2cos (problem 3.3.5)

Percentage Accurate: 51.7% → 99.8%

Time: 15.4s

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: 51.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (- (cos (+ x eps)) (cos x)))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return Math.cos((x + eps)) - Math.cos(x);
}

Python

def code(x, eps):
	return math.cos((x + eps)) - math.cos(x)

Julia

function code(x, eps)
	return Float64(cos(Float64(x + eps)) - cos(x))
end

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right), \mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(0.0006944444444444445, \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.25 \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (* (sin x) (cos (* 0.5 eps)))
   (sin (* 0.5 eps))
   (*
    (fma
     (* (cos x) (fma 0.0006944444444444445 (* eps eps) -0.020833333333333332))
     (* eps eps)
     (* 0.25 (cos x)))
    (* eps eps)))
  -2.0))

C

double code(double x, double eps) {
	return fma((sin(x) * cos((0.5 * eps))), sin((0.5 * eps)), (fma((cos(x) * fma(0.0006944444444444445, (eps * eps), -0.020833333333333332)), (eps * eps), (0.25 * cos(x))) * (eps * eps))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(fma(Float64(sin(x) * cos(Float64(0.5 * eps))), sin(Float64(0.5 * eps)), Float64(fma(Float64(cos(x) * fma(0.0006944444444444445, Float64(eps * eps), -0.020833333333333332)), Float64(eps * eps), Float64(0.25 * cos(x))) * Float64(eps * eps))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(0.0006944444444444445 * N[(eps * eps), $MachinePrecision] + -0.020833333333333332), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + N[(0.25 * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 45.8%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   4. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\varepsilon + 2 \cdot x\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   5. distribute-rgt-in N/A

      \[\leadsto \left(\sin \color{blue}{\left(\varepsilon \cdot \frac{1}{2} + \left(2 \cdot x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\color{blue}{\frac{1}{2} \cdot \varepsilon} + \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   7. sin-sum N/A

      \[\leadsto \left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   8. +-lft-identity N/A

      \[\leadsto \left(\left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(0 + \varepsilon\right)}\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   9. lift-+.f64 N/A

      \[\leadsto \left(\left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(0 + \varepsilon\right)}\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)} \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   11. lift-sin.f64 N/A

       \[\leadsto \left(\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)} \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   12. lower-fma.f64 N/A

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

6. Applied rewrites 99.8%

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sin \left(\varepsilon \cdot 0.5\right), \cos \left(1 \cdot x\right), \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(1 \cdot x\right)\right)} \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(1 \cdot x\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right)} \cdot -2 \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right) \cdot \mathsf{fma}\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(1 \cdot x\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right)\right)\right)} \cdot -2 \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right) \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(1 \cdot x\right) + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right)\right)}\right) \cdot -2 \]

   4. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right) \cdot \color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right) + \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(1 \cdot x\right)\right)}\right) \cdot -2 \]

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right) + \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(1 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right)} \cdot -2 \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right), \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right), \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(1 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right)} \cdot -2 \]

8. Applied rewrites 100.0%

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

9. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right), \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{4} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{48} \cdot \cos x + \frac{1}{1440} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right)}\right) \cdot -2 \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right), \color{blue}{\left(\frac{1}{4} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{48} \cdot \cos x + \frac{1}{1440} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right) \cdot {\varepsilon}^{2}}\right) \cdot -2 \]

    2. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right), \color{blue}{\left(\frac{1}{4} \cdot \cos x + {\varepsilon}^{2} \cdot \left(\frac{-1}{48} \cdot \cos x + \frac{1}{1440} \cdot \left({\varepsilon}^{2} \cdot \cos x\right)\right)\right) \cdot {\varepsilon}^{2}}\right) \cdot -2 \]

11. Applied rewrites 100.0%

    \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right), \color{blue}{\mathsf{fma}\left(\cos x \cdot \mathsf{fma}\left(0.0006944444444444445, \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.25 \cdot \cos x\right) \cdot \left(\varepsilon \cdot \varepsilon\right)}\right) \cdot -2 \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right), \left(\left(\cos x \cdot \mathsf{fma}\left(\varepsilon, -0.020833333333333332 \cdot \varepsilon, 0.25\right)\right) \cdot \varepsilon\right) \cdot \varepsilon\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma
   (* (sin x) (cos (* 0.5 eps)))
   (sin (* 0.5 eps))
   (* (* (* (cos x) (fma eps (* -0.020833333333333332 eps) 0.25)) eps) eps))
  -2.0))

