2cos (problem 3.3.5)

Percentage Accurate: 38.1% → 99.4%

Time: 17.4s

Alternatives: 13

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 38.1% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. diff-cos 43.0%

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

   2. div-inv 43.0%

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

   3. metadata-eval 43.0%

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

   4. div-inv 43.0%

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

   5. +-commutative 43.0%

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

   6. metadata-eval 43.0%

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

3. Applied egg-rr 43.0%

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

   1. *-commutative 43.0%

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

   2. +-commutative 43.0%

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

   3. associate--l+ 77.7%

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

   4. *-commutative 77.7%

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

   5. associate-+r+ 78.0%

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

   6. +-commutative 78.0%

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

5. Simplified 78.0%

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

   1. distribute-lft-in 78.0%

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

   2. sin-sum 99.4%

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

7. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. count-2 99.4%

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

   4. *-commutative 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

   7. count-2 99.4%

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

   8. *-commutative 99.4%

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

   9. count-2 99.4%

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

   10. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.4%

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

    1. +-commutative 99.4%

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

    2. fma-def 99.5%

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

    3. *-commutative 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return Float64(-2.0 * Float64(t_0 * fma(cos(x), t_0, Float64(sin(x) * cos(Float64(eps * -0.5))))))
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[Cos[x], $MachinePrecision] * t$95$0 + N[(N[Sin[x], $MachinePrecision] * N[Cos[N[(eps * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 37.1%

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

   1. diff-cos 43.0%

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

   2. div-inv 43.0%

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

   3. metadata-eval 43.0%

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

   4. div-inv 43.0%

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

   5. +-commutative 43.0%

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

   6. metadata-eval 43.0%

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

3. Applied egg-rr 43.0%

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

   1. *-commutative 43.0%

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

   2. +-commutative 43.0%

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

   3. associate--l+ 77.7%

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

   4. *-commutative 77.7%

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

   5. associate-+r+ 78.0%

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

   6. +-commutative 78.0%

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

5. Simplified 78.0%

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

   1. distribute-lft-in 78.0%

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

   2. sin-sum 99.4%

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

7. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. count-2 99.4%

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

   4. *-commutative 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

   7. count-2 99.4%

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

   8. *-commutative 99.4%

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

   9. count-2 99.4%

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

   10. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.4%

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

    1. +-commutative 99.4%

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

    2. fma-def 99.5%

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

    3. *-commutative 99.5%

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

    4. fma-def 99.4%

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

    5. +-commutative 99.4%

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

    6. fma-def 99.4%

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

    7. metadata-eval 99.4%

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

    8. associate-*r* 99.4%

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

    9. mul-1-neg 99.4%

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

    10. distribute-rgt-neg-out 99.4%

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

    11. metadata-eval 99.4%

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

    12. distribute-lft-neg-in 99.4%

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

    13. cos-neg 99.4%

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

    14. *-commutative 99.4%

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

    15. distribute-rgt-neg-in 99.4%

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

    16. metadata-eval 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. diff-cos 43.0%

