2cos (problem 3.3.5)

Percentage Accurate: 53.1% → 99.5%

Time: 14.2s

Alternatives: 12

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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

Alternative 2: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. sin-+PI/2-rev N/A

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

   4. lift-cos.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. sin-+PI/2-rev N/A

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

   7. diff-sin N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. cos-neg-rev N/A

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

   10. lower-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-PI.f64 7.5

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

7. Applied rewrites 7.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \cos \left(\left(\mathsf{PI}\left(\right) + \varepsilon\right) \cdot -0.5\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 7.4%

   \[\leadsto 2 \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) + \varepsilon, 0.5, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

10. Taylor expanded in eps around 0

    \[\leadsto 2 \cdot \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\sin \mathsf{PI}\left(\right) + \color{blue}{\varepsilon \cdot \left(\frac{1}{2} \cdot \cos \mathsf{PI}\left(\right) + \varepsilon \cdot \left(\frac{-1}{8} \cdot \sin \mathsf{PI}\left(\right) + \frac{-1}{48} \cdot \left(\varepsilon \cdot \cos \mathsf{PI}\left(\right)\right)\right)\right)}\right)\right) \]
11. Step-by-step derivation

12. Applied rewrites 99.7%

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

Alternative 3: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. sin-+PI/2-rev N/A

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

   4. lift-cos.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. sin-+PI/2-rev N/A

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

   7. diff-sin N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. cos-neg-rev N/A

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

   10. lower-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-PI.f64 7.5

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

7. Applied rewrites 7.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \cos \left(\left(\mathsf{PI}\left(\right) + \varepsilon\right) \cdot -0.5\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 7.4%

   \[\leadsto 2 \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) + \varepsilon, 0.5, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

10. Taylor expanded in eps around 0

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

12. Applied rewrites 99.5%

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

Alternative 4: 98.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{-1}{2} \cdot \cos x + \frac{1}{6} \cdot \left(\varepsilon \cdot \sin x\right)\right) - \sin x\right)} \]
7. 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

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-sin.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 99.3%

    \[\leadsto \left(-0.5 \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 5: 98.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(0.008333333333333333 \cdot x\right) \cdot x - 0.16666666666666666, x\right)\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (* (- (* 0.041666666666666664 (* eps eps)) 0.5) eps)
   (fma
    (* x (* x x))
    (- (* (* 0.008333333333333333 x) x) 0.16666666666666666)
    x))
  eps))

C

double code(double x, double eps) {
	return ((((0.041666666666666664 * (eps * eps)) - 0.5) * eps) - fma((x * (x * x)), (((0.008333333333333333 * x) * x) - 0.16666666666666666), x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(Float64(0.041666666666666664 * Float64(eps * eps)) - 0.5) * eps) - fma(Float64(x * Float64(x * x)), Float64(Float64(Float64(0.008333333333333333 * x) * x) - 0.16666666666666666), x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.041666666666666664 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * eps), $MachinePrecision] - N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.008333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{24} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 99.1%

    \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot x\right) \cdot x - 0.16666666666666666, x\right)\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

    \[\leadsto \left(\left(\frac{1}{24} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2}\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(\frac{1}{120} \cdot x\right) \cdot x - \frac{1}{6}, x\right)\right) \cdot \varepsilon \]
13. Step-by-step derivation

14. Applied rewrites 99.1%

    \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(0.008333333333333333 \cdot x\right) \cdot x - 0.16666666666666666, x\right)\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 6: 98.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (* (- (* 0.041666666666666664 (* eps eps)) 0.5) eps)
   (*
    (fma (- (* 0.008333333333333333 (* x x)) 0.16666666666666666) (* x x) 1.0)
    x))
  eps))

C

double code(double x, double eps) {
	return ((((0.041666666666666664 * (eps * eps)) - 0.5) * eps) - (fma(((0.008333333333333333 * (x * x)) - 0.16666666666666666), (x * x), 1.0) * x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(Float64(0.041666666666666664 * Float64(eps * eps)) - 0.5) * eps) - Float64(fma(Float64(Float64(0.008333333333333333 * Float64(x * x)) - 0.16666666666666666), Float64(x * x), 1.0) * x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.041666666666666664 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * eps), $MachinePrecision] - N[(N[(N[(N[(0.008333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{24} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{120} \cdot {x}^{2} - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 99.1%

    \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(0.008333333333333333 \cdot \left(x \cdot x\right) - 0.16666666666666666, x \cdot x, 1\right) \cdot x\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 7: 97.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{24} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{120} + \frac{-1}{5040} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 99.1%