C

double code(double x, double eps) {
	return fma((sin(x) * cos((0.5 * eps))), sin((0.5 * eps)), (((cos(x) * fma(eps, (-0.020833333333333332 * eps), 0.25)) * eps) * eps)) * -2.0;
}

Julia

function code(x, eps)
	return Float64(fma(Float64(sin(x) * cos(Float64(0.5 * eps))), sin(Float64(0.5 * eps)), Float64(Float64(Float64(cos(x) * fma(eps, Float64(-0.020833333333333332 * eps), 0.25)) * eps) * eps)) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(N[Cos[x], $MachinePrecision] * N[(eps * N[(-0.020833333333333332 * eps), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 45.8%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

   1. lift-sin.f64 N/A

      \[\leadsto \left(\color{blue}{\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\sin \color{blue}{\left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(2 \cdot x + \varepsilon\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   4. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(\varepsilon + 2 \cdot x\right)}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   5. distribute-rgt-in N/A

      \[\leadsto \left(\sin \color{blue}{\left(\varepsilon \cdot \frac{1}{2} + \left(2 \cdot x\right) \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   6. *-commutative N/A

      \[\leadsto \left(\sin \left(\color{blue}{\frac{1}{2} \cdot \varepsilon} + \left(2 \cdot x\right) \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   7. sin-sum N/A

      \[\leadsto \left(\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   8. +-lft-identity N/A

      \[\leadsto \left(\left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(0 + \varepsilon\right)}\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   9. lift-+.f64 N/A

      \[\leadsto \left(\left(\sin \left(\frac{1}{2} \cdot \color{blue}{\left(0 + \varepsilon\right)}\right) \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   10. lift-*.f64 N/A

       \[\leadsto \left(\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)} \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   11. lift-sin.f64 N/A

       \[\leadsto \left(\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)} \cdot \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right) + \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

   12. lower-fma.f64 N/A

       \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right), \cos \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right), \cos \left(\frac{1}{2} \cdot \varepsilon\right) \cdot \sin \left(\left(2 \cdot x\right) \cdot \frac{1}{2}\right)\right)} \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]

6. Applied rewrites 99.8%

   \[\leadsto \left(\color{blue}{\mathsf{fma}\left(\sin \left(\varepsilon \cdot 0.5\right), \cos \left(1 \cdot x\right), \cos \left(\varepsilon \cdot 0.5\right) \cdot \sin \left(1 \cdot x\right)\right)} \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2 \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(1 \cdot x\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right)} \cdot -2 \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right) \cdot \mathsf{fma}\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right), \cos \left(1 \cdot x\right), \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right)\right)\right)} \cdot -2 \]

   3. lift-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right) \cdot \color{blue}{\left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(1 \cdot x\right) + \cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right)\right)}\right) \cdot -2 \]

   4. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right) \cdot \color{blue}{\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right) + \sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(1 \cdot x\right)\right)}\right) \cdot -2 \]

   5. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right) + \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(1 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right)} \cdot -2 \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \sin \left(1 \cdot x\right), \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right), \left(\sin \left(\varepsilon \cdot \frac{1}{2}\right) \cdot \cos \left(1 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(0 + \varepsilon\right)\right)\right)} \cdot -2 \]

8. Applied rewrites 100.0%

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

9. Taylor expanded in eps around 0

   \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right), \color{blue}{{\varepsilon}^{2} \cdot \left(\frac{-1}{48} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{4} \cdot \cos x\right)}\right) \cdot -2 \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right), \color{blue}{\left(\frac{-1}{48} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{4} \cdot \cos x\right) \cdot {\varepsilon}^{2}}\right) \cdot -2 \]

    2. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right), \left(\frac{-1}{48} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{4} \cdot \cos x\right) \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \cdot -2 \]

    3. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right), \color{blue}{\left(\left(\frac{-1}{48} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{4} \cdot \cos x\right) \cdot \varepsilon\right) \cdot \varepsilon}\right) \cdot -2 \]

