2cos (problem 3.3.5)

Percentage Accurate: 52.0% → 99.8%

Time: 18.8s

Alternatives: 6

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. sin-lowering-sin.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.5%

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

   1. *-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-rgt-in N/A

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

   4. +-rgt-identity N/A

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

   5. sin-sum N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. cos-lowering-cos.f64 N/A

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

   11. +-rgt-identity N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. cos-lowering-cos.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. +-lowering-+.f64 N/A

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

   17. sin-lowering-sin.f64 N/A

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

   18. +-rgt-identity N/A

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

   19. *-lowering-*.f64 99.8

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

6. Applied egg-rr 99.8%

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

   1. +-rgt-identity 99.8

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

8. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. count-2 N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. *-lowering-*.f64 99.8

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. sin-lowering-sin.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.5%

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

   1. associate-+r+ N/A

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

   2. count-2 N/A

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

   3. accelerator-lowering-fma.f64 99.6

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

6. Applied egg-rr 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. sin-lowering-sin.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.5%

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-lowering-*.f64 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sin-lowering-sin.f64 N/A

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

   6. +-rgt-identity N/A

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

   7. *-lowering-*.f64 N/A

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

   8. sin-lowering-sin.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-lowering-+.f64 N/A

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

   11. +-lowering-+.f64 99.5

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 57.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. sin-lowering-sin.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.5%

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

   1. *-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-rgt-in N/A

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

   4. +-rgt-identity N/A

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

   5. sin-sum N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. cos-lowering-cos.f64 N/A

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

   11. +-rgt-identity N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. cos-lowering-cos.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. +-lowering-+.f64 N/A

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

   17. sin-lowering-sin.f64 N/A

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

   18. +-rgt-identity N/A

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

   19. *-lowering-*.f64 99.8

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

6. Applied egg-rr 99.8%

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

   1. +-rgt-identity 99.8

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

8. Applied egg-rr 99.8%

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

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

10. Applied egg-rr 56.7%

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

11. Final simplification 56.7%

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

Alternative 5: 52.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

Alternative 6: 50.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.1%

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

   1. diff-cos N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. *-lowering-*.f64 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. sin-lowering-sin.f64 N/A

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

   7. div-inv N/A

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

   8. metadata-eval N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. +-lowering-+.f64 N/A

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

   12. +-lowering-+.f64 N/A

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

   13. sin-lowering-sin.f64 N/A

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

   14. div-inv N/A

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

   15. metadata-eval N/A

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

   16. *-lowering-*.f64 N/A

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

4. Applied egg-rr 99.5%

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

   1. *-commutative N/A

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

   2. associate-+r+ N/A

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

   3. distribute-rgt-in N/A

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

   4. +-rgt-identity N/A

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

   5. sin-sum N/A

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. sin-lowering-sin.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. +-lowering-+.f64 N/A

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

   10. cos-lowering-cos.f64 N/A

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

   11. +-rgt-identity N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. cos-lowering-cos.f64 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. +-lowering-+.f64 N/A

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

   17. sin-lowering-sin.f64 N/A

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

   18. +-rgt-identity N/A

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

   19. *-lowering-*.f64 99.8

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

6. Applied egg-rr 99.8%

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

8. Applied egg-rr 50.3%

   \[\leadsto \color{blue}{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \varepsilon\right)\right)\right) \cdot -2} \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

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

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

   6. accelerator-lowering-fma.f64 N/A

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

   7. associate-*r* N/A

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

   8. metadata-eval N/A

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

   9. *-lft-identity N/A

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

   10. cos-lowering-cos.f64 N/A

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

   11. metadata-eval 50.3

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

10. Applied egg-rr 50.3%

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

11. Final simplification 50.3%

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

Developer Target 1: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 98.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

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

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

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

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