2cos (problem 3.3.5)

Percentage Accurate: 51.6% → 99.8%

Time: 18.4s

Alternatives: 16

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: 51.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 99.5%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]
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 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 99.7

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity 99.7

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. lower-*.f64 N/A

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

   12. lower-sin.f64 99.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 99.5%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]
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 99.7%

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

7. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-cos.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 99.7

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

9. Applied rewrites 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 99.5%

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

5. Taylor expanded in eps around inf

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

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

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

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

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

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

   7. metadata-eval N/A

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

   8. distribute-rgt-in N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. metadata-eval N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 4: 99.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.00026041666666666666\right), \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right) \cdot -2 \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (*
   (*
    (fma
     (fma
      (fma -1.5500992063492063e-6 (* eps eps) 0.00026041666666666666)
      (* eps eps)
      -0.020833333333333332)
     (* eps eps)
     0.5)
    eps)
   (sin (fma 0.5 eps x)))
  -2.0))

C

double code(double x, double eps) {
	return ((fma(fma(fma(-1.5500992063492063e-6, (eps * eps), 0.00026041666666666666), (eps * eps), -0.020833333333333332), (eps * eps), 0.5) * eps) * sin(fma(0.5, eps, x))) * -2.0;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(fma(fma(-1.5500992063492063e-6, Float64(eps * eps), 0.00026041666666666666), Float64(eps * eps), -0.020833333333333332), Float64(eps * eps), 0.5) * eps) * sin(fma(0.5, eps, x))) * -2.0)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(N[(-1.5500992063492063e-6 * N[(eps * eps), $MachinePrecision] + 0.00026041666666666666), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] * N[(eps * eps), $MachinePrecision] + 0.5), $MachinePrecision] * eps), $MachinePrecision] * N[Sin[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 99.5%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]
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 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. sin-sum N/A

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

   12. *-lft-identity N/A

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

   13. lift-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-lft-identity N/A

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

   19. lower-fma.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in eps around 0

   \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right)}\right) \cdot -2 \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

    2. lower-*.f64 N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \color{blue}{\left(\left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

    3. +-commutative N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\color{blue}{\left({\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right) + \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

    4. *-commutative N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\left(\color{blue}{\left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right) \cdot {\varepsilon}^{2}} + \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    5. lower-fma.f64 N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\color{blue}{\mathsf{fma}\left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}, {\varepsilon}^{2}, \frac{1}{2}\right)} \cdot \varepsilon\right)\right) \cdot -2 \]

    6. sub-neg N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{{\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right)}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    7. *-commutative N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2}} + \left(\mathsf{neg}\left(\frac{1}{48}\right)\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    8. metadata-eval N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) \cdot {\varepsilon}^{2} + \color{blue}{\frac{-1}{48}}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    9. lower-fma.f64 N/A

       \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}, {\varepsilon}^{2}, \frac{-1}{48}\right)}, {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    10. +-commutative N/A

        \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-1}{645120} \cdot {\varepsilon}^{2} + \frac{1}{3840}}, {\varepsilon}^{2}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    11. lower-fma.f64 N/A

        \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{645120}, {\varepsilon}^{2}, \frac{1}{3840}\right)}, {\varepsilon}^{2}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    12. unpow2 N/A

        \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{645120}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{3840}\right), {\varepsilon}^{2}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    13. lower-*.f64 N/A

        \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{645120}, \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{3840}\right), {\varepsilon}^{2}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    14. unpow2 N/A

        \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{645120}, \varepsilon \cdot \varepsilon, \frac{1}{3840}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    15. lower-*.f64 N/A

        \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{645120}, \varepsilon \cdot \varepsilon, \frac{1}{3840}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{-1}{48}\right), {\varepsilon}^{2}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    16. unpow2 N/A

        \[\leadsto \left(\sin \left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{645120}, \varepsilon \cdot \varepsilon, \frac{1}{3840}\right), \varepsilon \cdot \varepsilon, \frac{-1}{48}\right), \color{blue}{\varepsilon \cdot \varepsilon}, \frac{1}{2}\right) \cdot \varepsilon\right)\right) \cdot -2 \]