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

   2. div-inv 43.0%

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

   3. metadata-eval 43.0%

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

   4. div-inv 43.0%

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

   5. +-commutative 43.0%

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

   6. metadata-eval 43.0%

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

3. Applied egg-rr 43.0%

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

   1. *-commutative 43.0%

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

   2. +-commutative 43.0%

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

   3. associate--l+ 77.7%

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

   4. *-commutative 77.7%

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

   5. associate-+r+ 78.0%

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

   6. +-commutative 78.0%

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

5. Simplified 78.0%

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

   1. distribute-lft-in 78.0%

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

   2. sin-sum 99.4%

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

7. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. count-2 99.4%

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

   4. *-commutative 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

   7. count-2 99.4%

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

   8. *-commutative 99.4%

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

   9. count-2 99.4%

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

   10. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.4%

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

11. Final simplification 99.4%

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

Alternative 4: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-8} \lor \neg \left(x \leq 1.25 \cdot 10^{-39}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \varepsilon + x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -2.5e-8) (not (<= x 1.25e-39)))
   (- (* (cos x) (+ -1.0 (cos eps))) (* (sin x) (sin eps)))
   (* -2.0 (* (sin (* 0.5 eps)) (sin (+ (* 0.5 eps) x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -2.5e-8) || !(x <= 1.25e-39)) {
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(x) * sin(eps));
	} else {
		tmp = -2.0 * (sin((0.5 * eps)) * sin(((0.5 * eps) + x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-2.5d-8)) .or. (.not. (x <= 1.25d-39))) then
        tmp = (cos(x) * ((-1.0d0) + cos(eps))) - (sin(x) * sin(eps))
    else
        tmp = (-2.0d0) * (sin((0.5d0 * eps)) * sin(((0.5d0 * eps) + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -2.5e-8) || !(x <= 1.25e-39)) {
		tmp = (Math.cos(x) * (-1.0 + Math.cos(eps))) - (Math.sin(x) * Math.sin(eps));
	} else {
		tmp = -2.0 * (Math.sin((0.5 * eps)) * Math.sin(((0.5 * eps) + x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -2.5e-8) or not (x <= 1.25e-39):
		tmp = (math.cos(x) * (-1.0 + math.cos(eps))) - (math.sin(x) * math.sin(eps))
	else:
		tmp = -2.0 * (math.sin((0.5 * eps)) * math.sin(((0.5 * eps) + x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -2.5e-8) || !(x <= 1.25e-39))
		tmp = Float64(Float64(cos(x) * Float64(-1.0 + cos(eps))) - Float64(sin(x) * sin(eps)));
	else
		tmp = Float64(-2.0 * Float64(sin(Float64(0.5 * eps)) * sin(Float64(Float64(0.5 * eps) + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -2.5e-8) || ~((x <= 1.25e-39)))
		tmp = (cos(x) * (-1.0 + cos(eps))) - (sin(x) * sin(eps));
	else
		tmp = -2.0 * (sin((0.5 * eps)) * sin(((0.5 * eps) + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -2.5e-8], N[Not[LessEqual[x, 1.25e-39]], $MachinePrecision]], N[(N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(0.5 * eps), $MachinePrecision] + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4999999999999999e-8 or 1.25e-39 < x

   1. Initial program 9.4%

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

      1. cos-sum 48.7%

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

      2. cancel-sign-sub-inv 48.7%

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

      3. fma-def 48.7%

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

   3. Applied egg-rr 48.7%

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

   4. Taylor expanded in x around inf 48.7%

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

      1. neg-mul-1 48.7%

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

      2. distribute-lft-neg-in 48.7%

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

      3. associate--l+ 99.0%

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

      4. *-commutative 99.0%

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

      5. +-commutative 99.0%

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

      6. distribute-lft-neg-in 99.0%

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

      7. unsub-neg 99.0%

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

      8. *-rgt-identity 99.0%

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

      9. distribute-lft-out-- 99.0%

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

      10. sub-neg 99.0%

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

      11. metadata-eval 99.0%

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

      12. +-commutative 99.0%

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

   6. Simplified 99.0%

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

   if -2.4999999999999999e-8 < x < 1.25e-39

   1. Initial program 73.9%

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

      1. diff-cos 89.2%

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

      2. div-inv 89.2%

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

      3. metadata-eval 89.2%

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

      4. div-inv 89.2%

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

      5. +-commutative 89.2%

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

      6. metadata-eval 89.2%

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

   3. Applied egg-rr 89.2%

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

      1. *-commutative 89.2%

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

      2. +-commutative 89.2%

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

      3. associate--l+ 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-+r+ 99.7%

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

      6. +-commutative 99.7%

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

   5. Simplified 99.7%

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

      1. distribute-lft-in 99.7%

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

      2. sin-sum 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. +-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. count-2 99.7%

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

      4. *-commutative 99.7%

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

      5. count-2 99.7%

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

      6. *-commutative 99.7%

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

      7. count-2 99.7%

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

      8. *-commutative 99.7%

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

      9. count-2 99.7%

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

      10. fma-def 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in eps around inf 99.7%

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

       1. +-commutative 99.7%

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

       2. sin-sum 99.7%

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

   12. Applied egg-rr 99.7%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{-8} \lor \neg \left(x \leq 1.25 \cdot 10^{-39}\right):\\ \;\;\;\;\cos x \cdot \left(-1 + \cos \varepsilon\right) - \sin x \cdot \sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \varepsilon + x\right)\right)\\ \end{array} \]

