2cos (problem 3.3.5)

Percentage Accurate: 38.4% → 99.4%

Time: 18.6s

Alternatives: 15

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: 38.4% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 36.7%

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

   1. diff-cos 45.8%

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

   2. div-inv 45.8%

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

   3. metadata-eval 45.8%

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

   4. div-inv 45.8%

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

   5. +-commutative 45.8%

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

   6. metadata-eval 45.8%

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

3. Applied egg-rr 45.8%

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

   1. *-commutative 45.8%

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

   2. +-commutative 45.8%

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

   3. associate--l+ 74.9%

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

   4. +-inverses 74.9%

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

   5. distribute-lft-in 74.9%

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

   6. metadata-eval 74.9%

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

   7. *-commutative 74.9%

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

   8. associate-+r+ 74.9%

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

   9. +-commutative 74.9%

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

5. Simplified 74.9%

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

6. Taylor expanded in x around -inf 74.9%

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

   1. cancel-sign-sub-inv 74.9%

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

   2. metadata-eval 74.9%

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

   3. count-2 74.9%

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

   4. distribute-lft-in 74.9%

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

   5. sin-sum 99.4%

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

8. Applied egg-rr 99.4%

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

   1. fma-def 99.4%

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

   2. distribute-rgt-in 99.4%

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

   3. distribute-lft-out 99.4%

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

   4. metadata-eval 99.4%

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

   5. distribute-rgt-in 99.4%

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

   6. distribute-lft-out 99.4%

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

   7. metadata-eval 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in eps around inf 99.4%

    \[\leadsto \color{blue}{-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\cos x \cdot \sin \left(0.5 \cdot \varepsilon\right) + \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\right)} \]
12. 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. *-commutative 99.4%

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

    3. *-commutative 99.4%

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

    4. fma-def 99.4%

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

    5. associate-*l* 99.4%

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

    6. fma-def 99.4%

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

    7. +-commutative 99.4%

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

    8. *-commutative 99.4%

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

    9. *-commutative 99.4%

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

    10. fma-def 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (* -2.0 (* t_0 (+ (* (cos x) t_0) (* (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 * ((cos(x) * t_0) + (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 * ((cos(x) * t_0) + (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 * ((Math.cos(x) * t_0) + (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 * ((math.cos(x) * t_0) + (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(cos(x) * t_0) + 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 * ((cos(x) * t_0) + (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[(N[Cos[x], $MachinePrecision] * t$95$0), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 36.7%

