2cos (problem 3.3.5)

Percentage Accurate: 37.6% → 99.4%

Time: 23.5s

Alternatives: 13

Speedup: 1.9×

Specification

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: 37.6% 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: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

   1. diff-cos 52.9%

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

   2. div-inv 52.9%

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

   3. metadata-eval 52.9%

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

   4. div-inv 52.9%

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

   5. +-commutative 52.9%

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

   6. metadata-eval 52.9%

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

3. Applied egg-rr 52.9%

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

   1. *-commutative 52.9%

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

   2. +-commutative 52.9%

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

   3. associate--l+ 78.1%

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

   4. +-inverses 78.1%

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

   5. distribute-lft-in 78.1%

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

   6. metadata-eval 78.1%

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

   7. *-commutative 78.1%

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

   8. associate-+r+ 78.1%

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

   9. +-commutative 78.1%

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

5. Simplified 78.1%

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

   1. distribute-lft-in 78.1%

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

   2. sin-sum 99.4%

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

7. Applied egg-rr 99.4%

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

   1. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.4%

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

    1. *-commutative 99.4%

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

    2. metadata-eval 99.4%

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

    3. distribute-lft-neg-in 99.4%

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

    4. distribute-lft-neg-in 99.4%

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

    5. metadata-eval 99.4%

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

    6. fma-def 99.5%

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

    7. *-commutative 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin((0.5d0 * eps))
    code = (-2.0d0) * (t_0 * ((t_0 * cos(x)) + (sin(x) * cos((0.5d0 * eps)))))
end function

Java

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

Python

def code(x, eps):
	t_0 = math.sin((0.5 * eps))
	return -2.0 * (t_0 * ((t_0 * math.cos(x)) + (math.sin(x) * math.cos((0.5 * eps)))))

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = sin((0.5 * eps));
	tmp = -2.0 * (t_0 * ((t_0 * cos(x)) + (sin(x) * cos((0.5 * eps)))));
end

Wolfram

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

Derivation

1. Initial program 43.9%

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

   1. diff-cos 52.9%

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

   2. div-inv 52.9%

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

   3. metadata-eval 52.9%

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

   4. div-inv 52.9%

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

   5. +-commutative 52.9%

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

   6. metadata-eval 52.9%

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

3. Applied egg-rr 52.9%

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

   1. *-commutative 52.9%

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

   2. +-commutative 52.9%

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

   3. associate--l+ 78.1%

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

   4. +-inverses 78.1%

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

   5. distribute-lft-in 78.1%

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

   6. metadata-eval 78.1%

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

   7. *-commutative 78.1%

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

   8. associate-+r+ 78.1%

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

   9. +-commutative 78.1%

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

5. Simplified 78.1%

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

   1. distribute-lft-in 78.1%

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

   2. sin-sum 99.4%

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

7. Applied egg-rr 99.4%

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

   1. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.4%

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

11. Final simplification 99.4%

    \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \cos x + \sin x \cdot \cos \left(0.5 \cdot \varepsilon\right)\right)\right) \]

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155 \lor \neg \left(\varepsilon \leq 0.00325\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00155) (not (<= eps 0.00325)))
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps))))
   (+
    (* (cos x) (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps))))
    (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00155) || !(eps <= 0.00325)) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	} else {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00155d0)) .or. (.not. (eps <= 0.00325d0))) then
        tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)))
    else
        tmp = (cos(x) * ((0.041666666666666664d0 * (eps ** 4.0d0)) + ((-0.5d0) * (eps * eps)))) + (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) - eps))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00155) || !(eps <= 0.00325)) {
		tmp = (Math.cos(x) * Math.cos(eps)) - (Math.cos(x) + (Math.sin(x) * Math.sin(eps)));
	} else {
		tmp = (Math.cos(x) * ((0.041666666666666664 * Math.pow(eps, 4.0)) + (-0.5 * (eps * eps)))) + (Math.sin(x) * ((0.16666666666666666 * Math.pow(eps, 3.0)) - eps));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00155) or not (eps <= 0.00325):
		tmp = (math.cos(x) * math.cos(eps)) - (math.cos(x) + (math.sin(x) * math.sin(eps)))
	else:
		tmp = (math.cos(x) * ((0.041666666666666664 * math.pow(eps, 4.0)) + (-0.5 * (eps * eps)))) + (math.sin(x) * ((0.16666666666666666 * math.pow(eps, 3.0)) - eps))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00155) || !(eps <= 0.00325))
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(x) * sin(eps))));
	else
		tmp = Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00155) || ~((eps <= 0.00325)))
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	else
		tmp = (cos(x) * ((0.041666666666666664 * (eps ^ 4.0)) + (-0.5 * (eps * eps)))) + (sin(x) * ((0.16666666666666666 * (eps ^ 3.0)) - eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00155], N[Not[LessEqual[eps, 0.00325]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00154999999999999995 or 0.00324999999999999985 < eps