Alternative 5: 67.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -5e-16)
   (* (tan (/ eps 2.0)) (- (sin eps)))
   (* -2.0 (* (sin (* 0.5 eps)) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -5e-16) {
		tmp = tan((eps / 2.0)) * -sin(eps);
	} else {
		tmp = -2.0 * (sin((0.5 * eps)) * sin(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((cos((eps + x)) - cos(x)) <= (-5d-16)) then
        tmp = tan((eps / 2.0d0)) * -sin(eps)
    else
        tmp = (-2.0d0) * (sin((0.5d0 * eps)) * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -5e-16) {
		tmp = Math.tan((eps / 2.0)) * -Math.sin(eps);
	} else {
		tmp = -2.0 * (Math.sin((0.5 * eps)) * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -5e-16:
		tmp = math.tan((eps / 2.0)) * -math.sin(eps)
	else:
		tmp = -2.0 * (math.sin((0.5 * eps)) * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -5e-16)
		tmp = Float64(tan(Float64(eps / 2.0)) * Float64(-sin(eps)));
	else
		tmp = Float64(-2.0 * Float64(sin(Float64(0.5 * eps)) * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -5e-16)
		tmp = tan((eps / 2.0)) * -sin(eps);
	else
		tmp = -2.0 * (sin((0.5 * eps)) * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -5e-16], N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision], N[(-2.0 * N[(N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -5.0000000000000004e-16

   1. Initial program 78.2%

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

   2. Taylor expanded in x around 0 78.4%

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

      1. flip-- 78.2%

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

      2. metadata-eval 78.2%

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

      3. sub-1-cos 80.4%

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

      4. pow1 80.4%

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

      5. pow1 80.4%

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

      6. pow-prod-up 80.4%

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

      7. metadata-eval 80.4%

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

   4. Applied egg-rr 80.4%

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

      1. distribute-frac-neg 80.4%

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

      2. /-rgt-identity 80.4%

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

      3. associate-/r* 80.4%

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

      4. unpow2 80.4%

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

      5. times-frac 80.7%

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

      6. +-commutative 80.7%

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

      7. hang-0p-tan 81.0%

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

      8. /-rgt-identity 81.0%

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

   6. Simplified 81.0%

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

   if -5.0000000000000004e-16 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 18.8%

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

      1. diff-cos 26.4%

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

      2. div-inv 26.4%

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

      3. metadata-eval 26.4%

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

      4. div-inv 26.4%

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

      5. +-commutative 26.4%

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

      6. metadata-eval 26.4%

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

   3. Applied egg-rr 26.4%

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

      1. *-commutative 26.4%

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

      2. +-commutative 26.4%

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

      3. associate--l+ 76.5%

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

      4. *-commutative 76.5%

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

      5. associate-+r+ 76.9%

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

      6. +-commutative 76.9%

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

   5. Simplified 76.9%

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

      1. distribute-lft-in 76.9%

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

      2. sin-sum 99.5%

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

   7. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. count-2 99.5%

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

      4. *-commutative 99.5%

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

      5. count-2 99.5%

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

      6. *-commutative 99.5%

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

      7. count-2 99.5%

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

      8. *-commutative 99.5%

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

      9. count-2 99.5%

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

      10. fma-def 99.5%

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

   9. Simplified 99.5%

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

   10. Taylor expanded in eps around inf 99.5%

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

       1. +-commutative 99.5%

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

       2. fma-def 99.5%

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

       3. *-commutative 99.5%

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

   12. Simplified 99.5%

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

   13. Taylor expanded in eps around 0 68.8%

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

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\\ \end{array} \]

Alternative 6: 66.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-\varepsilon \cdot \sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -5e-16)
   (* (tan (/ eps 2.0)) (- (sin eps)))
   (log1p (- (* eps (sin x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -5e-16) {
		tmp = tan((eps / 2.0)) * -sin(eps);
	} else {
		tmp = log1p(-(eps * sin(x)));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -5e-16) {
		tmp = Math.tan((eps / 2.0)) * -Math.sin(eps);
	} else {
		tmp = Math.log1p(-(eps * Math.sin(x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -5e-16:
		tmp = math.tan((eps / 2.0)) * -math.sin(eps)
	else:
		tmp = math.log1p(-(eps * math.sin(x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -5e-16)
		tmp = Float64(tan(Float64(eps / 2.0)) * Float64(-sin(eps)));
	else
		tmp = log1p(Float64(-Float64(eps * sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -5e-16], N[(N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision], N[Log[1 + (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -5.0000000000000004e-16

