(sin (* (+ PI PI) z0))

Percentage Accurate: 51.5% → 51.5%

Time: 57.9s

Alternatives: 4

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 51.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (z0)
  :precision binary64
  (* 2 (* (sin (* z0 PI)) (sin (- (* 1/2 PI) (fabs (* PI z0)))))))

C

double code(double z0) {
	return 2.0 * (sin((z0 * ((double) M_PI))) * sin(((0.5 * ((double) M_PI)) - fabs((((double) M_PI) * z0)))));
}

Java

public static double code(double z0) {
	return 2.0 * (Math.sin((z0 * Math.PI)) * Math.sin(((0.5 * Math.PI) - Math.abs((Math.PI * z0)))));
}

Python

def code(z0):
	return 2.0 * (math.sin((z0 * math.pi)) * math.sin(((0.5 * math.pi) - math.fabs((math.pi * z0)))))

Julia

function code(z0)
	return Float64(2.0 * Float64(sin(Float64(z0 * pi)) * sin(Float64(Float64(0.5 * pi) - abs(Float64(pi * z0))))))
end

MATLAB

function tmp = code(z0)
	tmp = 2.0 * (sin((z0 * pi)) * sin(((0.5 * pi) - abs((pi * z0)))));
end

Wolfram

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

Derivation

1. Initial program 51.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2 N/A

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

   5. associate-*l* N/A

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

   6. sin-2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 51.5%

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

3. Applied rewrites 51.5%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-fabs-rev N/A

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

   3. cos-neg-rev N/A

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

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

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

   5. lower-sin.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-neg.f64 N/A

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

   8. lower-fabs.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. mult-flip N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval 51.5%

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

5. Applied rewrites 51.5%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. metadata-eval N/A

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

   5. mult-flip-rev N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-neg.f64 N/A

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

   8. sub-flip-reverse N/A

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

   9. lower--.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. mult-flip-rev N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 51.5%

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

7. Applied rewrites 51.5%

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

Alternative 2: 51.5% accurate, 0.3× speedup

Math

\[\mathsf{copysign}\left(1, z0\right) \cdot \left(2 \cdot \left(\sin \left(\left|z0\right| \cdot \pi\right) \cdot \sin \left(\pi \cdot \left(\left(-\left|z0\right|\right) + \frac{1}{2}\right)\right)\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (*
 (copysign 1 z0)
 (* 2 (* (sin (* (fabs z0) PI)) (sin (* PI (+ (- (fabs z0)) 1/2)))))))

C

double code(double z0) {
	return copysign(1.0, z0) * (2.0 * (sin((fabs(z0) * ((double) M_PI))) * sin((((double) M_PI) * (-fabs(z0) + 0.5)))));
}

Java

public static double code(double z0) {
	return Math.copySign(1.0, z0) * (2.0 * (Math.sin((Math.abs(z0) * Math.PI)) * Math.sin((Math.PI * (-Math.abs(z0) + 0.5)))));
}

Python

def code(z0):
	return math.copysign(1.0, z0) * (2.0 * (math.sin((math.fabs(z0) * math.pi)) * math.sin((math.pi * (-math.fabs(z0) + 0.5)))))

Julia

function code(z0)
	return Float64(copysign(1.0, z0) * Float64(2.0 * Float64(sin(Float64(abs(z0) * pi)) * sin(Float64(pi * Float64(Float64(-abs(z0)) + 0.5))))))
end

MATLAB

function tmp = code(z0)
	tmp = (sign(z0) * abs(1.0)) * (2.0 * (sin((abs(z0) * pi)) * sin((pi * (-abs(z0) + 0.5)))));
end

Wolfram

code[z0_] := N[(N[With[{TMP1 = Abs[1], TMP2 = Sign[z0]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * N[(2 * N[(N[Sin[N[(N[Abs[z0], $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(Pi * N[((-N[Abs[z0], $MachinePrecision]) + 1/2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2 N/A

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

   5. associate-*l* N/A

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

   6. sin-2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 51.5%

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

3. Applied rewrites 51.5%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]
4. Step-by-step derivation

   1. lift-cos.f64 N/A

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

   2. cos-neg-rev N/A

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

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

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

   4. lower-sin.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. lift-PI.f64 N/A

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

   9. mult-flip N/A

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

   10. distribute-lft-out N/A

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

   11. lower-*.f64 N/A

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

   12. lower-+.f64 N/A

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

   13. lower-neg.f64 N/A

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

   14. metadata-eval 51.5%

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

5. Applied rewrites 51.5%

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

Alternative 3: 51.5% accurate, 0.5× speedup

Math

\[2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right) \]

FPCore

(FPCore (z0)
  :precision binary64
  (* 2 (* (sin (* z0 PI)) (cos (* z0 PI)))))

C

double code(double z0) {
	return 2.0 * (sin((z0 * ((double) M_PI))) * cos((z0 * ((double) M_PI))));
}

Java

public static double code(double z0) {
	return 2.0 * (Math.sin((z0 * Math.PI)) * Math.cos((z0 * Math.PI)));
}

Python

def code(z0):
	return 2.0 * (math.sin((z0 * math.pi)) * math.cos((z0 * math.pi)))

Julia

function code(z0)
	return Float64(2.0 * Float64(sin(Float64(z0 * pi)) * cos(Float64(z0 * pi))))
end

MATLAB

function tmp = code(z0)
	tmp = 2.0 * (sin((z0 * pi)) * cos((z0 * pi)));
end

Wolfram

code[z0_] := N[(2 * N[(N[Sin[N[(z0 * Pi), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(z0 * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.5%

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

   1. lift-sin.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-+.f64 N/A

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

   4. count-2 N/A

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

   5. associate-*l* N/A

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

   6. sin-2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-*.f64 51.5%

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

3. Applied rewrites 51.5%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(z0 \cdot \pi\right) \cdot \cos \left(z0 \cdot \pi\right)\right)} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2025271 -o generate:taylor -o generate:evaluate
(FPCore (z0)
  :name "(sin (* (+ PI PI) z0))"
  :precision binary64
  (sin (* (+ PI PI) z0)))