2cos (problem 3.3.5)

Percentage Accurate: 51.7% → 99.3%

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: 51.7% 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: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \sin x\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * ((-0.5d0) * cos(x))) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * (-0.5 * Math.cos(x))) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * (-0.5 * math.cos(x))) - math.sin(x))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * (-0.5 * cos(x))) - sin(x));
end

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.8

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \color{blue}{\sin x}\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \sin x\right)} \]
6. Add Preprocessing

Alternative 2: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) - sin(x))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) - Math.sin(x));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) - math.sin(x))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) - sin(x));
end

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.8

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \color{blue}{\sin x}\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \sin x\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.6%

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

Alternative 3: 98.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -\varepsilon\right), -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  x
  (fma x (* eps (fma eps 0.25 (* x 0.16666666666666666))) (- eps))
  (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	return fma(x, fma(x, (eps * fma(eps, 0.25, (x * 0.16666666666666666))), -eps), (-0.5 * (eps * eps)));
}

Julia

function code(x, eps)
	return fma(x, fma(x, Float64(eps * fma(eps, 0.25, Float64(x * 0.16666666666666666))), Float64(-eps)), Float64(-0.5 * Float64(eps * eps)))
end

Wolfram

code[x_, eps_] := N[(x * N[(x * N[(eps * N[(eps * 0.25 + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-eps)), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.8

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \color{blue}{\sin x}\right) \]

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{2} \cdot {\varepsilon}^{2} + \color{blue}{x \cdot \left(-1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 99.1%

    \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -\varepsilon\right)}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]
11. Add Preprocessing

Alternative 4: 98.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -1\right), -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma x (* eps (fma x (* x 0.16666666666666666) -1.0)) (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	return fma(x, (eps * fma(x, (x * 0.16666666666666666), -1.0)), (-0.5 * (eps * eps)));
}

Julia

function code(x, eps)
	return fma(x, Float64(eps * fma(x, Float64(x * 0.16666666666666666), -1.0)), Float64(-0.5 * Float64(eps * eps)))
end

Wolfram

code[x_, eps_] := N[(x * N[(eps * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.8

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \color{blue}{\sin x}\right) \]

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{2} \cdot {\varepsilon}^{2} + \color{blue}{x \cdot \left(-1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 99.1%

    \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -\varepsilon\right)}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

11. Taylor expanded in eps around 0

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

13. Applied rewrites 99.0%

    \[\leadsto \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.16666666666666666}, -1\right), -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]
14. Add Preprocessing

Alternative 5: 98.3% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -1\right), \varepsilon \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (fma x (fma x (* x 0.16666666666666666) -1.0) (* eps -0.5))))

C

double code(double x, double eps) {
	return eps * fma(x, fma(x, (x * 0.16666666666666666), -1.0), (eps * -0.5));
}

Julia

function code(x, eps)
	return Float64(eps * fma(x, fma(x, Float64(x * 0.16666666666666666), -1.0), Float64(eps * -0.5)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision] + -1.0), $MachinePrecision] + N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.8

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \color{blue}{\sin x}\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \sin x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \left(\frac{-1}{2} \cdot \varepsilon + \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right)}\right) \]
7. Step-by-step derivation

8. Applied rewrites 98.9%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -1\right)}, \varepsilon \cdot -0.5\right) \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.9%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.16666666666666666, -1\right), \varepsilon \cdot -0.5\right) \]
12. Add Preprocessing

Alternative 6: 98.1% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.8

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \color{blue}{\sin x}\right) \]

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.7%

   \[\leadsto \frac{\varepsilon}{\color{blue}{\frac{1}{\cos x \cdot \left(\varepsilon \cdot -0.5\right) - \sin x}}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{2} \cdot {\varepsilon}^{2} + \color{blue}{x \cdot \left(-1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right)} \]
9. Step-by-step derivation

10. Applied rewrites 99.1%

    \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, 0.25, x \cdot 0.16666666666666666\right), -\varepsilon\right)}, -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

11. Taylor expanded in x around 0

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

13. Applied rewrites 98.6%

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

Alternative 7: 97.9% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) - x)
end function

Java

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

Python

def code(x, eps):
	return eps * ((eps * -0.5) - x)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) - x);
end

Wolfram

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

Derivation

1. Initial program 49.7%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.8

      \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \color{blue}{\sin x}\right) \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(-0.5 \cdot \cos x\right) - \sin x\right)} \]

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

Alternative 8: 78.5% accurate, 25.9× speedup

Math

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

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(x * Float64(-eps))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 78.4

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

5. Applied rewrites 78.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.5%

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

9. Final simplification 77.5%

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

Alternative 9: 50.4% 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 49.7%

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

3. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. lower-+.f64 N/A

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

   4. lower-cos.f64 49.3

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

5. Applied rewrites 49.3%

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

6. Taylor expanded in eps around 0

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

8. Applied rewrites 49.2%

   \[\leadsto 1 + -1 \]

9. Final simplification 49.2%

   \[\leadsto -1 + 1 \]
10. 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 2024237 
(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)))