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

   1. diff-cos 45.8%

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

   2. div-inv 45.8%

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

   3. metadata-eval 45.8%

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

   4. div-inv 45.8%

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

   5. +-commutative 45.8%

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

   6. metadata-eval 45.8%

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

3. Applied egg-rr 45.8%

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

   1. *-commutative 45.8%

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

   2. +-commutative 45.8%

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

   3. associate--l+ 74.9%

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

   4. +-inverses 74.9%

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

   5. distribute-lft-in 74.9%

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

   6. metadata-eval 74.9%

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

   7. *-commutative 74.9%

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

   8. associate-+r+ 74.9%

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

   9. +-commutative 74.9%

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

5. Simplified 74.9%

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

6. Taylor expanded in x around -inf 74.9%

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

   1. cancel-sign-sub-inv 74.9%

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

   2. metadata-eval 74.9%

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

   3. count-2 74.9%

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

   4. distribute-lft-in 74.9%

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

   5. sin-sum 99.4%

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

8. Applied egg-rr 99.4%

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

   1. fma-def 99.4%

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

   2. distribute-rgt-in 99.4%

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

   3. distribute-lft-out 99.4%

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

   4. metadata-eval 99.4%

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

   5. distribute-rgt-in 99.4%

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

   6. distribute-lft-out 99.4%

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

   7. metadata-eval 99.4%

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

10. Simplified 99.4%

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

    1. fma-udef 99.4%

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

    2. *-rgt-identity 99.4%

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

    3. *-rgt-identity 99.4%

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

12. Applied egg-rr 99.4%

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

13. Final simplification 99.4%

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

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145 \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000145) (not (<= eps 0.00013)))
   (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps))))
   (+
    (* (cos x) (* (* eps eps) -0.5))
    (* (sin x) (- (* (pow eps 3.0) 0.16666666666666666) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000145) || !(eps <= 0.00013)) {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	} else {
		tmp = (cos(x) * ((eps * eps) * -0.5)) + (sin(x) * ((pow(eps, 3.0) * 0.16666666666666666) - 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 <= (-0.000145d0)) .or. (.not. (eps <= 0.00013d0))) then
        tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)))
    else
        tmp = (cos(x) * ((eps * eps) * (-0.5d0))) + (sin(x) * (((eps ** 3.0d0) * 0.16666666666666666d0) - eps))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000145) || !(eps <= 0.00013)) {
		tmp = (Math.cos(x) * Math.cos(eps)) - (Math.cos(x) + (Math.sin(x) * Math.sin(eps)));
	} else {
		tmp = (Math.cos(x) * ((eps * eps) * -0.5)) + (Math.sin(x) * ((Math.pow(eps, 3.0) * 0.16666666666666666) - eps));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.000145) or not (eps <= 0.00013):
		tmp = (math.cos(x) * math.cos(eps)) - (math.cos(x) + (math.sin(x) * math.sin(eps)))
	else:
		tmp = (math.cos(x) * ((eps * eps) * -0.5)) + (math.sin(x) * ((math.pow(eps, 3.0) * 0.16666666666666666) - eps))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000145) || !(eps <= 0.00013))
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(x) * sin(eps))));
	else
		tmp = Float64(Float64(cos(x) * Float64(Float64(eps * eps) * -0.5)) + Float64(sin(x) * Float64(Float64((eps ^ 3.0) * 0.16666666666666666) - eps)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.000145) || ~((eps <= 0.00013)))
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	else
		tmp = (cos(x) * ((eps * eps) * -0.5)) + (sin(x) * (((eps ^ 3.0) * 0.16666666666666666) - eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000145], N[Not[LessEqual[eps, 0.00013]], $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[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.45e-4 or 1.29999999999999989e-4 < eps

   1. Initial program 50.9%

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

      1. sub-neg 50.9%

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

      2. cos-sum 98.7%

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

      3. associate-+l- 98.7%

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

      4. fma-neg 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around inf 98.7%

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

   if -1.45e-4 < eps < 1.29999999999999989e-4

   1. Initial program 21.9%

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

      1. expm1-log1p-u 21.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right)\right)\right)} - \cos x \]

   3. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right)\right)\right)} - \cos x \]

   4. Taylor expanded in eps around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-*r* 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-*r* 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. unpow2 99.9%

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

      12. +-commutative 99.9%

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

      13. sub-neg 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145 \lor \neg \left(\varepsilon \leq 0.00013\right):\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \cos \varepsilon\\ t_1 := \sin x \cdot \sin \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;\left(t_0 - t_1\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00017:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 - \left(\cos x + t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (cos x) (cos eps))) (t_1 (* (sin x) (sin eps))))
   (if (<= eps -0.000145)
     (- (- t_0 t_1) (cos x))
     (if (<= eps 0.00017)
       (+
        (* (cos x) (* (* eps eps) -0.5))
        (* (sin x) (- (* (pow eps 3.0) 0.16666666666666666) eps)))
       (- t_0 (+ (cos x) t_1))))))

C

double code(double x, double eps) {
	double t_0 = cos(x) * cos(eps);
	double t_1 = sin(x) * sin(eps);
	double tmp;
	if (eps <= -0.000145) {
		tmp = (t_0 - t_1) - cos(x);
	} else if (eps <= 0.00017) {
		tmp = (cos(x) * ((eps * eps) * -0.5)) + (sin(x) * ((pow(eps, 3.0) * 0.16666666666666666) - eps));
	} else {
		tmp = t_0 - (cos(x) + t_1);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(x) * cos(eps)
    t_1 = sin(x) * sin(eps)
    if (eps <= (-0.000145d0)) then
        tmp = (t_0 - t_1) - cos(x)
    else if (eps <= 0.00017d0) then
        tmp = (cos(x) * ((eps * eps) * (-0.5d0))) + (sin(x) * (((eps ** 3.0d0) * 0.16666666666666666d0) - eps))
    else
        tmp = t_0 - (cos(x) + t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(x) * Math.cos(eps);
	double t_1 = Math.sin(x) * Math.sin(eps);
	double tmp;
	if (eps <= -0.000145) {
		tmp = (t_0 - t_1) - Math.cos(x);
	} else if (eps <= 0.00017) {
		tmp = (Math.cos(x) * ((eps * eps) * -0.5)) + (Math.sin(x) * ((Math.pow(eps, 3.0) * 0.16666666666666666) - eps));
	} else {
		tmp = t_0 - (Math.cos(x) + t_1);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(x) * math.cos(eps)
	t_1 = math.sin(x) * math.sin(eps)
	tmp = 0
	if eps <= -0.000145:
		tmp = (t_0 - t_1) - math.cos(x)
	elif eps <= 0.00017:
		tmp = (math.cos(x) * ((eps * eps) * -0.5)) + (math.sin(x) * ((math.pow(eps, 3.0) * 0.16666666666666666) - eps))
	else:
		tmp = t_0 - (math.cos(x) + t_1)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(x) * cos(eps))
	t_1 = Float64(sin(x) * sin(eps))
	tmp = 0.0
	if (eps <= -0.000145)
		tmp = Float64(Float64(t_0 - t_1) - cos(x));
	elseif (eps <= 0.00017)
		tmp = Float64(Float64(cos(x) * Float64(Float64(eps * eps) * -0.5)) + Float64(sin(x) * Float64(Float64((eps ^ 3.0) * 0.16666666666666666) - eps)));
	else
		tmp = Float64(t_0 - Float64(cos(x) + t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(x) * cos(eps);
	t_1 = sin(x) * sin(eps);
	tmp = 0.0;
	if (eps <= -0.000145)
		tmp = (t_0 - t_1) - cos(x);
	elseif (eps <= 0.00017)
		tmp = (cos(x) * ((eps * eps) * -0.5)) + (sin(x) * (((eps ^ 3.0) * 0.16666666666666666) - eps));
	else
		tmp = t_0 - (cos(x) + t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.000145], N[(N[(t$95$0 - t$95$1), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00017], N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[Cos[x], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.45e-4

