(cos (* (+ PI PI) z0))

Percentage Accurate: 57.5% → 98.5%

Time: 1.9s

Alternatives: 4

Speedup: 109.0×

Specification

Math

\[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 57.5% accurate, 1.0× speedup

Math

\[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\sin \left(\pi \cdot \left|z0\right|\right) \cdot \cos \left(\pi \cdot \left(-0.5 - \left|z0\right|\right)\right) + \left(0.5 + 0.5 \cdot \sin \left(\pi \cdot \left(\left|z0\right| + \left(\left|z0\right| - -0.5\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= (fabs z0) 85000.0)
  (+
   (* (sin (* PI (fabs z0))) (cos (* PI (- -0.5 (fabs z0)))))
   (+ 0.5 (* 0.5 (sin (* PI (+ (fabs z0) (- (fabs z0) -0.5)))))))
  1.0))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = (sin((((double) M_PI) * fabs(z0))) * cos((((double) M_PI) * (-0.5 - fabs(z0))))) + (0.5 + (0.5 * sin((((double) M_PI) * (fabs(z0) + (fabs(z0) - -0.5))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = (Math.sin((Math.PI * Math.abs(z0))) * Math.cos((Math.PI * (-0.5 - Math.abs(z0))))) + (0.5 + (0.5 * Math.sin((Math.PI * (Math.abs(z0) + (Math.abs(z0) - -0.5))))));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = (math.sin((math.pi * math.fabs(z0))) * math.cos((math.pi * (-0.5 - math.fabs(z0))))) + (0.5 + (0.5 * math.sin((math.pi * (math.fabs(z0) + (math.fabs(z0) - -0.5))))))
	else:
		tmp = 1.0
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = Float64(Float64(sin(Float64(pi * abs(z0))) * cos(Float64(pi * Float64(-0.5 - abs(z0))))) + Float64(0.5 + Float64(0.5 * sin(Float64(pi * Float64(abs(z0) + Float64(abs(z0) - -0.5)))))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = (sin((pi * abs(z0))) * cos((pi * (-0.5 - abs(z0))))) + (0.5 + (0.5 * sin((pi * (abs(z0) + (abs(z0) - -0.5))))));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[(N[(N[Sin[N[(Pi * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(Pi * N[(-0.5 - N[Abs[z0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(0.5 + N[(0.5 * N[Sin[N[(Pi * N[(N[Abs[z0], $MachinePrecision] + N[(N[Abs[z0], $MachinePrecision] - -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-rgt-in N/A

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

      7. associate-+l+ N/A

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

      8. sin-sum N/A

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

      9. sin-+PI/2-rev N/A

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

      10. lower-+.f64 N/A

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

   3. Applied rewrites 59.6%

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

      1. lift-cos.f64 N/A

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

      2. sin-+PI/2-rev N/A

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

      3. lower-sin.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-PI.f64 N/A

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

      10. mult-flip N/A

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

      11. metadata-eval N/A

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

      12. *-commutative N/A

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

      13. lift-*.f64 N/A

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

      14. associate-+l+ N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lift-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. distribute-rgt-out N/A

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

      20. distribute-lft-out N/A

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

      21. lower-*.f64 N/A

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

   5. Applied rewrites 58.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 58.7%

         \[\leadsto \sin \color{blue}{\left(\pi \cdot z0\right)} \cdot \cos \left(z0 \cdot \pi + 0.5 \cdot \pi\right) + \left(0.5 + 0.5 \cdot \sin \left(\pi \cdot \left(z0 + \left(z0 - -0.5\right)\right)\right)\right) \]

      4. lift-cos.f64 N/A

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

      5. cos-neg-rev N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. distribute-rgt-out N/A

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

      11. metadata-eval N/A

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

      12. sub-flip N/A

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

      13. lift--.f64 N/A

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

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

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

      15. lower-*.f64 N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 58.7%

          \[\leadsto \sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot \color{blue}{\left(-0.5 - z0\right)}\right) + \left(0.5 + 0.5 \cdot \sin \left(\pi \cdot \left(z0 + \left(z0 - -0.5\right)\right)\right)\right) \]