   1. Initial program 57.5%

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

      1. sub-neg 57.5%

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

      2. cos-sum 98.6%

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

      3. associate-+l- 98.5%

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

      4. fma-neg 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. fma-neg 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. fma-neg 98.5%

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

      5. remove-double-neg 98.5%

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

   5. Simplified 98.5%

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

      1. fma-udef 98.5%

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

   7. Applied egg-rr 98.5%

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

   if -0.00154999999999999995 < eps < 0.00324999999999999985

   1. Initial program 28.8%

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

   2. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)} \]

      2. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} + -1 \cdot \left(\varepsilon \cdot \sin x\right) \]

      3. associate-+l+ 99.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      5. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4}\right) \cdot \cos x}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      7. unpow2 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      8. associate-*r* 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(\color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x} + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      9. associate-*r* 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}\right) \]

      10. distribute-rgt-out 99.8%

          \[\leadsto \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + -1 \cdot \varepsilon\right)} \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155 \lor \neg \left(\varepsilon \leq 0.00325\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155 \lor \neg \left(\varepsilon \leq 0.003\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00155) (not (<= eps 0.003)))
   (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x))
   (+
    (* (cos x) (+ (* 0.041666666666666664 (pow eps 4.0)) (* -0.5 (* eps eps))))
    (* (sin x) (- (* 0.16666666666666666 (pow eps 3.0)) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00155) || !(eps <= 0.003)) {
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	} else {
		tmp = (cos(x) * ((0.041666666666666664 * pow(eps, 4.0)) + (-0.5 * (eps * eps)))) + (sin(x) * ((0.16666666666666666 * pow(eps, 3.0)) - eps));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00155d0)) .or. (.not. (eps <= 0.003d0))) then
        tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x)
    else
        tmp = (cos(x) * ((0.041666666666666664d0 * (eps ** 4.0d0)) + ((-0.5d0) * (eps * eps)))) + (sin(x) * ((0.16666666666666666d0 * (eps ** 3.0d0)) - eps))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00155) || !(eps <= 0.003)) {
		tmp = ((Math.cos(x) * Math.cos(eps)) - (Math.sin(x) * Math.sin(eps))) - Math.cos(x);
	} else {
		tmp = (Math.cos(x) * ((0.041666666666666664 * Math.pow(eps, 4.0)) + (-0.5 * (eps * eps)))) + (Math.sin(x) * ((0.16666666666666666 * Math.pow(eps, 3.0)) - eps));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00155) or not (eps <= 0.003):
		tmp = ((math.cos(x) * math.cos(eps)) - (math.sin(x) * math.sin(eps))) - math.cos(x)
	else:
		tmp = (math.cos(x) * ((0.041666666666666664 * math.pow(eps, 4.0)) + (-0.5 * (eps * eps)))) + (math.sin(x) * ((0.16666666666666666 * math.pow(eps, 3.0)) - eps))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00155) || !(eps <= 0.003))
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - Float64(sin(x) * sin(eps))) - cos(x));
	else
		tmp = Float64(Float64(cos(x) * Float64(Float64(0.041666666666666664 * (eps ^ 4.0)) + Float64(-0.5 * Float64(eps * eps)))) + Float64(sin(x) * Float64(Float64(0.16666666666666666 * (eps ^ 3.0)) - eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00155) || ~((eps <= 0.003)))
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	else
		tmp = (cos(x) * ((0.041666666666666664 * (eps ^ 4.0)) + (-0.5 * (eps * eps)))) + (sin(x) * ((0.16666666666666666 * (eps ^ 3.0)) - eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00155], N[Not[LessEqual[eps, 0.003]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.041666666666666664 * N[Power[eps, 4.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00154999999999999995 or 0.0030000000000000001 < eps