    17. lower-*.f64 99.0

        \[\leadsto \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.00026041666666666666\right), \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \color{blue}{\varepsilon \cdot \varepsilon}, 0.5\right) \cdot \varepsilon\right)\right) \cdot -2 \]

11. Applied rewrites 99.0%

    \[\leadsto \left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.00026041666666666666\right), \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right)}\right) \cdot -2 \]

12. Final simplification 99.0%

    \[\leadsto \left(\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.5500992063492063 \cdot 10^{-6}, \varepsilon \cdot \varepsilon, 0.00026041666666666666\right), \varepsilon \cdot \varepsilon, -0.020833333333333332\right), \varepsilon \cdot \varepsilon, 0.5\right) \cdot \varepsilon\right) \cdot \sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right) \cdot -2 \]
13. Add Preprocessing

Alternative 5: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 99.5%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]
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 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. sin-sum N/A

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

   12. *-lft-identity N/A

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

   13. lift-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-lft-identity N/A

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

   19. lower-fma.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. sub-neg N/A

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

    7. metadata-eval N/A

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

    8. lower-fma.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 98.8

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

11. Applied rewrites 98.8%

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

12. Final simplification 98.8%

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

Alternative 6: 99.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 99.5%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]
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 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. sin-sum N/A

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

   12. *-lft-identity N/A

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

   13. lift-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-lft-identity N/A

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

   19. lower-fma.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in eps around 0

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

    1. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 98.8

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

11. Applied rewrites 98.8%

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

12. Final simplification 98.8%

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

Alternative 7: 99.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
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 99.5%

   \[\leadsto \color{blue}{\left(\sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \sin \left(0.5 \cdot \left(0 + \varepsilon\right)\right)\right) \cdot -2} \]
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 99.7%

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

   1. lift-fma.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-lft-identity N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. sin-sum N/A

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

   12. *-lft-identity N/A

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

   13. lift-*.f64 N/A

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

   14. lower-sin.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-lft-identity N/A

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

   19. lower-fma.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Taylor expanded in eps around 0

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

    1. lower-*.f64 98.1

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

11. Applied rewrites 98.1%

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

12. Final simplification 98.1%

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

Alternative 8: 97.9% accurate, 3.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.1%

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

Alternative 9: 97.9% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.1%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 96.1%

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

Alternative 10: 97.9% accurate, 5.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.1%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 96.0%

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

Alternative 11: 97.9% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.1%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 96.0%

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

12. Taylor expanded in eps around 0

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

14. Applied rewrites 96.0%

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

Alternative 12: 97.8% accurate, 7.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.1%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 95.8%

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

Alternative 13: 97.6% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot x\right) \cdot \varepsilon, 0.25, \mathsf{fma}\left({\varepsilon}^{3}, 0.16666666666666666, -\varepsilon\right)\right), \color{blue}{x}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 95.5%

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

12. Final simplification 95.5%

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

Alternative 14: 97.4% accurate, 12.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.1%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 96.0%

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

12. Taylor expanded in x around 0

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

14. Applied rewrites 95.3%

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

Alternative 15: 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 53.4%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 98.5%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 95.4%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\left(\varepsilon \cdot x\right) \cdot \varepsilon, 0.25, \mathsf{fma}\left({\varepsilon}^{3}, 0.16666666666666666, -\varepsilon\right)\right), \color{blue}{x}, \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) \]

9. Taylor expanded in eps around 0

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

11. Applied rewrites 77.8%

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

Alternative 16: 50.1% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 1.0 1.0))

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 1.0d0 - 1.0d0
end function

Java

public static double code(double x, double eps) {
	return 1.0 - 1.0;
}

Python

def code(x, eps):
	return 1.0 - 1.0

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = 1.0 - 1.0;
end

Wolfram

code[x_, eps_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 53.4%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

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

   2. lower-cos.f64 50.3

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

5. Applied rewrites 50.3%

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

6. Taylor expanded in eps around 0

   \[\leadsto 1 - 1 \]
7. Step-by-step derivation

8. Applied rewrites 50.1%

   \[\leadsto 1 - 1 \]
9. Add Preprocessing

Developer Target 1: 98.7% 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 2024259 
(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)))