2cos (problem 3.3.5)

Percentage Accurate: 53.5% → 99.5%

Time: 23.4s

Alternatives: 12

Speedup: 51.3×

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: 53.5% 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: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (0.5 + ((eps * eps) * (-0.020833333333333332 + ((eps * eps) * 0.00026041666666666666)))) * (sin((x + (eps * 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 * eps) * ((-0.020833333333333332d0) + ((eps * eps) * 0.00026041666666666666d0)))) * (sin((x + (eps * 0.5d0))) * (eps * (-2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 eps around inf 0

    \[\leadsto expr\]

12. Simplified 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 2: 99.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 inf 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]
11. Add Preprocessing

Alternative 3: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 eps around 0 0

    \[\leadsto expr\]

12. Simplified 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 4: 99.4% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

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

Alternative 5: 99.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 eps around 0 0

    \[\leadsto expr\]

12. Simplified 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 6: 79.9% 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 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 eps around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]

12. Applied egg-rr 0

    \[\leadsto expr\]
13. Add Preprocessing

Alternative 7: 79.4% accurate, 8.2× speedup

Math

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

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(Float64(x * x) * Float64(-0.16666666666666666 + Float64(Float64(x * 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[(N[(x * x), $MachinePrecision] * N[(-0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(x * x), $MachinePrecision] * -0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 eps around 0 0

   \[\leadsto expr\]

10. Simplified 0

    \[\leadsto expr\]

11. Taylor expanded in x around 0 0

    \[\leadsto expr\]

13. Simplified 0

    \[\leadsto expr\]
14. Add Preprocessing

Alternative 8: 79.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \left(\varepsilon \cdot \left(x \cdot \left(0.5 + x \cdot \left(x \cdot \left(-0.08333333333333333 + \left(x \cdot x\right) \cdot \left(0.004166666666666667 + \left(x \cdot x\right) \cdot -9.92063492063492 \cdot 10^{-5}\right)\right)\right)\right)\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   eps
   (*
    x
    (+
     0.5
     (*
      x
      (*
       x
       (+
        -0.08333333333333333
        (*
         (* x x)
         (+ 0.004166666666666667 (* (* x x) -9.92063492063492e-5)))))))))
  -2.0))

C

double code(double x, double eps) {
	return (eps * (x * (0.5 + (x * (x * (-0.08333333333333333 + ((x * x) * (0.004166666666666667 + ((x * x) * -9.92063492063492e-5))))))))) * -2.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (eps * (x * (0.5d0 + (x * (x * ((-0.08333333333333333d0) + ((x * x) * (0.004166666666666667d0 + ((x * x) * (-9.92063492063492d-5)))))))))) * (-2.0d0)
end function

Java

public static double code(double x, double eps) {
	return (eps * (x * (0.5 + (x * (x * (-0.08333333333333333 + ((x * x) * (0.004166666666666667 + ((x * x) * -9.92063492063492e-5))))))))) * -2.0;
}

Python

def code(x, eps):
	return (eps * (x * (0.5 + (x * (x * (-0.08333333333333333 + ((x * x) * (0.004166666666666667 + ((x * x) * -9.92063492063492e-5))))))))) * -2.0

Julia

function code(x, eps)
	return Float64(Float64(eps * Float64(x * Float64(0.5 + Float64(x * Float64(x * Float64(-0.08333333333333333 + Float64(Float64(x * x) * Float64(0.004166666666666667 + Float64(Float64(x * x) * -9.92063492063492e-5))))))))) * -2.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps * (x * (0.5 + (x * (x * (-0.08333333333333333 + ((x * x) * (0.004166666666666667 + ((x * x) * -9.92063492063492e-5))))))))) * -2.0;
end

Wolfram

code[x_, eps_] := N[(N[(eps * N[(x * N[(0.5 + N[(x * N[(x * N[(-0.08333333333333333 + N[(N[(x * x), $MachinePrecision] * N[(0.004166666666666667 + N[(N[(x * x), $MachinePrecision] * -9.92063492063492e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 9: 79.4% accurate, 12.1× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return x * (((eps * (x * x)) * (((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 * (((eps * (x * x)) * (((x * x) * (-0.008333333333333333d0)) + 0.16666666666666666d0)) - eps)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 10: 79.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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: 79.1% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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: 52.1% 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 54.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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: 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 2024110 
(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
  (* (* -2.0 (sin (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

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