   1. Initial program 51.9%

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

      1. cos-sum 98.7%

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

   3. Applied egg-rr 98.7%

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

   if -1.45e-4 < eps < 1.7e-4

   1. Initial program 21.9%

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

      1. expm1-log1p-u 21.9%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right)\right)\right)} - \cos x \]

   3. Applied egg-rr 21.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right)\right)\right)} - \cos x \]

   4. Taylor expanded in eps around 0 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+l+ 99.9%

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

      3. associate-*r* 99.9%

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

      4. mul-1-neg 99.9%

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

      5. distribute-lft-neg-out 99.9%

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

      6. distribute-rgt-in 99.9%

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

      7. +-commutative 99.9%

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

      8. associate-*r* 99.9%

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

      9. *-commutative 99.9%

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

      10. *-commutative 99.9%

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

      11. unpow2 99.9%

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

      12. +-commutative 99.9%

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

      13. sub-neg 99.9%

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

   6. Simplified 99.9%

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

   if 1.7e-4 < eps

   1. Initial program 50.0%

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

      1. sub-neg 50.0%

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

      2. cos-sum 98.8%

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

      3. associate-+l- 98.8%

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

      4. fma-neg 98.9%

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

   3. Applied egg-rr 98.9%

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

   4. Taylor expanded in x around inf 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{elif}\;\varepsilon \leq 0.00017:\\ \;\;\;\;\cos x \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 5: 78.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

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 * (sin(x) + (t_0 * cos((0.5d0 * (x + x))))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	t_0 = sin((0.5 * eps));
	tmp = -2.0 * (t_0 * (sin(x) + (t_0 * cos((0.5 * (x + x))))));
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[Sin[x], $MachinePrecision] + N[(t$95$0 * N[Cos[N[(0.5 * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 36.7%

