2cos (problem 3.3.5)

Percentage Accurate: 52.5% → 99.7%

Time: 16.6s

Alternatives: 9

Speedup: 25.9×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

\[\begin{array}{l} \\ \cos \left(x + \varepsilon\right) - \cos x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

Python

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos \left(x + \varepsilon\right) - \cos x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

C

double code(double x, double eps) {
	return cos((x + eps)) - cos(x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = cos((x + eps)) - cos(x)
end function

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

Python

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

MATLAB

function tmp = code(x, eps)
	tmp = cos((x + eps)) - cos(x);
end

Wolfram

code[x_, eps_] := N[(N[Cos[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* -2.0 (* (sin (* eps 0.5)) (sin (* (fma 2.0 x eps) 0.5)))))

C

double code(double x, double eps) {
	return -2.0 * (sin((eps * 0.5)) * sin((fma(2.0, x, eps) * 0.5)));
}

Julia

function code(x, eps)
	return Float64(-2.0 * Float64(sin(Float64(eps * 0.5)) * sin(Float64(fma(2.0, x, eps) * 0.5))))
end

Wolfram

code[x_, eps_] := N[(-2.0 * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right)} - \cos x \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{\cos x} \]

   4. diff-cos N/A

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]

5. Final simplification 99.7%

   \[\leadsto -2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \]
6. Add Preprocessing

Alternative 2: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6} \cdot \varepsilon, \varepsilon, 0.00026041666666666666\right) \cdot \varepsilon, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (*
    (fma
     (*
      (fma
       (* (fma (* -1.5500992063492063e-6 eps) eps 0.00026041666666666666) eps)
       eps
       -0.020833333333333332)
      eps)
     eps
     0.5)
    eps)
   (sin (* (fma 2.0 x eps) 0.5)))
  -2.0))

C

double code(double x, double eps) {
	return ((fma((fma((fma((-1.5500992063492063e-6 * eps), eps, 0.00026041666666666666) * eps), eps, -0.020833333333333332) * eps), eps, 0.5) * eps) * sin((fma(2.0, x, eps) * 0.5))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(fma(Float64(fma(Float64(-1.5500992063492063e-6 * eps), eps, 0.00026041666666666666) * eps), eps, -0.020833333333333332) * eps), eps, 0.5) * eps) * sin(Float64(fma(2.0, x, eps) * 0.5))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(-1.5500992063492063e-6 * eps), $MachinePrecision] * eps + 0.00026041666666666666), $MachinePrecision] * eps), $MachinePrecision] * eps + -0.020833333333333332), $MachinePrecision] * eps), $MachinePrecision] * eps + 0.5), $MachinePrecision] * eps), $MachinePrecision] * N[Sin[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right)} - \cos x \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{\cos x} \]

   4. diff-cos N/A

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]

5. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)}\right) \cdot -2 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

7. Applied rewrites 99.5%

   \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6} \cdot \varepsilon, \varepsilon, 0.00026041666666666666\right) \cdot \varepsilon, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

8. Final simplification 99.5%

   \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6} \cdot \varepsilon, \varepsilon, 0.00026041666666666666\right) \cdot \varepsilon, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2 \]
9. Add Preprocessing

Alternative 3: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666 \cdot \varepsilon, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(\varepsilon, 0.5, x\right)\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (*
    (fma
     (* (fma (* 0.00026041666666666666 eps) eps -0.020833333333333332) eps)
     eps
     0.5)
    eps)
   (sin (fma eps 0.5 x)))
  -2.0))

C

double code(double x, double eps) {
	return ((fma((fma((0.00026041666666666666 * eps), eps, -0.020833333333333332) * eps), eps, 0.5) * eps) * sin(fma(eps, 0.5, x))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(fma(Float64(0.00026041666666666666 * eps), eps, -0.020833333333333332) * eps), eps, 0.5) * eps) * sin(fma(eps, 0.5, x))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(N[(0.00026041666666666666 * eps), $MachinePrecision] * eps + -0.020833333333333332), $MachinePrecision] * eps), $MachinePrecision] * eps + 0.5), $MachinePrecision] * eps), $MachinePrecision] * N[Sin[N[(eps * 0.5 + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right)} - \cos x \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{\cos x} \]

   4. diff-cos N/A

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]

5. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)}\right) \cdot -2 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   3. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) + \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   4. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) \cdot {\varepsilon}^{2}} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   5. unpow2 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   6. associate-*r* N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\color{blue}{\left(\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) \cdot \varepsilon\right) \cdot \varepsilon} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) \cdot \varepsilon, \varepsilon, \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) \cdot \varepsilon}, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   9. sub-neg N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right)} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   10. unpow2 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\left(\frac{1}{3840} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   11. associate-*r* N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\left(\color{blue}{\left(\frac{1}{3840} \cdot \varepsilon\right) \cdot \varepsilon} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)\right) \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   12. metadata-eval N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\left(\left(\frac{1}{3840} \cdot \varepsilon\right) \cdot \varepsilon + \color{blue}{\frac{-1}{48}}\right) \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   13. lower-fma.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3840} \cdot \varepsilon, \varepsilon, \frac{-1}{48}\right)} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   14. lower-*.f64 99.5

       \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{0.00026041666666666666 \cdot \varepsilon}, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

7. Applied rewrites 99.5%

   \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666 \cdot \varepsilon, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

8. Taylor expanded in x around 0

   \[\leadsto \left(\sin \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840} \cdot \varepsilon, \varepsilon, \frac{-1}{48}\right) \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \varepsilon + x\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840} \cdot \varepsilon, \varepsilon, \frac{-1}{48}\right) \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   2. *-commutative N/A

      \[\leadsto \left(\sin \left(\color{blue}{\varepsilon \cdot \frac{1}{2}} + x\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840} \cdot \varepsilon, \varepsilon, \frac{-1}{48}\right) \cdot \varepsilon, \varepsilon, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   3. lower-fma.f64 99.5

      \[\leadsto \left(\sin \color{blue}{\left(\mathsf{fma}\left(\varepsilon, 0.5, x\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666 \cdot \varepsilon, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

10. Applied rewrites 99.5%

    \[\leadsto \left(\sin \color{blue}{\left(\mathsf{fma}\left(\varepsilon, 0.5, x\right)\right)} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666 \cdot \varepsilon, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

11. Final simplification 99.5%

    \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666 \cdot \varepsilon, \varepsilon, -0.020833333333333332\right) \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(\varepsilon, 0.5, x\right)\right)\right) \cdot -2 \]
12. Add Preprocessing

Alternative 4: 99.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(-0.020833333333333332 \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (* (fma (* -0.020833333333333332 eps) eps 0.5) eps)
   (sin (* (fma 2.0 x eps) 0.5)))
  -2.0))

C

double code(double x, double eps) {
	return ((fma((-0.020833333333333332 * eps), eps, 0.5) * eps) * sin((fma(2.0, x, eps) * 0.5))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(-0.020833333333333332 * eps), eps, 0.5) * eps) * sin(Float64(fma(2.0, x, eps) * 0.5))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(-0.020833333333333332 * eps), $MachinePrecision] * eps + 0.5), $MachinePrecision] * eps), $MachinePrecision] * N[Sin[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right)} - \cos x \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{\cos x} \]

   4. diff-cos N/A

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]

5. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right)}\right) \cdot -2 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   3. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\left(\frac{-1}{48} \cdot {\varepsilon}^{2} + \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   4. unpow2 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\frac{-1}{48} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   5. associate-*r* N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{-1}{48} \cdot \varepsilon\right) \cdot \varepsilon} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   6. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{-1}{48} \cdot \varepsilon, \varepsilon, \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   7. lower-*.f64 99.4

      \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{-0.020833333333333332 \cdot \varepsilon}, \varepsilon, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

7. Applied rewrites 99.4%

   \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(-0.020833333333333332 \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

8. Final simplification 99.4%

   \[\leadsto \left(\left(\mathsf{fma}\left(-0.020833333333333332 \cdot \varepsilon, \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2 \]
9. Add Preprocessing

Alternative 5: 99.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* (* eps 0.5) (sin (* (fma 2.0 x eps) 0.5))) -2.0))

C

double code(double x, double eps) {
	return ((eps * 0.5) * sin((fma(2.0, x, eps) * 0.5))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(eps * 0.5) * sin(Float64(fma(2.0, x, eps) * 0.5))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(eps * 0.5), $MachinePrecision] * N[Sin[N[(N[(2.0 * x + eps), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right) - \cos x} \]