    4. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(\frac{1}{2} \cdot \varepsilon\right), \sin \left(\frac{1}{2} \cdot \varepsilon\right), \color{blue}{\left(\left(\frac{-1}{48} \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \frac{1}{4} \cdot \cos x\right) \cdot \varepsilon\right) \cdot \varepsilon}\right) \cdot -2 \]

11. Applied rewrites 99.9%

    \[\leadsto \mathsf{fma}\left(\sin x \cdot \cos \left(0.5 \cdot \varepsilon\right), \sin \left(0.5 \cdot \varepsilon\right), \color{blue}{\left(\left(\cos x \cdot \mathsf{fma}\left(\varepsilon, -0.020833333333333332 \cdot \varepsilon, 0.25\right)\right) \cdot \varepsilon\right) \cdot \varepsilon}\right) \cdot -2 \]
12. Add Preprocessing

Alternative 3: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.8%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right)}\right) \cdot -2 \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

   3. +-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) + \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   4. *-commutative N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right) \cdot {\varepsilon}^{2}} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   5. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

   6. sub-neg N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\frac{1}{3840} \cdot {\varepsilon}^{2} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   7. metadata-eval N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\frac{1}{3840} \cdot {\varepsilon}^{2} + \color{blue}{\frac{-1}{48}}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3840}, {\varepsilon}^{2}, \frac{-1}{48}\right)}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   9. unpow2 N/A

      \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   10. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   11. unpow2 N/A

       \[\leadsto \left(\sin \left(\frac{1}{2} \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{3840}, \varepsilon \cdot \varepsilon, \frac{-1}{48}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

   12. lower-*.f64 99.7

       \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666, \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \color{blue}{\varepsilon \cdot \varepsilon}, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

7. Applied rewrites 99.7%

   \[\leadsto \left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.00026041666666666666, \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot -2 \]
8. Add Preprocessing

Alternative 4: 99.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.8%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 5: 79.2% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.8%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 78.1

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

7. Applied rewrites 78.1%

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

Alternative 6: 79.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.8%

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

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 78.1

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

5. Applied rewrites 78.1%

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

Alternative 7: 78.6% accurate, 4.7× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (fma
    (fma
     (fma 0.0001984126984126984 (* x x) -0.008333333333333333)
     (* x x)
     0.16666666666666666)
    (* x x)
    -1.0)
   x)
  eps))

C

double code(double x, double eps) {
	return (fma(fma(fma(0.0001984126984126984, (x * x), -0.008333333333333333), (x * x), 0.16666666666666666), (x * x), -1.0) * x) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(fma(fma(fma(0.0001984126984126984, Float64(x * x), -0.008333333333333333), Float64(x * x), 0.16666666666666666), Float64(x * x), -1.0) * x) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.0001984126984126984 * N[(x * x), $MachinePrecision] + -0.008333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -1.0), $MachinePrecision] * x), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 45.8%

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

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 78.1

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

5. Applied rewrites 78.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.5%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0001984126984126984, x \cdot x, -0.008333333333333333\right), x \cdot x, 0.16666666666666666\right), x \cdot x, -1\right) \cdot x\right) \cdot \varepsilon \]
9. Add Preprocessing

Alternative 8: 78.5% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.8%

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

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 78.1

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

5. Applied rewrites 78.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.4%

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

Alternative 9: 78.4% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.8%

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

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 78.1

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

5. Applied rewrites 78.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.4%

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

Alternative 10: 78.2% accurate, 12.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return ((0.5 * x) * eps) * -2.0;
}

Python

def code(x, eps):
	return ((0.5 * x) * eps) * -2.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.8%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.7%

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

5. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 78.1

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

7. Applied rewrites 78.1%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 77.3%

    \[\leadsto \left(\left(0.5 \cdot x\right) \cdot \varepsilon\right) \cdot -2 \]
11. Add Preprocessing

Alternative 11: 78.2% 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 45.8%

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

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

   2. *-commutative N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-sin.f64 78.1

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

5. Applied rewrites 78.1%

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

6. Taylor expanded in x around 0

   \[\leadsto \left(-1 \cdot x\right) \cdot \varepsilon \]
7. Step-by-step derivation

8. Applied rewrites 77.3%

   \[\leadsto \left(-x\right) \cdot \varepsilon \]
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 2024308 
(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)))