2cos (problem 3.3.5)

Percentage Accurate: 52.7% → 99.5%

Time: 17.3s

Alternatives: 11

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: 52.7% 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: 99.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\cos x, -0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right), \varepsilon \cdot \left(0.16666666666666666 \cdot \sin x\right)\right), \sin \left(-x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma
    (cos x)
    (+ -0.5 (* eps (* eps 0.041666666666666664)))
    (* eps (* 0.16666666666666666 (sin x))))
   (sin (- x)))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(cos(x), (-0.5 + (eps * (eps * 0.041666666666666664))), (eps * (0.16666666666666666 * sin(x)))), sin(-x));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(cos(x), Float64(-0.5 + Float64(eps * Float64(eps * 0.041666666666666664))), Float64(eps * Float64(0.16666666666666666 * sin(x)))), sin(Float64(-x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[(eps * N[(eps * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(0.16666666666666666 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sin[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \cos x\right) - \frac{-1}{6} \cdot \sin x\right)\right) - \sin x\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \cos x\right) - \frac{-1}{6} \cdot \sin x\right)\right) - \sin x\right)} \]

   2. sub-neg N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \cos x\right) - \frac{-1}{6} \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \cos x + \varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \cos x\right) - \frac{-1}{6} \cdot \sin x\right), \mathsf{neg}\left(\sin x\right)\right)} \]

5. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\cos x, -0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right), \varepsilon \cdot \left(0.16666666666666666 \cdot \sin x\right)\right), \sin \left(-x\right)\right)} \]
6. Add Preprocessing

Alternative 2: 99.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\cos x, -0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right), \varepsilon \cdot \left(x \cdot 0.16666666666666666\right)\right), \sin \left(-x\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (fma
    (cos x)
    (+ -0.5 (* eps (* eps 0.041666666666666664)))
    (* eps (* x 0.16666666666666666)))
   (sin (- x)))))

C

double code(double x, double eps) {
	return eps * fma(eps, fma(cos(x), (-0.5 + (eps * (eps * 0.041666666666666664))), (eps * (x * 0.16666666666666666))), sin(-x));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, fma(cos(x), Float64(-0.5 + Float64(eps * Float64(eps * 0.041666666666666664))), Float64(eps * Float64(x * 0.16666666666666666))), sin(Float64(-x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(N[Cos[x], $MachinePrecision] * N[(-0.5 + N[(eps * N[(eps * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Sin[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \cos x\right) - \frac{-1}{6} \cdot \sin x\right)\right) - \sin x\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \cos x\right) - \frac{-1}{6} \cdot \sin x\right)\right) - \sin x\right)} \]

   2. sub-neg N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \cos x\right) - \frac{-1}{6} \cdot \sin x\right)\right) + \left(\mathsf{neg}\left(\sin x\right)\right)\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \cos x + \varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \cos x\right) - \frac{-1}{6} \cdot \sin x\right), \mathsf{neg}\left(\sin x\right)\right)} \]

5. Applied rewrites 99.5%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\cos x, -0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right), \varepsilon \cdot \left(0.16666666666666666 \cdot \sin x\right)\right), \sin \left(-x\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\cos x, \frac{-1}{2} + \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{24}\right), \varepsilon \cdot \left(\frac{1}{6} \cdot x\right)\right), \sin \left(\mathsf{neg}\left(x\right)\right)\right) \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(\cos x, -0.5 + \varepsilon \cdot \left(\varepsilon \cdot 0.041666666666666664\right), \varepsilon \cdot \left(x \cdot 0.16666666666666666\right)\right), \sin \left(-x\right)\right) \]
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: 98.8% 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 2024230 
(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)))