   1. Initial program 78.2%

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

   2. Taylor expanded in x around 0 78.4%

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

      1. flip-- 78.2%

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

      2. metadata-eval 78.2%

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

      3. sub-1-cos 80.4%

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

      4. pow1 80.4%

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

      5. pow1 80.4%

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

      6. pow-prod-up 80.4%

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

      7. metadata-eval 80.4%

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

   4. Applied egg-rr 80.4%

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

      1. distribute-frac-neg 80.4%

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

      2. /-rgt-identity 80.4%

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

      3. associate-/r* 80.4%

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

      4. unpow2 80.4%

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

      5. times-frac 80.7%

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

      6. +-commutative 80.7%

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

      7. hang-0p-tan 81.0%

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

      8. /-rgt-identity 81.0%

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

   6. Simplified 81.0%

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

   if -5.0000000000000004e-16 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 18.8%

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

      1. log1p-expm1-u 18.8%

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

   3. Applied egg-rr 18.8%

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

   4. Taylor expanded in eps around 0 67.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)}\right) \]
   5. Step-by-step derivation

      1. mul-1-neg 67.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{-\varepsilon \cdot \sin x}\right) \]

      2. *-commutative 67.1%

         \[\leadsto \mathsf{log1p}\left(-\color{blue}{\sin x \cdot \varepsilon}\right) \]

      3. distribute-rgt-neg-in 67.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(-\varepsilon\right)}\right) \]

   6. Simplified 67.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(-\varepsilon\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(-\sin \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-\varepsilon \cdot \sin x\right)\\ \end{array} \]

Alternative 7: 66.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-\varepsilon \cdot \sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -5e-16)
   (- (cos eps) (cos x))
   (log1p (- (* eps (sin x))))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -5e-16) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = log1p(-(eps * sin(x)));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((Math.cos((eps + x)) - Math.cos(x)) <= -5e-16) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = Math.log1p(-(eps * Math.sin(x)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -5e-16:
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = math.log1p(-(eps * math.sin(x)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -5e-16)
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = log1p(Float64(-Float64(eps * sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -5e-16], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Log[1 + (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -5.0000000000000004e-16

   1. Initial program 78.2%

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

   2. Taylor expanded in x around 0 78.6%

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

   if -5.0000000000000004e-16 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 18.8%

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

      1. log1p-expm1-u 18.8%

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

   3. Applied egg-rr 18.8%

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

   4. Taylor expanded in eps around 0 67.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)}\right) \]
   5. Step-by-step derivation

      1. mul-1-neg 67.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{-\varepsilon \cdot \sin x}\right) \]

      2. *-commutative 67.1%

         \[\leadsto \mathsf{log1p}\left(-\color{blue}{\sin x \cdot \varepsilon}\right) \]

      3. distribute-rgt-neg-in 67.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(-\varepsilon\right)}\right) \]

   6. Simplified 67.1%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(-\varepsilon\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -5 \cdot 10^{-16}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(-\varepsilon \cdot \sin x\right)\\ \end{array} \]

Alternative 8: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. diff-cos 43.0%

