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

Percentage Accurate: 37.0% → 99.0%

Time: 13.6s

Alternatives: 2

Speedup: 1.0×

• could not determine a ground truth

  1. z0 = 6.933255579494087e+50
  2. z1 = 1.4239788943817417e+126
  3. z2 = 6.179446916570151e-97

Specification

Math

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

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (atan (* (tan (- (* z0 (+ PI PI)) (* -1/2 PI))) (/ z1 z2))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z0_, z1_, z2_] := N[ArcTan[N[(N[Tan[N[(N[(z0 * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] - N[(-1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(z1 / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Initial Program: 37.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (atan (* (tan (- (* z0 (+ PI PI)) (* -1/2 PI))) (/ z1 z2))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z0_, z1_, z2_] := N[ArcTan[N[(N[Tan[N[(N[(z0 * N[(Pi + Pi), $MachinePrecision]), $MachinePrecision] - N[(-1/2 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(z1 / z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\tan^{-1} \left(\frac{1}{z2} \cdot \frac{z1}{\tan \left(\left(-2 \cdot z0 - 0\right) \cdot \pi\right)}\right) \]

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (atan (* (/ 1 z2) (/ z1 (tan (* (- (* -2 z0) 0) PI))))))

C

double code(double z0, double z1, double z2) {
	return atan(((1.0 / z2) * (z1 / tan((((-2.0 * z0) - 0.0) * ((double) M_PI))))));
}

Java

public static double code(double z0, double z1, double z2) {
	return Math.atan(((1.0 / z2) * (z1 / Math.tan((((-2.0 * z0) - 0.0) * Math.PI)))));
}

Python

def code(z0, z1, z2):
	return math.atan(((1.0 / z2) * (z1 / math.tan((((-2.0 * z0) - 0.0) * math.pi)))))

Julia

function code(z0, z1, z2)
	return atan(Float64(Float64(1.0 / z2) * Float64(z1 / tan(Float64(Float64(Float64(-2.0 * z0) - 0.0) * pi)))))
end

MATLAB

function tmp = code(z0, z1, z2)
	tmp = atan(((1.0 / z2) * (z1 / tan((((-2.0 * z0) - 0.0) * pi)))));
end

Wolfram

code[z0_, z1_, z2_] := N[ArcTan[N[(N[(1 / z2), $MachinePrecision] * N[(z1 / N[Tan[N[(N[(N[(-2 * z0), $MachinePrecision] - 0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 37.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. mult-flip N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 37.1%

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

   10. lift--.f64 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. +-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lift-+.f64 N/A

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

   16. count-2 N/A

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

3. Applied rewrites 37.0%

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

5. Applied rewrites 36.8%

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

7. Applied rewrites 99.0%

   \[\leadsto \tan^{-1} \left(\frac{1}{z2} \cdot \color{blue}{\frac{z1}{\tan \left(\left(-2 \cdot z0 - 0\right) \cdot \pi\right)}}\right) \]
8. Add Preprocessing

Alternative 2: 99.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (z0 z1 z2)
  :precision binary64
  (atan (/ z1 (* (tan (* (* PI z0) -2)) z2))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[z0_, z1_, z2_] := N[ArcTan[N[(z1 / N[(N[Tan[N[(N[(Pi * z0), $MachinePrecision] * -2), $MachinePrecision]], $MachinePrecision] * z2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 37.0%

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

   1. lift-tan.f64 N/A

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

   2. tan-+PI-rev N/A

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

   3. tan-+PI-rev N/A

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

   4. lower-tan.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. associate-+l+ N/A

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

   8. lift-+.f64 N/A

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

   9. lower-+.f64 36.8%

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

3. Applied rewrites 36.8%

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

5. Applied rewrites 99.0%

   \[\leadsto \tan^{-1} \color{blue}{\left(\frac{z1}{\tan \left(\left(-2 \cdot z0 - 0\right) \cdot \pi\right) \cdot z2}\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{z1}{\tan \color{blue}{\left(\left(-2 \cdot z0 - 0\right) \cdot \pi\right)} \cdot z2}\right) \]

   2. lift--.f64 N/A

      \[\leadsto \tan^{-1} \left(\frac{z1}{\tan \left(\color{blue}{\left(-2 \cdot z0 - 0\right)} \cdot \pi\right) \cdot z2}\right) \]

   3. --rgt-identity N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 99.0%

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

7. Applied rewrites 99.0%

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

Reproduce

herbie shell --seed 2025277 -o generate:taylor -o generate:evaluate
(FPCore (z0 z1 z2)
  :name "(atan (* (tan (- (* z0 (+ PI PI)) (* -1/2 PI))) (/ z1 z2)))"
  :precision binary64
  (atan (* (tan (- (* z0 (+ PI PI)) (* -1/2 PI))) (/ z1 z2))))