2cos (problem 3.3.5)

Percentage Accurate: 52.0% → 99.8%

Time: 18.0s

Alternatives: 13

Speedup: 25.9×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.0% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := 0.5 \cdot \left(x + x\right)\\ \left(t\_0 \cdot \mathsf{fma}\left(\sin t\_1, \cos \left(\varepsilon \cdot 0.5\right), \cos t\_1 \cdot t\_0\right)\right) \cdot -2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))) (t_1 (* 0.5 (+ x x))))
   (* (* t_0 (fma (sin t_1) (cos (* eps 0.5)) (* (cos t_1) t_0))) -2.0)))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = 0.5 * (x + x);
	return (t_0 * fma(sin(t_1), cos((eps * 0.5)), (cos(t_1) * t_0))) * -2.0;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(0.5 * Float64(x + x))
	return Float64(Float64(t_0 * fma(sin(t_1), cos(Float64(eps * 0.5)), Float64(cos(t_1) * t_0))) * -2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 * N[(x + x), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$0 * N[(N[Sin[t$95$1], $MachinePrecision] * N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[t$95$1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]]]

Derivation

1. Initial program 52.1%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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 egg-rr 99.5%

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

   1. lift-+.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-+.f64 N/A

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

   5. lift-+.f64 N/A

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

   6. associate-+r+ N/A

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

   7. distribute-rgt-in N/A

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

   8. +-rgt-identity N/A

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

   9. lift-+.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. sin-sum N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-+.f64 N/A

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

   16. lower-cos.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-rgt-identity N/A

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

   19. lift-sin.f64 N/A

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

6. Applied egg-rr 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. lift-+.f64 N/A

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

   2. diff-cos N/A

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

   3. *-commutative N/A

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

   4. 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 egg-rr 99.5%

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

5. Taylor expanded in eps around inf

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

   1. metadata-eval N/A

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

   2. cancel-sign-sub-inv N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. metadata-eval N/A

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

   9. distribute-lft-in N/A

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

   10. associate-*r* N/A

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

   11. metadata-eval N/A

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

   12. *-lft-identity N/A

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

   13. lower-fma.f64 99.6

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 3: 99.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.2

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

5. Simplified 99.2%

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

Alternative 4: 98.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. lower-fma.f64 N/A

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

5. Simplified 99.5%

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

6. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 98.5

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

8. Simplified 98.5%

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

Alternative 5: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

8. Simplified 98.5%

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

Alternative 6: 98.5% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) + \frac{-1}{2} \cdot {\varepsilon}^{2}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right)} \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right) + -1 \cdot \varepsilon}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right)}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \varepsilon\right) \cdot x} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot \varepsilon\right)} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot \varepsilon, \frac{1}{4} \cdot {\varepsilon}^{2}\right)}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{1}{6}}, \frac{1}{4} \cdot {\varepsilon}^{2}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{1}{6}}, \frac{1}{4} \cdot {\varepsilon}^{2}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{4}}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{4}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot \varepsilon\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\varepsilon \cdot \left(\frac{1}{4} \cdot \varepsilon\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   17. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \color{blue}{\mathsf{neg}\left(\varepsilon\right)}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   18. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \color{blue}{\mathsf{neg}\left(\varepsilon\right)}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}}\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   21. lower-*.f64 97.9

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

8. Simplified 97.9%

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

9. Taylor expanded in x around 0

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

    1. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6} \cdot \color{blue}{\left(x \cdot \varepsilon\right)} + \frac{1}{4} \cdot {\varepsilon}^{2}, \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    2. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{6} \cdot x\right) \cdot \varepsilon} + \frac{1}{4} \cdot {\varepsilon}^{2}, \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\frac{1}{6} \cdot x\right) \cdot \varepsilon + \frac{1}{4} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}, \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    4. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \left(\frac{1}{6} \cdot x\right) \cdot \varepsilon + \color{blue}{\left(\frac{1}{4} \cdot \varepsilon\right) \cdot \varepsilon}, \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    5. distribute-rgt-in N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)}, \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    6. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right)}, \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    7. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot \varepsilon + \frac{1}{6} \cdot x\right)}, \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot \frac{1}{4}} + \frac{1}{6} \cdot x\right), \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{1}{4}, \frac{1}{6} \cdot x\right)}, \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{1}{4}, \color{blue}{x \cdot \frac{1}{6}}\right), \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \]

    11. lower-*.f64 97.9

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

11. Simplified 97.9%

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

Alternative 7: 98.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1\right) + \frac{-1}{2} \cdot \varepsilon\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) - 1, \frac{-1}{2} \cdot \varepsilon\right)} \]

   3. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + \left(\mathsf{neg}\left(1\right)\right)}, \frac{-1}{2} \cdot \varepsilon\right) \]

   4. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon\right) + \color{blue}{-1}, \frac{-1}{2} \cdot \varepsilon\right) \]

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot x + \frac{1}{4} \cdot \varepsilon, -1\right)}, \frac{-1}{2} \cdot \varepsilon\right) \]

   6. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{6}} + \frac{1}{4} \cdot \varepsilon, -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6}, \frac{1}{4} \cdot \varepsilon\right)}, -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   8. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \color{blue}{\varepsilon \cdot \frac{1}{4}}\right), -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \frac{1}{6}, \color{blue}{\varepsilon \cdot \frac{1}{4}}\right), -1\right), \frac{-1}{2} \cdot \varepsilon\right) \]