   2. lift-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \left(x + \varepsilon\right)} - \cos x \]

   3. lift-cos.f64 N/A

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{\cos x} \]

   4. diff-cos N/A

      \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{\left(x + \varepsilon\right) - x}{2}\right) \cdot \sin \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \cdot -2} \]

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]

5. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)}\right) \cdot -2 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)}\right) \cdot -2 \]

   2. lower-*.f64 99.2

      \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot 0.5\right)}\right) \cdot -2 \]

7. Applied rewrites 99.2%

   \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot 0.5\right)}\right) \cdot -2 \]

8. Final simplification 99.2%

   \[\leadsto \left(\left(\varepsilon \cdot 0.5\right) \cdot \sin \left(\mathsf{fma}\left(2, x, \varepsilon\right) \cdot 0.5\right)\right) \cdot -2 \]
9. Add Preprocessing

Alternative 6: 97.7% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, \varepsilon, -0.5\right), \varepsilon, -x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (fma (fma (* 0.16666666666666666 x) eps -0.5) eps (- x)) eps))

C

double code(double x, double eps) {
	return fma(fma((0.16666666666666666 * x), eps, -0.5), eps, -x) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(Float64(0.16666666666666666 * x), eps, -0.5), eps, Float64(-x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.16666666666666666 * x), $MachinePrecision] * eps + -0.5), $MachinePrecision] * eps + (-x)), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\cos \varepsilon + -1 \cdot \left(x \cdot \sin \varepsilon\right)\right) - 1} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\cos \varepsilon + \color{blue}{\left(\mathsf{neg}\left(x \cdot \sin \varepsilon\right)\right)}\right) - 1 \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\cos \varepsilon - x \cdot \sin \varepsilon\right)} - 1 \]

   3. associate--l- N/A

      \[\leadsto \color{blue}{\cos \varepsilon - \left(x \cdot \sin \varepsilon + 1\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{\cos \varepsilon - \left(x \cdot \sin \varepsilon + 1\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \varepsilon} - \left(x \cdot \sin \varepsilon + 1\right) \]

   6. *-commutative N/A

      \[\leadsto \cos \varepsilon - \left(\color{blue}{\sin \varepsilon \cdot x} + 1\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \cos \varepsilon - \color{blue}{\mathsf{fma}\left(\sin \varepsilon, x, 1\right)} \]

   8. lower-sin.f64 50.4

      \[\leadsto \cos \varepsilon - \mathsf{fma}\left(\color{blue}{\sin \varepsilon}, x, 1\right) \]

5. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, x, 1\right)} \]

6. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) - \frac{1}{2}\right) - x\right)} \]
7. Step-by-step derivation

8. Applied rewrites 97.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666 \cdot x, \varepsilon, -0.5\right), \varepsilon, -x\right) \cdot \color{blue}{\varepsilon} \]
9. Add Preprocessing

Alternative 7: 97.7% accurate, 14.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \varepsilon, -x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (fma -0.5 eps (- x)) eps))

C

double code(double x, double eps) {
	return fma(-0.5, eps, -x) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(-0.5, eps, Float64(-x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(-0.5 * eps + (-x)), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\cos \varepsilon + -1 \cdot \left(x \cdot \sin \varepsilon\right)\right) - 1} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \left(\cos \varepsilon + \color{blue}{\left(\mathsf{neg}\left(x \cdot \sin \varepsilon\right)\right)}\right) - 1 \]

   2. unsub-neg N/A

      \[\leadsto \color{blue}{\left(\cos \varepsilon - x \cdot \sin \varepsilon\right)} - 1 \]

   3. associate--l- N/A

      \[\leadsto \color{blue}{\cos \varepsilon - \left(x \cdot \sin \varepsilon + 1\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \color{blue}{\cos \varepsilon - \left(x \cdot \sin \varepsilon + 1\right)} \]

   5. lower-cos.f64 N/A

      \[\leadsto \color{blue}{\cos \varepsilon} - \left(x \cdot \sin \varepsilon + 1\right) \]

   6. *-commutative N/A

      \[\leadsto \cos \varepsilon - \left(\color{blue}{\sin \varepsilon \cdot x} + 1\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \cos \varepsilon - \color{blue}{\mathsf{fma}\left(\sin \varepsilon, x, 1\right)} \]