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

   1. diff-cos 45.8%

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

   2. div-inv 45.8%

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

   3. metadata-eval 45.8%

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

   4. div-inv 45.8%

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

   5. +-commutative 45.8%

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

   6. metadata-eval 45.8%

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

3. Applied egg-rr 45.8%

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

   1. *-commutative 45.8%

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

   2. +-commutative 45.8%

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

   3. associate--l+ 74.9%

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

   4. +-inverses 74.9%

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

   5. distribute-lft-in 74.9%

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

   6. metadata-eval 74.9%

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

   7. *-commutative 74.9%

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

   8. associate-+r+ 74.9%

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

   9. +-commutative 74.9%

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

5. Simplified 74.9%

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

6. Taylor expanded in x around -inf 74.9%

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

   1. cancel-sign-sub-inv 74.9%

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

   2. metadata-eval 74.9%

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

   3. count-2 74.9%

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

   4. distribute-rgt-in 74.9%

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

   5. *-commutative 74.9%

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

   6. sin-sum 99.4%

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

8. Applied egg-rr 99.4%

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

9. Taylor expanded in eps around 0 76.5%

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

10. Final simplification 76.5%

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

Alternative 6: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00056:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.185:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00056)
   (* -2.0 (pow (sin (* 0.5 eps)) 2.0))
   (if (<= eps 0.185)
     (- (* -0.5 (* eps (* eps (cos x)))) (* (sin x) eps))
     (- (cos eps) (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00056) {
		tmp = -2.0 * pow(sin((0.5 * eps)), 2.0);
	} else if (eps <= 0.185) {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (sin(x) * eps);
	} 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 <= (-0.00056d0)) then
        tmp = (-2.0d0) * (sin((0.5d0 * eps)) ** 2.0d0)
    else if (eps <= 0.185d0) then
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (sin(x) * eps)
    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 <= -0.00056) {
		tmp = -2.0 * Math.pow(Math.sin((0.5 * eps)), 2.0);
	} else if (eps <= 0.185) {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (Math.sin(x) * eps);
	} else {
		tmp = Math.cos(eps) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00056:
		tmp = -2.0 * math.pow(math.sin((0.5 * eps)), 2.0)
	elif eps <= 0.185:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (math.sin(x) * eps)
	else:
		tmp = math.cos(eps) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00056)
		tmp = Float64(-2.0 * (sin(Float64(0.5 * eps)) ^ 2.0));
	elseif (eps <= 0.185)
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(sin(x) * eps));
	else
		tmp = Float64(cos(eps) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00056)
		tmp = -2.0 * (sin((0.5 * eps)) ^ 2.0);
	elseif (eps <= 0.185)
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (sin(x) * eps);
	else
		tmp = cos(eps) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00056], N[(-2.0 * N[Power[N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.185], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * eps), $MachinePrecision]), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -5.5999999999999995e-4

   1. Initial program 51.9%

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

      1. diff-cos 52.7%

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

      2. div-inv 52.7%

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

      3. metadata-eval 52.7%

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

      4. div-inv 52.7%

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

      5. +-commutative 52.7%

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

      6. metadata-eval 52.7%

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

   3. Applied egg-rr 52.7%

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

      1. *-commutative 52.7%

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

      2. +-commutative 52.7%

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

      3. associate--l+ 53.7%

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

      4. +-inverses 53.7%

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

      5. distribute-lft-in 53.7%

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

      6. metadata-eval 53.7%

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

      7. *-commutative 53.7%

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

      8. associate-+r+ 53.6%

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

      9. +-commutative 53.6%

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

   5. Simplified 53.6%

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

   6. Taylor expanded in x around 0 54.2%

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

   if -5.5999999999999995e-4 < eps < 0.185

   1. Initial program 21.8%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. mul-1-neg 99.2%

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

      3. unsub-neg 99.2%

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

      4. unpow2 99.2%

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

      5. associate-*l* 99.2%

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

   4. Simplified 99.2%

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

   if 0.185 < eps

   1. Initial program 50.7%

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

   2. Taylor expanded in x around 0 53.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00056:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{elif}\;\varepsilon \leq 0.185:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \sin x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 7: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (-2.0d0))))))
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 * -2.0)))));
}

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = -2.0 * (sin((0.5 * eps)) * sin((0.5 * (eps - (x * -2.0)))));
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 * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.7%

