (* (+ PI PI) z0) (* -1/2 PI))))