2cos (problem 3.3.5)

Percentage Accurate: 53.1% → 99.6%

Time: 15.2s

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: 53.1% 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: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 83.5%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

7. Applied rewrites 99.7%

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

Alternative 3: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

Alternative 4: 99.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.5%

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

Alternative 5: 98.9% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.3%

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

Alternative 6: 98.3% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

Alternative 7: 98.3% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

Alternative 8: 98.3% accurate, 6.9× speedup

Math

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

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)) * Float64(-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 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 99.0%

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

9. Taylor expanded in x around inf

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

11. Applied rewrites 99.0%

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

Alternative 9: 98.1% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.5%

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

11. Taylor expanded in x around 0

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

13. Applied rewrites 99.0%

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

14. Final simplification 99.0%

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

Alternative 10: 97.9% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.8%

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

Alternative 11: 79.2% 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 54.2%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. mul-1-neg N/A

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

   3. lower-*.f64 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 79.0

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

5. Applied rewrites 79.0%

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

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 78.5%

   \[\leadsto \left(-x\right) \cdot \color{blue}{\varepsilon} \]
9. 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 2024340 
(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)))