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

   1. diff-cos 45.8%

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

   2. div-inv 45.8%

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

   3. metadata-eval 45.8%

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

   4. div-inv 45.8%

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

   5. +-commutative 45.8%

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

   6. metadata-eval 45.8%

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

3. Applied egg-rr 45.8%

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

   1. *-commutative 45.8%

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

   2. +-commutative 45.8%

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

   3. associate--l+ 74.9%

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

   4. +-inverses 74.9%

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

   5. distribute-lft-in 74.9%

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

   6. metadata-eval 74.9%

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

   7. *-commutative 74.9%

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

   8. associate-+r+ 74.9%

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

   9. +-commutative 74.9%

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

5. Simplified 74.9%

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

6. Taylor expanded in x around -inf 74.9%

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

7. Final simplification 74.9%

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

Alternative 8: 66.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 235000000000:\\ \;\;\;\;\frac{-{\sin \varepsilon}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.4e-6)
   (+ (cos eps) -1.0)
   (if (<= eps 1.8e-112)
     (* (sin x) (- eps))
     (if (<= eps 235000000000.0)
       (/ (- (pow (sin eps) 2.0)) 2.0)
       (- (cos eps) (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.4e-6) {
		tmp = cos(eps) + -1.0;
	} else if (eps <= 1.8e-112) {
		tmp = sin(x) * -eps;
	} else if (eps <= 235000000000.0) {
		tmp = -pow(sin(eps), 2.0) / 2.0;
	} 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 <= (-1.4d-6)) then
        tmp = cos(eps) + (-1.0d0)
    else if (eps <= 1.8d-112) then
        tmp = sin(x) * -eps
    else if (eps <= 235000000000.0d0) then
        tmp = -(sin(eps) ** 2.0d0) / 2.0d0
    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 <= -1.4e-6) {
		tmp = Math.cos(eps) + -1.0;
	} else if (eps <= 1.8e-112) {
		tmp = Math.sin(x) * -eps;
	} else if (eps <= 235000000000.0) {
		tmp = -Math.pow(Math.sin(eps), 2.0) / 2.0;
	} else {
		tmp = Math.cos(eps) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.4e-6:
		tmp = math.cos(eps) + -1.0
	elif eps <= 1.8e-112:
		tmp = math.sin(x) * -eps
	elif eps <= 235000000000.0:
		tmp = -math.pow(math.sin(eps), 2.0) / 2.0
	else:
		tmp = math.cos(eps) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.4e-6)
		tmp = Float64(cos(eps) + -1.0);
	elseif (eps <= 1.8e-112)
		tmp = Float64(sin(x) * Float64(-eps));
	elseif (eps <= 235000000000.0)
		tmp = Float64(Float64(-(sin(eps) ^ 2.0)) / 2.0);
	else
		tmp = Float64(cos(eps) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.4e-6)
		tmp = cos(eps) + -1.0;
	elseif (eps <= 1.8e-112)
		tmp = sin(x) * -eps;
	elseif (eps <= 235000000000.0)
		tmp = -(sin(eps) ^ 2.0) / 2.0;
	else
		tmp = cos(eps) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.4e-6], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[eps, 1.8e-112], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], If[LessEqual[eps, 235000000000.0], N[((-N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision]) / 2.0), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < -1.39999999999999994e-6

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 53.8%

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

   if -1.39999999999999994e-6 < eps < 1.8e-112

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 88.4%

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

      1. mul-1-neg 88.4%

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

      2. *-commutative 88.4%

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

      3. distribute-rgt-neg-in 88.4%

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

   4. Simplified 88.4%

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

   if 1.8e-112 < eps < 2.35e11

   1. Initial program 4.1%

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

   2. Taylor expanded in x around 0 5.4%

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

      1. flip-- 5.4%

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

      2. metadata-eval 5.4%

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

   4. Applied egg-rr 5.4%

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

   5. Taylor expanded in eps around inf 5.4%

      \[\leadsto \frac{\color{blue}{{\cos \varepsilon}^{2} - 1}}{\cos \varepsilon + 1} \]
   6. Step-by-step derivation

      1. unpow2 5.4%

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

      2. sub-1-cos 59.1%

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

      3. unpow2 59.1%

         \[\leadsto \frac{-\color{blue}{{\sin \varepsilon}^{2}}}{\cos \varepsilon + 1} \]

   7. Simplified 59.1%

      \[\leadsto \frac{\color{blue}{-{\sin \varepsilon}^{2}}}{\cos \varepsilon + 1} \]

   8. Taylor expanded in eps around 0 59.1%

      \[\leadsto \frac{-{\sin \varepsilon}^{2}}{\color{blue}{2}} \]

   if 2.35e11 < eps

   1. Initial program 51.4%

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

   2. Taylor expanded in x around 0 54.1%

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.4 \cdot 10^{-6}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 235000000000:\\ \;\;\;\;\frac{-{\sin \varepsilon}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 9: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -8 \cdot 10^{-87} \lor \neg \left(x \leq 4.7 \cdot 10^{-7}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (if (or (<= x -8e-87) (not (<= x 4.7e-7)))
     (* -2.0 (* (sin x) t_0))
     (* -2.0 (pow t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	double tmp;
	if ((x <= -8e-87) || !(x <= 4.7e-7)) {
		tmp = -2.0 * (sin(x) * t_0);
	} else {
		tmp = -2.0 * pow(t_0, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((0.5d0 * eps))
    if ((x <= (-8d-87)) .or. (.not. (x <= 4.7d-7))) then
        tmp = (-2.0d0) * (sin(x) * t_0)
    else
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((0.5 * eps));
	double tmp;
	if ((x <= -8e-87) || !(x <= 4.7e-7)) {
		tmp = -2.0 * (Math.sin(x) * t_0);
	} else {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((0.5 * eps))
	tmp = 0
	if (x <= -8e-87) or not (x <= 4.7e-7):
		tmp = -2.0 * (math.sin(x) * t_0)
	else:
		tmp = -2.0 * math.pow(t_0, 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	tmp = 0.0
	if ((x <= -8e-87) || !(x <= 4.7e-7))
		tmp = Float64(-2.0 * Float64(sin(x) * t_0));
	else
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((0.5 * eps));
	tmp = 0.0;
	if ((x <= -8e-87) || ~((x <= 4.7e-7)))
		tmp = -2.0 * (sin(x) * t_0);
	else
		tmp = -2.0 * (t_0 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -8e-87], N[Not[LessEqual[x, 4.7e-7]], $MachinePrecision]], N[(-2.0 * N[(N[Sin[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000014e-87 or 4.7e-7 < x

