2cos (problem 3.3.5)

Percentage Accurate: 38.3% → 99.4%

Time: 16.9s

Alternatives: 14

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: 38.3% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

   1. diff-cos 44.5%

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

   2. div-inv 44.5%

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

   3. metadata-eval 44.5%

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

   4. div-inv 44.5%

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

   5. +-commutative 44.5%

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

   6. metadata-eval 44.5%

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

3. Applied egg-rr 44.5%

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

   1. *-commutative 44.5%

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

   2. +-commutative 44.5%

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

   3. associate--l+ 76.1%

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

   4. *-commutative 76.1%

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

   5. associate-+r+ 76.3%

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

   6. +-commutative 76.3%

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

5. Simplified 76.3%

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

   1. log1p-expm1-u 76.2%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\right)}\right) \]

7. Applied egg-rr 76.2%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. log1p-expm1-u 76.3%

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

   2. distribute-lft-in 76.3%

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

   3. *-commutative 76.3%

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

   4. sin-sum 99.5%

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

   5. *-commutative 99.5%

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

   6. *-commutative 99.5%

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

9. Applied egg-rr 99.5%

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

    1. fma-def 99.5%

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

    2. distribute-rgt-in 99.5%

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

    3. distribute-lft-out 99.5%

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

    4. metadata-eval 99.5%

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

    5. distribute-rgt-in 99.5%

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

    6. distribute-lft-out 99.5%

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

    7. metadata-eval 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

   1. diff-cos 44.5%

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

   2. div-inv 44.5%

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

   3. metadata-eval 44.5%

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

   4. div-inv 44.5%

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

   5. +-commutative 44.5%

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

   6. metadata-eval 44.5%

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

3. Applied egg-rr 44.5%

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

   1. *-commutative 44.5%

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

   2. +-commutative 44.5%

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

   3. associate--l+ 76.1%

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

   4. *-commutative 76.1%

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

   5. associate-+r+ 76.3%

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

   6. +-commutative 76.3%

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

5. Simplified 76.3%

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

   1. log1p-expm1-u 76.2%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\right)}\right) \]

7. Applied egg-rr 76.2%

   \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x - x\right)\right)\right) \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\varepsilon + \left(x + x\right)\right)\right)\right)\right)}\right) \]
8. Step-by-step derivation

   1. log1p-expm1-u 76.3%

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

   2. distribute-rgt-in 76.3%

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

   3. sin-sum 99.5%

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

   4. *-commutative 99.5%

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

   5. *-commutative 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \mathsf{fma}\left({\varepsilon}^{3}, 0.16666666666666666, -\varepsilon\right), \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.000145)
   (fma (sin x) (- (sin eps)) (* (cos x) (+ -1.0 (cos eps))))
   (if (<= eps 0.00015)
     (fma
      (sin x)
      (fma (pow eps 3.0) 0.16666666666666666 (- eps))
      (* (cos x) (* eps (* eps -0.5))))
     (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.000145) {
		tmp = fma(sin(x), -sin(eps), (cos(x) * (-1.0 + cos(eps))));
	} else if (eps <= 0.00015) {
		tmp = fma(sin(x), fma(pow(eps, 3.0), 0.16666666666666666, -eps), (cos(x) * (eps * (eps * -0.5))));
	} else {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.000145)
		tmp = fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	elseif (eps <= 0.00015)
		tmp = fma(sin(x), fma((eps ^ 3.0), 0.16666666666666666, Float64(-eps)), Float64(cos(x) * Float64(eps * Float64(eps * -0.5))));
	else
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(x) * sin(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.000145], N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00015], N[(N[Sin[x], $MachinePrecision] * N[(N[Power[eps, 3.0], $MachinePrecision] * 0.16666666666666666 + (-eps)), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(eps * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.45e-4

   1. Initial program 44.4%

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

      1. cos-sum 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. fma-def 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. associate--l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

