2cos (problem 3.3.5)

Percentage Accurate: 53.0% → 100.0%

Time: 18.3s

Alternatives: 16

Speedup: 41.0×

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: 53.0% 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: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\varepsilon}{2}\right)\\ \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{\varepsilon}{2}\right), t\_0, \left(t\_0 \cdot \cos x\right) \cdot t\_0\right) \cdot -2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (/ eps 2.0))))
   (* (fma (* (sin x) (cos (/ eps 2.0))) t_0 (* (* t_0 (cos x)) t_0)) -2.0)))

C

double code(double x, double eps) {
	double t_0 = sin((eps / 2.0));
	return fma((sin(x) * cos((eps / 2.0))), t_0, ((t_0 * cos(x)) * t_0)) * -2.0;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps / 2.0))
	return Float64(fma(Float64(sin(x) * cos(Float64(eps / 2.0))), t_0, Float64(Float64(t_0 * cos(x)) * t_0)) * -2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[(N[(t$95$0 * N[Cos[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 2: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (sin((0.5d0 * eps)) * sin((x + (0.5d0 * eps)))) * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return (Math.sin((0.5 * eps)) * Math.sin((x + (0.5 * eps)))) * -2.0;
}

Python

def code(x, eps):
	return (math.sin((0.5 * eps)) * math.sin((x + (0.5 * eps)))) * -2.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (sin((0.5 * eps)) * sin((x + (0.5 * eps)))) * -2.0;
end

Wolfram

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

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]
8. Add Preprocessing

Alternative 3: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * sin(((0.5d0 * eps) + x))) * ((0.5d0 + (eps * (eps * ((-0.020833333333333332d0) + (eps * (eps * (0.00026041666666666666d0 + ((eps * eps) * (-1.5500992063492063d-6))))))))) * (-2.0d0))
end function

Java

public static double code(double x, double eps) {
	return (eps * Math.sin(((0.5 * eps) + x))) * ((0.5 + (eps * (eps * (-0.020833333333333332 + (eps * (eps * (0.00026041666666666666 + ((eps * eps) * -1.5500992063492063e-6)))))))) * -2.0);
}

Python

def code(x, eps):
	return (eps * math.sin(((0.5 * eps) + x))) * ((0.5 + (eps * (eps * (-0.020833333333333332 + (eps * (eps * (0.00026041666666666666 + ((eps * eps) * -1.5500992063492063e-6)))))))) * -2.0)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (eps * sin(((0.5 * eps) + x))) * ((0.5 + (eps * (eps * (-0.020833333333333332 + (eps * (eps * (0.00026041666666666666 + ((eps * eps) * -1.5500992063492063e-6)))))))) * -2.0);
end

Wolfram

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

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]

10. Taylor expanded in x around inf 0

    \[\leadsto expr\]

12. Simplified 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 4: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((eps * (0.5d0 + ((eps * eps) * ((-0.020833333333333332d0) + (eps * (eps * 0.00026041666666666666d0)))))) * sin((x + (0.5d0 * eps)))) * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return ((eps * (0.5 + ((eps * eps) * (-0.020833333333333332 + (eps * (eps * 0.00026041666666666666)))))) * Math.sin((x + (0.5 * eps)))) * -2.0;
}

Python

def code(x, eps):
	return ((eps * (0.5 + ((eps * eps) * (-0.020833333333333332 + (eps * (eps * 0.00026041666666666666)))))) * math.sin((x + (0.5 * eps)))) * -2.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = ((eps * (0.5 + ((eps * eps) * (-0.020833333333333332 + (eps * (eps * 0.00026041666666666666)))))) * sin((x + (0.5 * eps)))) * -2.0;
end

Wolfram

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

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 5: 99.4% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((eps * (0.5d0 + (eps * (eps * (-0.020833333333333332d0))))) * sin((x + (0.5d0 * eps)))) * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return ((eps * (0.5 + (eps * (eps * -0.020833333333333332)))) * Math.sin((x + (0.5 * eps)))) * -2.0;
}

Python

def code(x, eps):
	return ((eps * (0.5 + (eps * (eps * -0.020833333333333332)))) * math.sin((x + (0.5 * eps)))) * -2.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = ((eps * (0.5 + (eps * (eps * -0.020833333333333332)))) * sin((x + (0.5 * eps)))) * -2.0;
end