   1. Initial program 57.5%

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

      1. cos-sum 98.6%

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

   3. Applied egg-rr 98.6%

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

   if -0.00154999999999999995 < eps < 0.0030000000000000001

   1. Initial program 28.8%

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

   2. Taylor expanded in eps around 0 99.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + \left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right)} \]
   3. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)} \]

      2. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + 0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right)\right)} + -1 \cdot \left(\varepsilon \cdot \sin x\right) \]

      3. associate-+l+ 99.8%

         \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right)} \]

      4. associate-*r* 99.8%

         \[\leadsto \left(\color{blue}{\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \cos x\right)\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      5. associate-*r* 99.8%

         \[\leadsto \left(\left(-0.5 \cdot {\varepsilon}^{2}\right) \cdot \cos x + \color{blue}{\left(0.041666666666666664 \cdot {\varepsilon}^{4}\right) \cdot \cos x}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot {\varepsilon}^{2} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right)} + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      7. unpow2 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \sin x\right) + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      8. associate-*r* 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(\color{blue}{\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x} + -1 \cdot \left(\varepsilon \cdot \sin x\right)\right) \]

      9. associate-*r* 99.8%

         \[\leadsto \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \left(\left(0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \sin x + \color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}\right) \]

      10. distribute-rgt-out 99.8%

          \[\leadsto \cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \color{blue}{\sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + -1 \cdot \varepsilon\right)} \]

   4. Simplified 99.8%

      \[\leadsto \color{blue}{\cos x \cdot \left(-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right) + 0.041666666666666664 \cdot {\varepsilon}^{4}\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} + \left(-\varepsilon\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00155 \lor \neg \left(\varepsilon \leq 0.003\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(0.041666666666666664 \cdot {\varepsilon}^{4} + -0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \sin x \cdot \left(0.16666666666666666 \cdot {\varepsilon}^{3} - \varepsilon\right)\\ \end{array} \]

Alternative 5: 77.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0102:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0029:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0102)
   (* (cos x) (+ (cos eps) -1.0))
   (if (<= eps 0.0029)
     (- (* (* eps eps) (* (cos x) -0.5)) (* eps (sin x)))
     (- (cos eps) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0102) {
		tmp = cos(x) * (cos(eps) + -1.0);
	} else if (eps <= 0.0029) {
		tmp = ((eps * eps) * (cos(x) * -0.5)) - (eps * sin(x));
	} else {
		tmp = cos(eps) - cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.0102d0)) then
        tmp = cos(x) * (cos(eps) + (-1.0d0))
    else if (eps <= 0.0029d0) then
        tmp = ((eps * eps) * (cos(x) * (-0.5d0))) - (eps * sin(x))
    else
        tmp = cos(eps) - cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0102) {
		tmp = Math.cos(x) * (Math.cos(eps) + -1.0);
	} else if (eps <= 0.0029) {
		tmp = ((eps * eps) * (Math.cos(x) * -0.5)) - (eps * Math.sin(x));
	} else {
		tmp = Math.cos(eps) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.0102:
		tmp = math.cos(x) * (math.cos(eps) + -1.0)
	elif eps <= 0.0029:
		tmp = ((eps * eps) * (math.cos(x) * -0.5)) - (eps * math.sin(x))
	else:
		tmp = math.cos(eps) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0102)
		tmp = Float64(cos(x) * Float64(cos(eps) + -1.0));
	elseif (eps <= 0.0029)
		tmp = Float64(Float64(Float64(eps * eps) * Float64(cos(x) * -0.5)) - Float64(eps * sin(x)));
	else
		tmp = Float64(cos(eps) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.0102)
		tmp = cos(x) * (cos(eps) + -1.0);
	elseif (eps <= 0.0029)
		tmp = ((eps * eps) * (cos(x) * -0.5)) - (eps * sin(x));
	else
		tmp = cos(eps) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0102], N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0029], N[(N[(N[(eps * eps), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.010200000000000001