   7. Applied rewrites 58.7%

      \[\leadsto \color{blue}{\sin \left(\pi \cdot z0\right) \cdot \cos \left(\pi \cdot \left(-0.5 - z0\right)\right)} + \left(0.5 + 0.5 \cdot \sin \left(\pi \cdot \left(z0 + \left(z0 - -0.5\right)\right)\right)\right) \]

   if 85000 < z0

   1. Initial program 57.5%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 2: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\sin \left(\left(-2 \cdot \pi\right) \cdot \left|z0\right| - -1.5707963267948966\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= (fabs z0) 85000.0)
  (sin (- (* (* -2.0 PI) (fabs z0)) -1.5707963267948966))
  1.0))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = sin((((-2.0 * ((double) M_PI)) * fabs(z0)) - -1.5707963267948966));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = Math.sin((((-2.0 * Math.PI) * Math.abs(z0)) - -1.5707963267948966));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = math.sin((((-2.0 * math.pi) * math.fabs(z0)) - -1.5707963267948966))
	else:
		tmp = 1.0
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = sin(Float64(Float64(Float64(-2.0 * pi) * abs(z0)) - -1.5707963267948966));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = sin((((-2.0 * pi) * abs(z0)) - -1.5707963267948966));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[Sin[N[(N[(N[(-2.0 * Pi), $MachinePrecision] * N[Abs[z0], $MachinePrecision]), $MachinePrecision] - -1.5707963267948966), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]
   2. Step-by-step derivation

      1. lift-cos.f64 N/A

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

      2. cos-neg-rev N/A

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

      3. sin-+PI/2-rev N/A

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

      4. lower-sin.f64 N/A

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

      5. add-flip N/A

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

      6. lower--.f64 N/A

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

      7. lift-*.f64 N/A

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

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

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

      9. lower-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. count-2 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-PI.f64 N/A

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

      16. mult-flip N/A

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

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

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

   3. Applied rewrites 57.5%

      \[\leadsto \color{blue}{\sin \left(\left(-2 \cdot \pi\right) \cdot z0 - \pi \cdot -0.5\right)} \]

   4. Evaluated real constant 57.5%

      \[\leadsto \sin \left(\left(-2 \cdot \pi\right) \cdot z0 - \color{blue}{-1.5707963267948966}\right) \]

   if 85000 < z0

   1. Initial program 57.5%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 3: 98.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|z0\right| \leq 85000:\\ \;\;\;\;\cos \left(6.283185307179586 \cdot \left|z0\right|\right)\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (z0)
  :precision binary64
  (if (<= (fabs z0) 85000.0) (cos (* 6.283185307179586 (fabs z0))) 1.0))

C

double code(double z0) {
	double tmp;
	if (fabs(z0) <= 85000.0) {
		tmp = cos((6.283185307179586 * fabs(z0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8) :: tmp
    if (abs(z0) <= 85000.0d0) then
        tmp = cos((6.283185307179586d0 * abs(z0)))
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double z0) {
	double tmp;
	if (Math.abs(z0) <= 85000.0) {
		tmp = Math.cos((6.283185307179586 * Math.abs(z0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(z0):
	tmp = 0
	if math.fabs(z0) <= 85000.0:
		tmp = math.cos((6.283185307179586 * math.fabs(z0)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(z0)
	tmp = 0.0
	if (abs(z0) <= 85000.0)
		tmp = cos(Float64(6.283185307179586 * abs(z0)));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0)
	tmp = 0.0;
	if (abs(z0) <= 85000.0)
		tmp = cos((6.283185307179586 * abs(z0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_] := If[LessEqual[N[Abs[z0], $MachinePrecision], 85000.0], N[Cos[N[(6.283185307179586 * N[Abs[z0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if z0 < 85000

   1. Initial program 57.5%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]

   2. Evaluated real constant 57.5%

      \[\leadsto \cos \left(\color{blue}{6.283185307179586} \cdot z0\right) \]

   if 85000 < z0

   1. Initial program 57.5%

      \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]

   2. Taylor expanded in z0 around 0

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

   4. Applied rewrites 97.2%

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

Alternative 4: 97.2% accurate, 109.0× speedup

Math

\[1 \]

FPCore

(FPCore (z0)
  :precision binary64
  1.0)

C

double code(double z0) {
	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(z0)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    code = 1.0d0
end function

Java

public static double code(double z0) {
	return 1.0;
}

Python

def code(z0):
	return 1.0

Julia

function code(z0)
	return 1.0
end

MATLAB

function tmp = code(z0)
	tmp = 1.0;
end

Wolfram

code[z0_] := 1.0

Derivation

1. Initial program 57.5%

   \[\cos \left(\left(\pi + \pi\right) \cdot z0\right) \]

2. Taylor expanded in z0 around 0

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

4. Applied rewrites 97.2%

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

Reproduce

herbie shell --seed 2025250 
(FPCore (z0)
  :name "(cos (* (+ PI PI) z0))"
  :precision binary64
  (cos (* (+ PI PI) z0)))