    \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot \left(x \cdot x\right), \left(\mathsf{fma}\left(-0.0001984126984126984, x \cdot x, 0.008333333333333333\right) \cdot x\right) \cdot x - 0.16666666666666666, x\right)\right) \cdot \varepsilon \]

12. Taylor expanded in x around 0

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

14. Applied rewrites 99.1%

    \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot \left(x \cdot x\right), -0.16666666666666666, x\right)\right) \cdot \varepsilon \]
15. Add Preprocessing

Alternative 8: 97.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (* (- (* 0.041666666666666664 (* eps eps)) 0.5) eps)
   (* (fma (* x x) -0.16666666666666666 1.0) x))
  eps))

C

double code(double x, double eps) {
	return ((((0.041666666666666664 * (eps * eps)) - 0.5) * eps) - (fma((x * x), -0.16666666666666666, 1.0) * x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(Float64(Float64(0.041666666666666664 * Float64(eps * eps)) - 0.5) * eps) - Float64(fma(Float64(x * x), -0.16666666666666666, 1.0) * x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.041666666666666664 * N[(eps * eps), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * eps), $MachinePrecision] - N[(N[(N[(x * x), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 99.3%

   \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]

9. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{24} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2}\right) \cdot \varepsilon - x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 99.0%

    \[\leadsto \left(\left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon - \mathsf{fma}\left(x \cdot x, -0.16666666666666666, 1\right) \cdot x\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 9: 97.4% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.5%

   \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.16666666666666666 - 1, x, \left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.5%

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

Alternative 10: 97.4% accurate, 14.8× speedup

Math

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

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, Float64(-x)) * eps)
end

Wolfram

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

Derivation

1. Initial program 50.9%

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

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.7%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(\varepsilon \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {\varepsilon}^{2} - 1\right)\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 98.5%

   \[\leadsto \mathsf{fma}\left(\left(\varepsilon \cdot \varepsilon\right) \cdot 0.16666666666666666 - 1, x, \left(0.041666666666666664 \cdot \left(\varepsilon \cdot \varepsilon\right) - 0.5\right) \cdot \varepsilon\right) \cdot \varepsilon \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.5%

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

Alternative 11: 78.7% 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 50.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 80.6

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

5. Applied rewrites 80.6%

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

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 80.2%

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

Alternative 12: 51.6% accurate, 34.5× speedup

Math

\[\begin{array}{l} \\ 2 \cdot 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* 2.0 0.0))

C

double code(double x, double eps) {
	return 2.0 * 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 2.0d0 * 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 2.0 * 0.0;
}

Python

def code(x, eps):
	return 2.0 * 0.0

Julia

function code(x, eps)
	return Float64(2.0 * 0.0)
end

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * 0.0;
end

Wolfram

code[x_, eps_] := N[(2.0 * 0.0), $MachinePrecision]

Derivation

1. Initial program 50.9%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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. sin-+PI/2-rev N/A

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

   4. lift-cos.f64 N/A

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

   5. cos-neg-rev N/A

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

   6. sin-+PI/2-rev N/A

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

   7. diff-sin N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 51.6%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sin.f64 N/A

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

   4. distribute-lft-in N/A

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

   5. associate-*r* N/A

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

   6. metadata-eval N/A

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

   7. *-lft-identity N/A

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

   8. lower-fma.f64 N/A

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

   9. cos-neg-rev N/A

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

   10. lower-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. distribute-rgt-neg-in N/A

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

   13. metadata-eval N/A

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-PI.f64 7.5

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

7. Applied rewrites 7.5%

   \[\leadsto 2 \cdot \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \cos \left(\left(\mathsf{PI}\left(\right) + \varepsilon\right) \cdot -0.5\right)\right)} \]
8. Step-by-step derivation

9. Applied rewrites 7.4%

   \[\leadsto 2 \cdot \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \sin \left(\mathsf{fma}\left(\mathsf{PI}\left(\right) + \varepsilon, 0.5, \frac{\mathsf{PI}\left(\right)}{2}\right)\right)\right) \]

10. Taylor expanded in eps around 0

    \[\leadsto 2 \cdot \left(\sin x \cdot \color{blue}{\sin \mathsf{PI}\left(\right)}\right) \]
11. Step-by-step derivation

12. Applied rewrites 50.3%

    \[\leadsto 2 \cdot 0 \]
13. 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 2024339 
(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)))