Wolfram

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

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 6: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = ((0.5d0 * eps) * sin((x + (0.5d0 * eps)))) * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return ((0.5 * eps) * Math.sin((x + (0.5 * eps)))) * -2.0;
}

Python

def code(x, eps):
	return ((0.5 * eps) * math.sin((x + (0.5 * eps)))) * -2.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = ((0.5 * eps) * sin((x + (0.5 * eps)))) * -2.0;
end

Wolfram

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

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 7: 80.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{1}{\frac{1}{\varepsilon}} \cdot \sin x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 0.0 (* (/ 1.0 (/ 1.0 eps)) (sin x))))

C

double code(double x, double eps) {
	return 0.0 - ((1.0 / (1.0 / eps)) * sin(x));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0 - ((1.0d0 / (1.0d0 / eps)) * sin(x))
end function

Java

public static double code(double x, double eps) {
	return 0.0 - ((1.0 / (1.0 / eps)) * Math.sin(x));
}

Python

def code(x, eps):
	return 0.0 - ((1.0 / (1.0 / eps)) * math.sin(x))

Julia

function code(x, eps)
	return Float64(0.0 - Float64(Float64(1.0 / Float64(1.0 / eps)) * sin(x)))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0 - ((1.0 / (1.0 / eps)) * sin(x));
end

Wolfram

code[x_, eps_] := N[(0.0 - N[(N[(1.0 / N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 8: 80.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (0.5d0 * sin(x))) * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return (eps * (0.5 * Math.sin(x))) * -2.0;
}

Python

def code(x, eps):
	return (eps * (0.5 * math.sin(x))) * -2.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (0.5 * sin(x))) * -2.0;
end

Wolfram

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

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around inf 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]

10. Taylor expanded in eps around 0 0

    \[\leadsto expr\]

12. Simplified 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 9: 80.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]
10. Add Preprocessing

Alternative 10: 79.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ 0 - \varepsilon \cdot \left(x \cdot \left(1 + x \cdot \left(x \cdot \left(-0.16666666666666666 + x \cdot \left(x \cdot \left(0.008333333333333333 + \left(x \cdot x\right) \cdot -0.0001984126984126984\right)\right)\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  0.0
  (*
   eps
   (*
    x
    (+
     1.0
     (*
      x
      (*
       x
       (+
        -0.16666666666666666
        (*
         x
         (*
          x
          (+ 0.008333333333333333 (* (* x x) -0.0001984126984126984))))))))))))

C

double code(double x, double eps) {
	return 0.0 - (eps * (x * (1.0 + (x * (x * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984))))))))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0 - (eps * (x * (1.0d0 + (x * (x * ((-0.16666666666666666d0) + (x * (x * (0.008333333333333333d0 + ((x * x) * (-0.0001984126984126984d0)))))))))))
end function

Java

public static double code(double x, double eps) {
	return 0.0 - (eps * (x * (1.0 + (x * (x * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984))))))))));
}

Python

def code(x, eps):
	return 0.0 - (eps * (x * (1.0 + (x * (x * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984))))))))))

Julia

function code(x, eps)
	return Float64(0.0 - Float64(eps * Float64(x * Float64(1.0 + Float64(x * Float64(x * Float64(-0.16666666666666666 + Float64(x * Float64(x * Float64(0.008333333333333333 + Float64(Float64(x * x) * -0.0001984126984126984)))))))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0 - (eps * (x * (1.0 + (x * (x * (-0.16666666666666666 + (x * (x * (0.008333333333333333 + ((x * x) * -0.0001984126984126984))))))))));
end

Wolfram

code[x_, eps_] := N[(0.0 - N[(eps * N[(x * N[(1.0 + N[(x * N[(x * N[(-0.16666666666666666 + N[(x * N[(x * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in x around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 11: 79.4% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot \left(x \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot -0.008333333333333333 + 0.16666666666666666\right)\right)\right) - \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  x
  (-
   (*
    x
    (* x (* eps (+ (* (* x x) -0.008333333333333333) 0.16666666666666666))))
   eps)))