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

   2. div-inv 43.0%

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

   3. metadata-eval 43.0%

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

   4. div-inv 43.0%

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

   5. +-commutative 43.0%

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

   6. metadata-eval 43.0%

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

3. Applied egg-rr 43.0%

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

   1. *-commutative 43.0%

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

   2. +-commutative 43.0%

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

   3. associate--l+ 77.7%

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

   4. *-commutative 77.7%

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

   5. associate-+r+ 78.0%

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

   6. +-commutative 78.0%

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

5. Simplified 78.0%

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

   1. distribute-lft-in 78.0%

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

   2. sin-sum 99.4%

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

7. Applied egg-rr 99.4%

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

   1. +-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. count-2 99.4%

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

   4. *-commutative 99.4%

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

   5. count-2 99.4%

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

   6. *-commutative 99.4%

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

   7. count-2 99.4%

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

   8. *-commutative 99.4%

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

   9. count-2 99.4%

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

   10. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.4%

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

    1. +-commutative 99.4%

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

    2. sin-sum 78.0%

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

12. Applied egg-rr 78.0%

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

13. Final simplification 78.0%

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

Alternative 9: 67.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 6.4 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.2e-8) (not (<= eps 6.4e-7)))
   (- (cos eps) (cos x))
   (- (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.2e-8) || !(eps <= 6.4e-7)) {
		tmp = cos(eps) - cos(x);
	} else {
		tmp = -(eps * sin(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 <= (-4.2d-8)) .or. (.not. (eps <= 6.4d-7))) then
        tmp = cos(eps) - cos(x)
    else
        tmp = -(eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.2e-8) || !(eps <= 6.4e-7)) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else {
		tmp = -(eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.2e-8) or not (eps <= 6.4e-7):
		tmp = math.cos(eps) - math.cos(x)
	else:
		tmp = -(eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.2e-8) || !(eps <= 6.4e-7))
		tmp = Float64(cos(eps) - cos(x));
	else
		tmp = Float64(-Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.2e-8) || ~((eps <= 6.4e-7)))
		tmp = cos(eps) - cos(x);
	else
		tmp = -(eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.2e-8], N[Not[LessEqual[eps, 6.4e-7]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if eps < -4.19999999999999989e-8 or 6.4000000000000001e-7 < eps

   1. Initial program 52.7%

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

   2. Taylor expanded in x around 0 54.5%

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

   if -4.19999999999999989e-8 < eps < 6.4000000000000001e-7

   1. Initial program 22.4%

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

   2. Taylor expanded in eps around 0 88.3%

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

      1. associate-*r* 88.3%

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

      2. mul-1-neg 88.3%

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

   4. Simplified 88.3%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 6.4 \cdot 10^{-7}\right):\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \]

Alternative 10: 45.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \cos \varepsilon\\ \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -3.8 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 900:\\ \;\;\;\;-\varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ -1.0 (cos eps))))
   (if (<= eps -0.00016)
     t_0
     (if (<= eps -3.8e-175)
       (* -0.5 (* eps eps))
       (if (<= eps 900.0) (- (* eps x)) t_0)))))

C

double code(double x, double eps) {
	double t_0 = -1.0 + cos(eps);
	double tmp;
	if (eps <= -0.00016) {
		tmp = t_0;
	} else if (eps <= -3.8e-175) {
		tmp = -0.5 * (eps * eps);
	} else if (eps <= 900.0) {
		tmp = -(eps * x);
	} 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 = (-1.0d0) + cos(eps)
    if (eps <= (-0.00016d0)) then
        tmp = t_0
    else if (eps <= (-3.8d-175)) then
        tmp = (-0.5d0) * (eps * eps)
    else if (eps <= 900.0d0) then
        tmp = -(eps * x)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = -1.0 + Math.cos(eps);
	double tmp;
	if (eps <= -0.00016) {
		tmp = t_0;
	} else if (eps <= -3.8e-175) {
		tmp = -0.5 * (eps * eps);
	} else if (eps <= 900.0) {
		tmp = -(eps * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = -1.0 + math.cos(eps)
	tmp = 0
	if eps <= -0.00016:
		tmp = t_0
	elif eps <= -3.8e-175:
		tmp = -0.5 * (eps * eps)
	elif eps <= 900.0:
		tmp = -(eps * x)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(-1.0 + cos(eps))
	tmp = 0.0
	if (eps <= -0.00016)
		tmp = t_0;
	elseif (eps <= -3.8e-175)
		tmp = Float64(-0.5 * Float64(eps * eps));
	elseif (eps <= 900.0)
		tmp = Float64(-Float64(eps * x));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = -1.0 + cos(eps);
	tmp = 0.0;
	if (eps <= -0.00016)
		tmp = t_0;
	elseif (eps <= -3.8e-175)
		tmp = -0.5 * (eps * eps);
	elseif (eps <= 900.0)
		tmp = -(eps * x);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[eps, -0.00016], t$95$0, If[LessEqual[eps, -3.8e-175], N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 900.0], (-N[(eps * x), $MachinePrecision]), t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.60000000000000013e-4 or 900 < eps

   1. Initial program 53.5%

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

   2. Taylor expanded in x around 0 53.4%

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

   if -1.60000000000000013e-4 < eps < -3.8e-175

   1. Initial program 9.8%

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

   2. Taylor expanded in x around 0 9.8%

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

   3. Taylor expanded in eps around 0 33.6%

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

      1. *-commutative 33.6%

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

      2. unpow2 33.6%

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

   5. Simplified 33.6%

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

   if -3.8e-175 < eps < 900

   1. Initial program 28.3%

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

   2. Taylor expanded in eps around 0 93.0%

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

      1. associate-*r* 93.0%

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

      2. mul-1-neg 93.0%

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

   4. Simplified 93.0%

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

   5. Taylor expanded in x around 0 41.3%

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

      1. associate-*r* 41.3%

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

      2. neg-mul-1 41.3%

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

   7. Simplified 41.3%

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

4. Final simplification 45.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00016:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{elif}\;\varepsilon \leq -3.8 \cdot 10^{-175}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \mathbf{elif}\;\varepsilon \leq 900:\\ \;\;\;\;-\varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \end{array} \]