   1. Initial program 58.7%

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

      1. sub-neg 58.7%

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

      2. cos-sum 98.5%

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

      3. associate-+l- 98.4%

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

      4. fma-neg 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. fma-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-neg 98.4%

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

      5. remove-double-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in eps around 0 61.8%

      \[\leadsto \cos \varepsilon \cdot \cos x - \color{blue}{\cos x} \]
   7. Step-by-step derivation

      1. sub-neg 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. neg-mul-1 61.8%

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

      2. distribute-rgt-out 61.8%

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

      3. +-commutative 61.8%

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

   10. Simplified 61.8%

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

   if -0.010200000000000001 < eps < 0.0029

   1. Initial program 28.3%

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

   2. Taylor expanded in eps around 0 98.7%

      \[\leadsto \color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right) + -0.5 \cdot \left({\varepsilon}^{2} \cdot \cos x\right)} \]
   3. Step-by-step derivation

      1. +-commutative 98.7%

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

      2. mul-1-neg 98.7%

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

      3. unsub-neg 98.7%

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

      4. *-commutative 98.7%

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

      5. associate-*l* 98.7%

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

      6. unpow2 98.7%

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

   4. Simplified 98.7%

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

   if 0.0029 < eps

   1. Initial program 57.9%

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

   2. Taylor expanded in x around 0 60.4%

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0102:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0029:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot \left(\cos x \cdot -0.5\right) - \varepsilon \cdot \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 6: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (eps + (x + x)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

   1. diff-cos 52.9%

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

   2. div-inv 52.9%

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

   3. metadata-eval 52.9%

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

   4. div-inv 52.9%

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

   5. +-commutative 52.9%

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

   6. metadata-eval 52.9%

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

3. Applied egg-rr 52.9%

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

   1. *-commutative 52.9%

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

   2. +-commutative 52.9%

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

   3. associate--l+ 78.1%

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

   4. +-inverses 78.1%

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

   5. distribute-lft-in 78.1%

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

   6. metadata-eval 78.1%

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

   7. *-commutative 78.1%

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

   8. associate-+r+ 78.1%

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

   9. +-commutative 78.1%

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

5. Simplified 78.1%

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

6. Final simplification 78.1%

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

Alternative 7: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0102 \lor \neg \left(\varepsilon \leq 0.0029\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0102) (not (<= eps 0.0029)))
   (- (cos eps) (cos x))
   (* -2.0 (* (* 0.5 eps) (sin (* 0.5 (+ eps (+ x x))))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0102) || !(eps <= 0.0029)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps + (x + x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0102d0)) .or. (.not. (eps <= 0.0029d0))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = (-2.0d0) * ((0.5d0 * eps) * sin((0.5d0 * (eps + (x + x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0102) || !(eps <= 0.0029)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = -2.0 * ((0.5 * eps) * Math.sin((0.5 * (eps + (x + x)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0102) or not (eps <= 0.0029):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = -2.0 * ((0.5 * eps) * math.sin((0.5 * (eps + (x + x)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0102) || !(eps <= 0.0029))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(-2.0 * Float64(Float64(0.5 * eps) * sin(Float64(0.5 * Float64(eps + Float64(x + x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0102) || ~((eps <= 0.0029)))
		tmp = cos(eps) - cos(x);
	else
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps + (x + x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0102], N[Not[LessEqual[eps, 0.0029]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[(0.5 * eps), $MachinePrecision] * N[Sin[N[(0.5 * N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.010200000000000001 or 0.0029 < eps