C

double code(double x, double eps) {
	return x * ((x * (x * (eps * (((x * x) * -0.008333333333333333) + 0.16666666666666666)))) - eps);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x * ((x * (x * (eps * (((x * x) * (-0.008333333333333333d0)) + 0.16666666666666666d0)))) - eps)
end function

Java

public static double code(double x, double eps) {
	return x * ((x * (x * (eps * (((x * x) * -0.008333333333333333) + 0.16666666666666666)))) - eps);
}

Python

def code(x, eps):
	return x * ((x * (x * (eps * (((x * x) * -0.008333333333333333) + 0.16666666666666666)))) - eps)

Julia

function code(x, eps)
	return Float64(x * Float64(Float64(x * Float64(x * Float64(eps * Float64(Float64(Float64(x * x) * -0.008333333333333333) + 0.16666666666666666)))) - eps))
end

MATLAB

function tmp = code(x, eps)
	tmp = x * ((x * (x * (eps * (((x * x) * -0.008333333333333333) + 0.16666666666666666)))) - eps);
end

Wolfram

code[x_, eps_] := N[(x * N[(N[(x * N[(x * N[(eps * N[(N[(N[(x * x), $MachinePrecision] * -0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in x around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 12: 79.3% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\varepsilon \cdot \left(\left(x \cdot x\right) \cdot 0.16666666666666666 + -1\right)\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = x * (eps * (((x * x) * 0.16666666666666666d0) + (-1.0d0)))
end function

Java

public static double code(double x, double eps) {
	return x * (eps * (((x * x) * 0.16666666666666666) + -1.0));
}

Python

def code(x, eps):
	return x * (eps * (((x * x) * 0.16666666666666666) + -1.0))

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = x * (eps * (((x * x) * 0.16666666666666666) + -1.0));
end

Wolfram

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

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in x around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 13: 79.1% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ 0 - \frac{1}{\frac{1}{\varepsilon}} \cdot x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 0.0 (* (/ 1.0 (/ 1.0 eps)) x)))

C

double code(double x, double eps) {
	return 0.0 - ((1.0 / (1.0 / eps)) * x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0 - ((1.0d0 / (1.0d0 / eps)) * x)
end function

Java

public static double code(double x, double eps) {
	return 0.0 - ((1.0 / (1.0 / eps)) * x);
}

Python

def code(x, eps):
	return 0.0 - ((1.0 / (1.0 / eps)) * x)

Julia

function code(x, eps)
	return Float64(0.0 - Float64(Float64(1.0 / Float64(1.0 / eps)) * x))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0 - ((1.0 / (1.0 / eps)) * x);
end

Wolfram

code[x_, eps_] := N[(0.0 - N[(N[(1.0 / N[(1.0 / eps), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

9. Applied egg-rr 0

   \[\leadsto expr\]

10. Taylor expanded in x around 0 0

    \[\leadsto expr\]

12. Simplified 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 14: 79.1% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ 0 - \varepsilon \cdot x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 0.0 (* eps x)))

C

double code(double x, double eps) {
	return 0.0 - (eps * x);
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0 - (eps * x)
end function

Java

public static double code(double x, double eps) {
	return 0.0 - (eps * x);
}

Python

def code(x, eps):
	return 0.0 - (eps * x)

Julia

function code(x, eps)
	return Float64(0.0 - Float64(eps * x))
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0 - (eps * x);
end

Wolfram

code[x_, eps_] := N[(0.0 - N[(eps * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.1%

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

4. Applied egg-rr 0

   \[\leadsto expr\]

5. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

7. Simplified 0

   \[\leadsto expr\]

8. Taylor expanded in x around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 15: 52.8% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in x around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]
9. Add Preprocessing

Alternative 16: 51.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 50.1%

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

3. Taylor expanded in x around 0 0

   \[\leadsto expr\]

5. Simplified 0

   \[\leadsto expr\]

6. Taylor expanded in eps around 0 0

   \[\leadsto expr\]

8. Simplified 0

   \[\leadsto expr\]

10. Applied egg-rr 0

    \[\leadsto expr\]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.0× 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]

Reproduce

herbie shell --seed 2024111 
(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))))

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