   1. Initial program 9.6%

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

      1. diff-cos 10.3%

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

      2. div-inv 10.3%

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

      3. metadata-eval 10.3%

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

      4. div-inv 10.3%

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

      5. +-commutative 10.3%

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

      6. metadata-eval 10.3%

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

   3. Applied egg-rr 10.3%

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

      1. *-commutative 10.3%

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

      2. +-commutative 10.3%

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

      3. associate--l+ 56.4%

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

      4. +-inverses 56.4%

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

      5. distribute-lft-in 56.4%

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

      6. metadata-eval 56.4%

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

      7. *-commutative 56.4%

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

      8. associate-+r+ 56.5%

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

      9. +-commutative 56.5%

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

   5. Simplified 56.5%

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

   6. Taylor expanded in x around -inf 56.5%

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

   7. Taylor expanded in eps around 0 52.2%

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

   if -8.00000000000000014e-87 < x < 4.7e-7

   1. Initial program 72.2%

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

      1. diff-cos 92.1%

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

      2. div-inv 92.1%

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

      3. metadata-eval 92.1%

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

      4. div-inv 92.1%

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

      5. +-commutative 92.1%

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

      6. metadata-eval 92.1%

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

   3. Applied egg-rr 92.1%

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

      1. *-commutative 92.1%

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

      2. +-commutative 92.1%

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

      3. associate--l+ 99.0%

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

      4. +-inverses 99.0%

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

      5. distribute-lft-in 99.0%

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

      6. metadata-eval 99.0%

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

      7. *-commutative 99.0%

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

      8. associate-+r+ 99.0%

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

      9. +-commutative 99.0%

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

   5. Simplified 99.0%

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

   6. Taylor expanded in x around 0 92.1%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{-87} \lor \neg \left(x \leq 4.7 \cdot 10^{-7}\right):\\ \;\;\;\;-2 \cdot \left(\sin x \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \end{array} \]

Alternative 10: 66.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.36 \cdot 10^{-13} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-112}\right):\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -1.36e-13) (not (<= eps 1.8e-112)))
   (* -2.0 (pow (sin (* 0.5 eps)) 2.0))
   (* (sin x) (- eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.36e-13) || !(eps <= 1.8e-112)) {
		tmp = -2.0 * pow(sin((0.5 * eps)), 2.0);
	} 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 <= (-1.36d-13)) .or. (.not. (eps <= 1.8d-112))) then
        tmp = (-2.0d0) * (sin((0.5d0 * eps)) ** 2.0d0)
    else
        tmp = sin(x) * -eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -1.36e-13) || !(eps <= 1.8e-112)) {
		tmp = -2.0 * Math.pow(Math.sin((0.5 * eps)), 2.0);
	} else {
		tmp = Math.sin(x) * -eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -1.36e-13) or not (eps <= 1.8e-112):
		tmp = -2.0 * math.pow(math.sin((0.5 * eps)), 2.0)
	else:
		tmp = math.sin(x) * -eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -1.36e-13) || !(eps <= 1.8e-112))
		tmp = Float64(-2.0 * (sin(Float64(0.5 * eps)) ^ 2.0));
	else
		tmp = Float64(sin(x) * Float64(-eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -1.36e-13) || ~((eps <= 1.8e-112)))
		tmp = -2.0 * (sin((0.5 * eps)) ^ 2.0);
	else
		tmp = sin(x) * -eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -1.36e-13], N[Not[LessEqual[eps, 1.8e-112]], $MachinePrecision]], N[(-2.0 * N[Power[N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.36000000000000001e-13 or 1.8e-112 < eps