   1. Initial program 58.3%

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

   2. Taylor expanded in x around 0 60.8%

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

   if -0.010200000000000001 < eps < 0.0029

   1. Initial program 28.3%

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

      1. diff-cos 47.6%

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

      2. div-inv 47.6%

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

      3. metadata-eval 47.6%

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

      4. div-inv 47.6%

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

      5. +-commutative 47.6%

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

      6. metadata-eval 47.6%

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

   3. Applied egg-rr 47.6%

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

      1. *-commutative 47.6%

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

      2. +-commutative 47.6%

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

      3. associate--l+ 98.5%

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

      4. +-inverses 98.5%

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

      5. distribute-lft-in 98.5%

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

      6. metadata-eval 98.5%

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

      7. *-commutative 98.5%

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

      8. associate-+r+ 98.6%

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

      9. +-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in eps around 0 98.4%

      \[\leadsto -2 \cdot \left(\color{blue}{\left(0.5 \cdot \varepsilon\right)} \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0102 \lor \neg \left(\varepsilon \leq 0.0029\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 8: 77.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0125:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0066:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0125)
   (* (cos x) (+ (cos eps) -1.0))
   (if (<= eps 0.0066)
     (* -2.0 (* (* 0.5 eps) (sin (* 0.5 (+ eps (+ x x))))))
     (- (cos eps) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0125) {
		tmp = cos(x) * (cos(eps) + -1.0);
	} else if (eps <= 0.0066) {
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps + (x + x)))));
	} else {
		tmp = cos(eps) - cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.0125d0)) then
        tmp = cos(x) * (cos(eps) + (-1.0d0))
    else if (eps <= 0.0066d0) then
        tmp = (-2.0d0) * ((0.5d0 * eps) * sin((0.5d0 * (eps + (x + x)))))
    else
        tmp = cos(eps) - cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0125) {
		tmp = Math.cos(x) * (Math.cos(eps) + -1.0);
	} else if (eps <= 0.0066) {
		tmp = -2.0 * ((0.5 * eps) * Math.sin((0.5 * (eps + (x + x)))));
	} else {
		tmp = Math.cos(eps) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.0125:
		tmp = math.cos(x) * (math.cos(eps) + -1.0)
	elif eps <= 0.0066:
		tmp = -2.0 * ((0.5 * eps) * math.sin((0.5 * (eps + (x + x)))))
	else:
		tmp = math.cos(eps) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0125)
		tmp = Float64(cos(x) * Float64(cos(eps) + -1.0));
	elseif (eps <= 0.0066)
		tmp = Float64(-2.0 * Float64(Float64(0.5 * eps) * sin(Float64(0.5 * Float64(eps + Float64(x + x))))));
	else
		tmp = Float64(cos(eps) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.0125)
		tmp = cos(x) * (cos(eps) + -1.0);
	elseif (eps <= 0.0066)
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps + (x + x)))));
	else
		tmp = cos(eps) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0125], N[(N[Cos[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.0066], N[(-2.0 * N[(N[(0.5 * eps), $MachinePrecision] * N[Sin[N[(0.5 * N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.012500000000000001

   1. Initial program 58.7%

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

      1. sub-neg 58.7%

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

      2. cos-sum 98.5%

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

      3. associate-+l- 98.4%

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

      4. fma-neg 98.4%

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

   3. Applied egg-rr 98.4%

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

      1. fma-neg 98.4%

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

      2. *-commutative 98.4%

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

      3. *-commutative 98.4%

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

      4. fma-neg 98.4%

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

      5. remove-double-neg 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in eps around 0 61.8%

      \[\leadsto \cos \varepsilon \cdot \cos x - \color{blue}{\cos x} \]
   7. Step-by-step derivation

