2cos (problem 3.3.5)

Percentage Accurate: 52.6% → 99.4%

Time: 10.6s

Alternatives: 10

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: 52.6% 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: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Final simplification 99.7%

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

Alternative 3: 98.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.16666666666666666, -1\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, t\_0 \cdot x, 0.25 \cdot \varepsilon\right), x, t\_0\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma (* eps eps) 0.16666666666666666 -1.0)))
   (*
    (fma
     (fma (fma -0.16666666666666666 (* t_0 x) (* 0.25 eps)) x t_0)
     x
     (* -0.5 eps))
    eps)))

C

double code(double x, double eps) {
	double t_0 = fma((eps * eps), 0.16666666666666666, -1.0);
	return fma(fma(fma(-0.16666666666666666, (t_0 * x), (0.25 * eps)), x, t_0), x, (-0.5 * eps)) * eps;
}

Julia

function code(x, eps)
	t_0 = fma(Float64(eps * eps), 0.16666666666666666, -1.0)
	return Float64(fma(fma(fma(-0.16666666666666666, Float64(t_0 * x), Float64(0.25 * eps)), x, t_0), x, Float64(-0.5 * eps)) * eps)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[(eps * eps), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision]}, N[(N[(N[(N[(-0.16666666666666666 * N[(t$95$0 * x), $MachinePrecision] + N[(0.25 * eps), $MachinePrecision]), $MachinePrecision] * x + t$95$0), $MachinePrecision] * x + N[(-0.5 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.7%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.16666666666666666, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.16666666666666666, -1\right) \cdot x, 0.25 \cdot \varepsilon\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.16666666666666666, -1\right)\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 4: 98.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \varepsilon, 0.16666666666666666 \cdot x\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.16666666666666666, -1\right)\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma
    (fma 0.25 eps (* 0.16666666666666666 x))
    x
    (fma (* eps eps) 0.16666666666666666 -1.0))
   x
   (* -0.5 eps))
  eps))

C

double code(double x, double eps) {
	return fma(fma(fma(0.25, eps, (0.16666666666666666 * x)), x, fma((eps * eps), 0.16666666666666666, -1.0)), x, (-0.5 * eps)) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(0.25, eps, Float64(0.16666666666666666 * x)), x, fma(Float64(eps * eps), 0.16666666666666666, -1.0)), x, Float64(-0.5 * eps)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.25 * eps + N[(0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] * x + N[(N[(eps * eps), $MachinePrecision] * 0.16666666666666666 + -1.0), $MachinePrecision]), $MachinePrecision] * x + N[(-0.5 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.7%

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

12. Taylor expanded in eps around 0

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{4}, \varepsilon, \frac{1}{6} \cdot x\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \frac{1}{6}, -1\right)\right), x, \frac{-1}{2} \cdot \varepsilon\right) \cdot \varepsilon \]
13. Step-by-step derivation

14. Applied rewrites 98.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, \varepsilon, 0.16666666666666666 \cdot x\right), x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.16666666666666666, -1\right)\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 5: 98.2% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \varepsilon, x, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, -1\right)\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (fma (* 0.25 eps) x (fma (* 0.16666666666666666 x) x -1.0))
   x
   (* -0.5 eps))
  eps))

C

double code(double x, double eps) {
	return fma(fma((0.25 * eps), x, fma((0.16666666666666666 * x), x, -1.0)), x, (-0.5 * eps)) * eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(Float64(0.25 * eps), x, fma(Float64(0.16666666666666666 * x), x, -1.0)), x, Float64(-0.5 * eps)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(0.25 * eps), $MachinePrecision] * x + N[(N[(0.16666666666666666 * x), $MachinePrecision] * x + -1.0), $MachinePrecision]), $MachinePrecision] * x + N[(-0.5 * eps), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.7%

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

12. Taylor expanded in eps around 0

    \[\leadsto \mathsf{fma}\left(\left(\frac{1}{6} \cdot {x}^{2} + \frac{1}{4} \cdot \left(\varepsilon \cdot x\right)\right) - 1, x, \frac{-1}{2} \cdot \varepsilon\right) \cdot \varepsilon \]
13. Step-by-step derivation

14. Applied rewrites 98.7%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.25 \cdot \varepsilon, x, \mathsf{fma}\left(0.16666666666666666 \cdot x, x, -1\right)\right), x, -0.5 \cdot \varepsilon\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 6: 98.1% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 98.7%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 98.6%

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

Alternative 7: 97.6% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.4%

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

Alternative 8: 79.1% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.6%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.3

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

5. Applied rewrites 80.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 79.7%

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

9. Final simplification 79.7%

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

Alternative 9: 78.9% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.6%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.3

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

5. Applied rewrites 80.3%

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

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 79.7%

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

Alternative 10: 50.5% accurate, 34.5× 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(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 55.6%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 80.3

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

5. Applied rewrites 80.3%

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

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 79.7%

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

10. Applied rewrites 54.1%

    \[\leadsto x \cdot \varepsilon \]

11. Final simplification 54.1%

    \[\leadsto \varepsilon \cdot x \]
12. Add Preprocessing

Developer Target 1: 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 2024332 
(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 (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

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