   8. lower-sin.f64 50.4

      \[\leadsto \cos \varepsilon - \mathsf{fma}\left(\color{blue}{\sin \varepsilon}, x, 1\right) \]

5. Applied rewrites 50.4%

   \[\leadsto \color{blue}{\cos \varepsilon - \mathsf{fma}\left(\sin \varepsilon, x, 1\right)} \]

6. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-1}{2} \cdot \varepsilon - x\right)} \]
7. Step-by-step derivation

8. Applied rewrites 97.3%

   \[\leadsto \mathsf{fma}\left(-0.5, \varepsilon, -x\right) \cdot \color{blue}{\varepsilon} \]
9. Add Preprocessing

Alternative 8: 78.9% accurate, 25.9× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* (- x) eps))

C

double code(double x, double eps) {
	return -x * eps;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = -x * eps
end function

Java

public static double code(double x, double eps) {
	return -x * eps;
}

Python

def code(x, eps):
	return -x * eps

Julia

function code(x, eps)
	return Float64(Float64(-x) * eps)
end

MATLAB

function tmp = code(x, eps)
	tmp = -x * eps;
end

Wolfram

code[x_, eps_] := N[((-x) * eps), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\varepsilon \cdot \sin x\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sin x \cdot \varepsilon}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \varepsilon} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right) \cdot \varepsilon} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sin x\right)\right)} \cdot \varepsilon \]

   6. lower-sin.f64 78.5

      \[\leadsto \left(-\color{blue}{\sin x}\right) \cdot \varepsilon \]

5. Applied rewrites 78.5%

   \[\leadsto \color{blue}{\left(-\sin x\right) \cdot \varepsilon} \]

6. Taylor expanded in x around 0

   \[\leadsto \left(-1 \cdot x\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 77.6%

   \[\leadsto \left(-x\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 9: 51.1% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 1.0 1.0))

C

double code(double x, double eps) {
	return 1.0 - 1.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0 - 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0 - 1.0;
}

Python

def code(x, eps):
	return 1.0 - 1.0

Julia

function code(x, eps)
	return Float64(1.0 - 1.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_, eps_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 51.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   2. lower-cos.f64 49.7

      \[\leadsto \color{blue}{\cos \varepsilon} - 1 \]

5. Applied rewrites 49.7%

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

6. Taylor expanded in eps around 0

   \[\leadsto 1 - 1 \]
7. Step-by-step derivation

8. Applied rewrites 49.5%

   \[\leadsto 1 - 1 \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \left(-2 \cdot \sin \left(x + \frac{\varepsilon}{2}\right)\right) \cdot \sin \left(\frac{\varepsilon}{2}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* (* -2.0 (sin (+ x (/ eps 2.0)))) (sin (/ eps 2.0))))

C

double code(double x, double eps) {
	return (-2.0 * sin((x + (eps / 2.0)))) * sin((eps / 2.0));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((-2.0d0) * sin((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

public static double code(double x, double eps) {
	return (-2.0 * Math.sin((x + (eps / 2.0)))) * Math.sin((eps / 2.0));
}

Python

def code(x, eps):
	return (-2.0 * math.sin((x + (eps / 2.0)))) * math.sin((eps / 2.0))

Julia

function code(x, eps)
	return Float64(Float64(-2.0 * sin(Float64(x + Float64(eps / 2.0)))) * sin(Float64(eps / 2.0)))
end

MATLAB

function tmp = code(x, eps)
	tmp = (-2.0 * sin((x + (eps / 2.0)))) * sin((eps / 2.0));
end

Wolfram

code[x_, eps_] := N[(N[(-2.0 * N[Sin[N[(x + N[(eps / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Developer Target 2: 98.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (pow (cbrt (* (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))) (sin (* 0.5 eps)))) 3.0))

C

double code(double x, double eps) {
	return pow(cbrt(((-2.0 * sin((0.5 * fma(2.0, x, eps)))) * sin((0.5 * eps)))), 3.0);
}

Julia

function code(x, eps)
	return cbrt(Float64(Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))) * sin(Float64(0.5 * eps)))) ^ 3.0
end

Wolfram

code[x_, eps_] := N[Power[N[Power[N[(N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Reproduce

herbie shell --seed 2024234 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* -2 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

  (- (cos (+ x eps)) (cos x)))