      1. sub-neg 61.8%

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

   8. Applied egg-rr 61.8%

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

      1. neg-mul-1 61.8%

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

      2. distribute-rgt-out 61.8%

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

      3. +-commutative 61.8%

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

   10. Simplified 61.8%

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

   if -0.012500000000000001 < eps < 0.0066

   1. Initial program 28.3%

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

      1. diff-cos 47.6%

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

      2. div-inv 47.6%

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

      3. metadata-eval 47.6%

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

      4. div-inv 47.6%

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

      5. +-commutative 47.6%

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

      6. metadata-eval 47.6%

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

   3. Applied egg-rr 47.6%

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

      1. *-commutative 47.6%

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

      2. +-commutative 47.6%

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

      3. associate--l+ 98.5%

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

      4. +-inverses 98.5%

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

      5. distribute-lft-in 98.5%

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

      6. metadata-eval 98.5%

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

      7. *-commutative 98.5%

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

      8. associate-+r+ 98.6%

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

      9. +-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in eps around 0 98.4%

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

   if 0.0066 < eps

   1. Initial program 57.9%

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

   2. Taylor expanded in x around 0 60.4%

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0125:\\ \;\;\;\;\cos x \cdot \left(\cos \varepsilon + -1\right)\\ \mathbf{elif}\;\varepsilon \leq 0.0066:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 9: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0102 \lor \neg \left(\varepsilon \leq 0.00186\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0102) (not (<= eps 0.00186)))
   (+ (cos eps) -1.0)
   (* -2.0 (* (* 0.5 eps) (sin (* 0.5 (+ eps (+ x x))))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0102) || !(eps <= 0.00186)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps + (x + x)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0102d0)) .or. (.not. (eps <= 0.00186d0))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = (-2.0d0) * ((0.5d0 * eps) * sin((0.5d0 * (eps + (x + x)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0102) || !(eps <= 0.00186)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = -2.0 * ((0.5 * eps) * Math.sin((0.5 * (eps + (x + x)))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0102) or not (eps <= 0.00186):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = -2.0 * ((0.5 * eps) * math.sin((0.5 * (eps + (x + x)))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0102) || !(eps <= 0.00186))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(-2.0 * Float64(Float64(0.5 * eps) * sin(Float64(0.5 * Float64(eps + Float64(x + x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0102) || ~((eps <= 0.00186)))
		tmp = cos(eps) + -1.0;
	else
		tmp = -2.0 * ((0.5 * eps) * sin((0.5 * (eps + (x + x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0102], N[Not[LessEqual[eps, 0.00186]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(-2.0 * N[(N[(0.5 * eps), $MachinePrecision] * N[Sin[N[(0.5 * N[(eps + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.010200000000000001 or 0.0018600000000000001 < eps

   1. Initial program 58.3%

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

   2. Taylor expanded in x around 0 58.8%

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

   if -0.010200000000000001 < eps < 0.0018600000000000001

   1. Initial program 28.3%

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

      1. diff-cos 47.6%

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

      2. div-inv 47.6%

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

      3. metadata-eval 47.6%

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

      4. div-inv 47.6%

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

      5. +-commutative 47.6%

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

      6. metadata-eval 47.6%

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

   3. Applied egg-rr 47.6%

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

      1. *-commutative 47.6%

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

      2. +-commutative 47.6%

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

      3. associate--l+ 98.5%

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

      4. +-inverses 98.5%

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

      5. distribute-lft-in 98.5%

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

      6. metadata-eval 98.5%

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

      7. *-commutative 98.5%

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

      8. associate-+r+ 98.6%

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

      9. +-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in eps around 0 98.4%

      \[\leadsto -2 \cdot \left(\color{blue}{\left(0.5 \cdot \varepsilon\right)} \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0102 \lor \neg \left(\varepsilon \leq 0.00186\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\\ \end{array} \]

Alternative 10: 67.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0102 \lor \neg \left(\varepsilon \leq 8.6 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0102) (not (<= eps 8.6e-8)))
   (+ (cos eps) -1.0)
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0102) || !(eps <= 8.6e-8)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = sin(x) * -eps;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0102d0)) .or. (.not. (eps <= 8.6d-8))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0102) || !(eps <= 8.6e-8)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0102) or not (eps <= 8.6e-8):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0102) || !(eps <= 8.6e-8))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0102) || ~((eps <= 8.6e-8)))
		tmp = cos(eps) + -1.0;
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0102], N[Not[LessEqual[eps, 8.6e-8]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.010200000000000001 or 8.6000000000000002e-8 < eps