   1. Initial program 43.9%

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

      1. diff-cos 52.8%

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

      2. div-inv 52.8%

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

      3. metadata-eval 52.8%

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

      4. div-inv 52.8%

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

      5. +-commutative 52.8%

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

      6. metadata-eval 52.8%

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

   3. Applied egg-rr 52.8%

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

      1. *-commutative 52.8%

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

      2. +-commutative 52.8%

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

      3. associate--l+ 58.5%

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

      4. +-inverses 58.5%

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

      5. distribute-lft-in 58.5%

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

      6. metadata-eval 58.5%

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

      7. *-commutative 58.5%

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

      8. associate-+r+ 58.5%

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

      9. +-commutative 58.5%

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

   5. Simplified 58.5%

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

   6. Taylor expanded in x around 0 53.7%

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

   if -1.36000000000000001e-13 < eps < 1.8e-112

   1. Initial program 25.9%

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

   2. Taylor expanded in eps around 0 90.4%

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

      1. mul-1-neg 90.4%

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

      2. *-commutative 90.4%

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

      3. distribute-rgt-neg-in 90.4%

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

   4. Simplified 90.4%

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

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.36 \cdot 10^{-13} \lor \neg \left(\varepsilon \leq 1.8 \cdot 10^{-112}\right):\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \end{array} \]

Alternative 11: 65.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 235000000000:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.3e-5)
   (+ (cos eps) -1.0)
   (if (<= eps 1.7e-112)
     (* (sin x) (- eps))
     (if (<= eps 235000000000.0) (* (* eps eps) -0.5) (- (cos eps) (cos x))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.3e-5) {
		tmp = cos(eps) + -1.0;
	} else if (eps <= 1.7e-112) {
		tmp = sin(x) * -eps;
	} else if (eps <= 235000000000.0) {
		tmp = (eps * eps) * -0.5;
	} 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 <= (-1.3d-5)) then
        tmp = cos(eps) + (-1.0d0)
    else if (eps <= 1.7d-112) then
        tmp = sin(x) * -eps
    else if (eps <= 235000000000.0d0) then
        tmp = (eps * eps) * (-0.5d0)
    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 <= -1.3e-5) {
		tmp = Math.cos(eps) + -1.0;
	} else if (eps <= 1.7e-112) {
		tmp = Math.sin(x) * -eps;
	} else if (eps <= 235000000000.0) {
		tmp = (eps * eps) * -0.5;
	} else {
		tmp = Math.cos(eps) - Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.3e-5:
		tmp = math.cos(eps) + -1.0
	elif eps <= 1.7e-112:
		tmp = math.sin(x) * -eps
	elif eps <= 235000000000.0:
		tmp = (eps * eps) * -0.5
	else:
		tmp = math.cos(eps) - math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.3e-5)
		tmp = Float64(cos(eps) + -1.0);
	elseif (eps <= 1.7e-112)
		tmp = Float64(sin(x) * Float64(-eps));
	elseif (eps <= 235000000000.0)
		tmp = Float64(Float64(eps * eps) * -0.5);
	else
		tmp = Float64(cos(eps) - cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -1.3e-5)
		tmp = cos(eps) + -1.0;
	elseif (eps <= 1.7e-112)
		tmp = sin(x) * -eps;
	elseif (eps <= 235000000000.0)
		tmp = (eps * eps) * -0.5;
	else
		tmp = cos(eps) - cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.3e-5], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[eps, 1.7e-112], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], If[LessEqual[eps, 235000000000.0], N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if eps < -1.29999999999999992e-5

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 53.8%

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

   if -1.29999999999999992e-5 < eps < 1.6999999999999999e-112

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 88.4%

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

      1. mul-1-neg 88.4%

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

      2. *-commutative 88.4%

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

      3. distribute-rgt-neg-in 88.4%

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

   4. Simplified 88.4%

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

   if 1.6999999999999999e-112 < eps < 2.35e11

   1. Initial program 4.1%

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

   2. Taylor expanded in x around 0 5.4%

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

   3. Taylor expanded in eps around 0 58.7%

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

      1. *-commutative 58.7%

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

      2. unpow2 58.7%

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

   5. Simplified 58.7%

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

   if 2.35e11 < eps

   1. Initial program 51.4%

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

   2. Taylor expanded in x around 0 54.1%

      \[\leadsto \color{blue}{\cos \varepsilon} - \cos x \]
3. Recombined 4 regimes into one program.