Alternative 11: 66.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 900\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -4.5e-8) (not (<= eps 900.0)))
   (+ -1.0 (cos eps))
   (- (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.5e-8) || !(eps <= 900.0)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = -(eps * sin(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 <= (-4.5d-8)) .or. (.not. (eps <= 900.0d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = -(eps * sin(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -4.5e-8) || !(eps <= 900.0)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = -(eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -4.5e-8) or not (eps <= 900.0):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = -(eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -4.5e-8) || !(eps <= 900.0))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(-Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -4.5e-8) || ~((eps <= 900.0)))
		tmp = -1.0 + cos(eps);
	else
		tmp = -(eps * sin(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -4.5e-8], N[Not[LessEqual[eps, 900.0]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], (-N[(eps * N[Sin[x], $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if eps < -4.49999999999999993e-8 or 900 < eps

   1. Initial program 53.1%

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

   2. Taylor expanded in x around 0 53.1%

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

   if -4.49999999999999993e-8 < eps < 900

   1. Initial program 22.3%

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

   2. Taylor expanded in eps around 0 87.7%

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

      1. associate-*r* 87.7%

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

      2. mul-1-neg 87.7%

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

   4. Simplified 87.7%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.5 \cdot 10^{-8} \lor \neg \left(\varepsilon \leq 900\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-\varepsilon \cdot \sin x\\ \end{array} \]

Alternative 12: 25.0% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-93} \lor \neg \left(x \leq 8 \cdot 10^{-93}\right):\\ \;\;\;\;-\varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -1.85e-93) (not (<= x 8e-93)))
   (- (* eps x))
   (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -1.85e-93) || !(x <= 8e-93)) {
		tmp = -(eps * x);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-1.85d-93)) .or. (.not. (x <= 8d-93))) then
        tmp = -(eps * x)
    else
        tmp = (-0.5d0) * (eps * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -1.85e-93) || !(x <= 8e-93)) {
		tmp = -(eps * x);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -1.85e-93) or not (x <= 8e-93):
		tmp = -(eps * x)
	else:
		tmp = -0.5 * (eps * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -1.85e-93) || !(x <= 8e-93))
		tmp = Float64(-Float64(eps * x));
	else
		tmp = Float64(-0.5 * Float64(eps * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -1.85e-93) || ~((x <= 8e-93)))
		tmp = -(eps * x);
	else
		tmp = -0.5 * (eps * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -1.85e-93], N[Not[LessEqual[x, 8e-93]], $MachinePrecision]], (-N[(eps * x), $MachinePrecision]), N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85000000000000001e-93 or 7.9999999999999992e-93 < x

   1. Initial program 20.1%

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

   2. Taylor expanded in eps around 0 53.0%

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

      1. associate-*r* 53.0%

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

      2. mul-1-neg 53.0%

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

   4. Simplified 53.0%

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

   5. Taylor expanded in x around 0 12.3%

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

      1. associate-*r* 12.3%

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

      2. neg-mul-1 12.3%

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

   7. Simplified 12.3%

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

   if -1.85000000000000001e-93 < x < 7.9999999999999992e-93

   1. Initial program 76.7%

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

   2. Taylor expanded in x around 0 76.7%

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

   3. Taylor expanded in eps around 0 52.4%

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

      1. *-commutative 52.4%

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

      2. unpow2 52.4%

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

   5. Simplified 52.4%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-93} \lor \neg \left(x \leq 8 \cdot 10^{-93}\right):\\ \;\;\;\;-\varepsilon \cdot x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 13: 17.9% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

2. Taylor expanded in eps around 0 47.4%

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

   1. associate-*r* 47.4%

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

   2. mul-1-neg 47.4%

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

4. Simplified 47.4%

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

5. Taylor expanded in x around 0 19.0%

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

   1. associate-*r* 19.0%

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

   2. neg-mul-1 19.0%

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

7. Simplified 19.0%

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

8. Final simplification 19.0%

   \[\leadsto -\varepsilon \cdot x \]

Reproduce

herbie shell --seed 2023222 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))