   1. Initial program 57.8%

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

   2. Taylor expanded in x around 0 58.3%

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

   if -0.010200000000000001 < eps < 8.6000000000000002e-8

   1. Initial program 28.2%

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

   2. Taylor expanded in eps around 0 80.7%

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

      1. mul-1-neg 80.7%

         \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

      2. *-commutative 80.7%

         \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

      3. distribute-rgt-neg-in 80.7%

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

   4. Simplified 80.7%

      \[\leadsto \color{blue}{\sin x \cdot \left(-\varepsilon\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0102 \lor \neg \left(\varepsilon \leq 8.6 \cdot 10^{-8}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 11: 46.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018 \lor \neg \left(\varepsilon \leq 0.000136\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00018) (not (<= eps 0.000136)))
   (+ (cos eps) -1.0)
   (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00018) || !(eps <= 0.000136)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.00018d0)) .or. (.not. (eps <= 0.000136d0))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = (-0.5d0) * (eps * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00018) || !(eps <= 0.000136)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00018) or not (eps <= 0.000136):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = -0.5 * (eps * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00018) || !(eps <= 0.000136))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(-0.5 * Float64(eps * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00018) || ~((eps <= 0.000136)))
		tmp = cos(eps) + -1.0;
	else
		tmp = -0.5 * (eps * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00018], N[Not[LessEqual[eps, 0.000136]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.80000000000000011e-4 or 1.36e-4 < eps

   1. Initial program 57.5%

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

   2. Taylor expanded in x around 0 57.9%

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

   if -1.80000000000000011e-4 < eps < 1.36e-4

   1. Initial program 28.8%

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

   2. Taylor expanded in x around 0 28.8%

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

   3. Taylor expanded in eps around 0 47.4%

      \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
   4. Step-by-step derivation

      1. *-commutative 47.4%

         \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

      2. unpow2 47.4%

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

   5. Simplified 47.4%

      \[\leadsto \color{blue}{\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00018 \lor \neg \left(\varepsilon \leq 0.000136\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 12: 21.3% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.9%

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

2. Taylor expanded in x around 0 44.1%

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

3. Taylor expanded in eps around 0 24.3%

   \[\leadsto \color{blue}{-0.5 \cdot {\varepsilon}^{2}} \]
4. Step-by-step derivation

   1. *-commutative 24.3%

      \[\leadsto \color{blue}{{\varepsilon}^{2} \cdot -0.5} \]

   2. unpow2 24.3%

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

5. Simplified 24.3%

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

6. Final simplification 24.3%

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

Alternative 13: 12.5% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 43.9%

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

   1. diff-cos 52.9%

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

   2. div-inv 52.9%

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

   3. metadata-eval 52.9%

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

   4. div-inv 52.9%

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

   5. +-commutative 52.9%

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

   6. metadata-eval 52.9%

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

3. Applied egg-rr 52.9%

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

   1. *-commutative 52.9%

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

   2. +-commutative 52.9%

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

   3. associate--l+ 78.1%

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

   4. +-inverses 78.1%

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

   5. distribute-lft-in 78.1%

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

   6. metadata-eval 78.1%

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

   7. *-commutative 78.1%

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

   8. associate-+r+ 78.1%

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

   9. +-commutative 78.1%

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

5. Simplified 78.1%

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

   1. distribute-lft-in 78.1%

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

   2. sin-sum 99.4%

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

7. Applied egg-rr 99.4%

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

   1. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.4%

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

    1. *-commutative 99.4%

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

    2. metadata-eval 99.4%

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

    3. distribute-lft-neg-in 99.4%

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

    4. distribute-lft-neg-in 99.4%

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

    5. metadata-eval 99.4%

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

    6. fma-def 99.5%

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

    7. *-commutative 99.5%

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

12. Simplified 99.5%

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

14. Applied egg-rr 14.9%

    \[\leadsto \color{blue}{0} \cdot -2 \]

15. Final simplification 14.9%

    \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))