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 235000000000:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \end{array} \]

Alternative 12: 66.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.18:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)))
   (if (<= eps -1.35e-5)
     t_0
     (if (<= eps 1.6e-112)
       (* (sin x) (- eps))
       (if (<= eps 0.18) (* (* eps eps) -0.5) t_0)))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double tmp;
	if (eps <= -1.35e-5) {
		tmp = t_0;
	} else if (eps <= 1.6e-112) {
		tmp = sin(x) * -eps;
	} else if (eps <= 0.18) {
		tmp = (eps * eps) * -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(eps) + (-1.0d0)
    if (eps <= (-1.35d-5)) then
        tmp = t_0
    else if (eps <= 1.6d-112) then
        tmp = sin(x) * -eps
    else if (eps <= 0.18d0) then
        tmp = (eps * eps) * (-0.5d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) + -1.0;
	double tmp;
	if (eps <= -1.35e-5) {
		tmp = t_0;
	} else if (eps <= 1.6e-112) {
		tmp = Math.sin(x) * -eps;
	} else if (eps <= 0.18) {
		tmp = (eps * eps) * -0.5;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	tmp = 0
	if eps <= -1.35e-5:
		tmp = t_0
	elif eps <= 1.6e-112:
		tmp = math.sin(x) * -eps
	elif eps <= 0.18:
		tmp = (eps * eps) * -0.5
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	tmp = 0.0
	if (eps <= -1.35e-5)
		tmp = t_0;
	elseif (eps <= 1.6e-112)
		tmp = Float64(sin(x) * Float64(-eps));
	elseif (eps <= 0.18)
		tmp = Float64(Float64(eps * eps) * -0.5);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	tmp = 0.0;
	if (eps <= -1.35e-5)
		tmp = t_0;
	elseif (eps <= 1.6e-112)
		tmp = sin(x) * -eps;
	elseif (eps <= 0.18)
		tmp = (eps * eps) * -0.5;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[eps, -1.35e-5], t$95$0, If[LessEqual[eps, 1.6e-112], N[(N[Sin[x], $MachinePrecision] * (-eps)), $MachinePrecision], If[LessEqual[eps, 0.18], N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.3499999999999999e-5 or 0.17999999999999999 < eps

   1. Initial program 51.2%

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

   2. Taylor expanded in x around 0 52.1%

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

   if -1.3499999999999999e-5 < eps < 1.59999999999999997e-112

   1. Initial program 25.3%

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

   2. Taylor expanded in eps around 0 88.4%

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

      1. mul-1-neg 88.4%

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

      2. *-commutative 88.4%

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

      3. distribute-rgt-neg-in 88.4%

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

   4. Simplified 88.4%

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

   if 1.59999999999999997e-112 < eps < 0.17999999999999999

   1. Initial program 4.1%

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

   2. Taylor expanded in x around 0 4.7%

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

   3. Taylor expanded in eps around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. unpow2 61.0%

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

   5. Simplified 61.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq 1.6 \cdot 10^{-112}:\\ \;\;\;\;\sin x \cdot \left(-\varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 0.18:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 13: 47.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.2 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 0.18\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -9.2e-7) (not (<= eps 0.18)))
   (+ (cos eps) -1.0)
   (* (* eps eps) -0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -9.2e-7) || !(eps <= 0.18)) {
		tmp = cos(eps) + -1.0;
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-9.2d-7)) .or. (.not. (eps <= 0.18d0))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = (eps * eps) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -9.2e-7) || !(eps <= 0.18)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -9.2e-7) or not (eps <= 0.18):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = (eps * eps) * -0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -9.2e-7) || !(eps <= 0.18))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(Float64(eps * eps) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -9.2e-7) || ~((eps <= 0.18)))
		tmp = cos(eps) + -1.0;
	else
		tmp = (eps * eps) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -9.2e-7], N[Not[LessEqual[eps, 0.18]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -9.1999999999999998e-7 or 0.17999999999999999 < eps

   1. Initial program 50.9%

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

   2. Taylor expanded in x around 0 51.7%

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

   if -9.1999999999999998e-7 < eps < 0.17999999999999999

   1. Initial program 21.9%

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

   2. Taylor expanded in x around 0 22.0%

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

   3. Taylor expanded in eps around 0 40.7%

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

      1. *-commutative 40.7%

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

      2. unpow2 40.7%

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

   5. Simplified 40.7%

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

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -9.2 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 0.18\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 14: 21.7% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 36.7%

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

2. Taylor expanded in x around 0 37.2%

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

3. Taylor expanded in eps around 0 21.6%

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

   1. *-commutative 21.6%

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

   2. unpow2 21.6%

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

5. Simplified 21.6%

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

6. Final simplification 21.6%

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

Alternative 15: 12.8% 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 36.7%

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

2. Taylor expanded in x around 0 37.2%

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

3. Taylor expanded in eps around 0 12.3%

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

4. Final simplification 12.3%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))