   10. lower-*.f64 97.7

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

8. Simplified 97.7%

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

9. Final simplification 97.7%

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

Alternative 8: 98.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) + \frac{-1}{2} \cdot {\varepsilon}^{2}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right)} \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right) + -1 \cdot \varepsilon}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right)}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \varepsilon\right) \cdot x} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot \varepsilon\right)} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot \varepsilon, \frac{1}{4} \cdot {\varepsilon}^{2}\right)}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{1}{6}}, \frac{1}{4} \cdot {\varepsilon}^{2}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{1}{6}}, \frac{1}{4} \cdot {\varepsilon}^{2}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{4}}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{4}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot \varepsilon\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\varepsilon \cdot \left(\frac{1}{4} \cdot \varepsilon\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   17. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \color{blue}{\mathsf{neg}\left(\varepsilon\right)}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   18. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \color{blue}{\mathsf{neg}\left(\varepsilon\right)}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}}\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   21. lower-*.f64 97.9

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

8. Simplified 97.9%

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

9. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \left(\frac{1}{4} \cdot {x}^{2} - \frac{1}{2}\right) + x \cdot \left(\frac{1}{6} \cdot {x}^{2} - 1\right)\right)} \]

11. Simplified 97.7%

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

Alternative 9: 98.4% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right)\right) + \frac{-1}{2} \cdot {\varepsilon}^{2}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, -1 \cdot \varepsilon + x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right)} \]

   3. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}\right) + -1 \cdot \varepsilon}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   4. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot \left(\varepsilon \cdot x\right) + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right)}, \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\left(\frac{1}{6} \cdot \varepsilon\right) \cdot x} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{6} \cdot \varepsilon\right)} + \frac{1}{4} \cdot {\varepsilon}^{2}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot \varepsilon, \frac{1}{4} \cdot {\varepsilon}^{2}\right)}, -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{1}{6}}, \frac{1}{4} \cdot {\varepsilon}^{2}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \color{blue}{\varepsilon \cdot \frac{1}{6}}, \frac{1}{4} \cdot {\varepsilon}^{2}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{{\varepsilon}^{2} \cdot \frac{1}{4}}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} \cdot \frac{1}{4}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   12. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\frac{1}{4} \cdot \varepsilon\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   14. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \color{blue}{\varepsilon \cdot \left(\frac{1}{4} \cdot \varepsilon\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \frac{1}{4}\right)}\right), -1 \cdot \varepsilon\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   17. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \color{blue}{\mathsf{neg}\left(\varepsilon\right)}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   18. lower-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \color{blue}{\mathsf{neg}\left(\varepsilon\right)}\right), \frac{-1}{2} \cdot {\varepsilon}^{2}\right) \]

   19. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \mathsf{neg}\left(\varepsilon\right)\right), \color{blue}{\frac{-1}{2} \cdot {\varepsilon}^{2}}\right) \]

   20. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, \varepsilon \cdot \frac{1}{6}, \varepsilon \cdot \left(\varepsilon \cdot \frac{1}{4}\right)\right), \mathsf{neg}\left(\varepsilon\right)\right), \frac{-1}{2} \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)}\right) \]

   21. lower-*.f64 97.9

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

8. Simplified 97.9%

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

9. Taylor expanded in eps around 0

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

    1. lower-*.f64 N/A

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

    2. sub-neg N/A

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

    3. unpow2 N/A

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

    4. associate-*r* N/A

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

    5. *-commutative N/A

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

    6. metadata-eval N/A

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

    7. lower-fma.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 97.7

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

11. Simplified 97.7%

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

Alternative 10: 98.0% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 96.9

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

8. Simplified 96.9%

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

Alternative 11: 97.8% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower--.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-cos.f64 N/A

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

   9. lower-sin.f64 99.2

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. neg-mul-1 N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. lower-*.f64 96.8

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

8. Simplified 96.8%

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

9. Final simplification 96.8%

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

Alternative 12: 78.5% accurate, 25.9× speedup

Math

\[\begin{array}{l} \\ -\varepsilon \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

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

   3. lower-*.f64 N/A

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

   4. lower-sin.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 79.9

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

5. Simplified 79.9%

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

6. Taylor expanded in x around 0

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f64 78.6

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

8. Simplified 78.6%

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

9. Final simplification 78.6%

   \[\leadsto -\varepsilon \cdot x \]
10. Add Preprocessing

Alternative 13: 50.7% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 52.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto \color{blue}{\cos \varepsilon + \left(\mathsf{neg}\left(1\right)\right)} \]

   2. metadata-eval N/A

      \[\leadsto \cos \varepsilon + \color{blue}{-1} \]

   3. lower-+.f64 N/A

      \[\leadsto \color{blue}{\cos \varepsilon + -1} \]

   4. lower-cos.f64 50.3

      \[\leadsto \color{blue}{\cos \varepsilon} + -1 \]

5. Simplified 50.3%

   \[\leadsto \color{blue}{\cos \varepsilon + -1} \]

6. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{1} + -1 \]
7. Step-by-step derivation

8. Simplified 50.2%

   \[\leadsto \color{blue}{1} + -1 \]
9. Step-by-step derivation

   1. metadata-eval 50.2

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 50.2%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (pow (cbrt (* (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))) (sin (* 0.5 eps)))) 3.0))

C

double code(double x, double eps) {
	return pow(cbrt(((-2.0 * sin((0.5 * fma(2.0, x, eps)))) * sin((0.5 * eps)))), 3.0);
}

Julia

function code(x, eps)
	return cbrt(Float64(Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))) * sin(Float64(0.5 * eps)))) ^ 3.0
end

Wolfram

code[x_, eps_] := N[Power[N[Power[N[(N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Reproduce

herbie shell --seed 2024207 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* -2 (sin (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

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