      5. fma-def 98.8%

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

      6. *-rgt-identity 98.8%

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

      7. distribute-lft-out-- 98.8%

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

      8. sub-neg 98.8%

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

      9. metadata-eval 98.8%

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

      10. +-commutative 98.8%

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

   6. Simplified 98.8%

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

   if -1.45e-4 < eps < 1.49999999999999987e-4

   1. Initial program 26.2%

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

      1. expm1-log1p-u 26.2%

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

   3. Applied egg-rr 26.2%

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

   4. Taylor expanded in eps around 0 99.8%

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

      1. fma-def 99.8%

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

      2. unpow2 99.8%

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

      3. associate-*r* 99.8%

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

      4. fma-def 99.8%

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

      5. +-commutative 99.8%

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

      6. associate-+r+ 99.8%

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

      7. +-commutative 99.8%

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

      8. mul-1-neg 99.8%

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

      9. distribute-lft-neg-out 99.8%

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

      10. associate-*r* 99.8%

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

      11. distribute-rgt-in 99.8%

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

   6. Simplified 99.8%

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

   if 1.49999999999999987e-4 < eps

   1. Initial program 50.3%

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

      1. cos-sum 98.5%

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

      2. associate--l- 98.6%

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

      3. fma-neg 98.6%

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

      4. fma-def 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around -inf 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;\mathsf{fma}\left(\sin x, \mathsf{fma}\left({\varepsilon}^{3}, 0.16666666666666666, -\varepsilon\right), \cos x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.5\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 4: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.000145)
   (fma (sin x) (- (sin eps)) (* (cos x) (+ -1.0 (cos eps))))
   (if (<= eps 0.00015)
     (+
      (* -0.5 (* eps (* eps (cos x))))
      (* (sin x) (- (* (pow eps 3.0) 0.16666666666666666) eps)))
     (- (* (cos x) (cos eps)) (+ (cos x) (* (sin x) (sin eps)))))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.000145) {
		tmp = fma(sin(x), -sin(eps), (cos(x) * (-1.0 + cos(eps))));
	} else if (eps <= 0.00015) {
		tmp = (-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((pow(eps, 3.0) * 0.16666666666666666) - eps));
	} else {
		tmp = (cos(x) * cos(eps)) - (cos(x) + (sin(x) * sin(eps)));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.000145)
		tmp = fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	elseif (eps <= 0.00015)
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64((eps ^ 3.0) * 0.16666666666666666) - eps)));
	else
		tmp = Float64(Float64(cos(x) * cos(eps)) - Float64(cos(x) + Float64(sin(x) * sin(eps))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.000145], N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00015], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.45e-4

   1. Initial program 44.4%

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

      1. cos-sum 98.7%

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

      2. cancel-sign-sub-inv 98.7%

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

      3. fma-def 98.8%

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

   3. Applied egg-rr 98.8%

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

   4. Taylor expanded in x around inf 98.7%

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

      1. neg-mul-1 98.7%

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

      2. associate--l+ 98.9%

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

      3. *-commutative 98.9%

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

      4. distribute-rgt-neg-in 98.9%

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

      5. fma-def 98.8%

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

      6. *-rgt-identity 98.8%

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

      7. distribute-lft-out-- 98.8%

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

      8. sub-neg 98.8%

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

      9. metadata-eval 98.8%

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

      10. +-commutative 98.8%

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

   6. Simplified 98.8%

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

   if -1.45e-4 < eps < 1.49999999999999987e-4

   1. Initial program 26.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. unpow2 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*r* 99.8%

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

      6. associate-*r* 99.8%

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

      7. distribute-rgt-out 99.8%

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

      8. mul-1-neg 99.8%

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

   4. Simplified 99.8%

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

   if 1.49999999999999987e-4 < eps

   1. Initial program 50.3%

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

      1. cos-sum 98.5%

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

      2. associate--l- 98.6%

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

      3. fma-neg 98.6%

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

      4. fma-def 98.7%

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

   3. Applied egg-rr 98.7%

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

   4. Taylor expanded in x around -inf 98.6%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145:\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{elif}\;\varepsilon \leq 0.00015:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \cos \varepsilon - \left(\cos x + \sin x \cdot \sin \varepsilon\right)\\ \end{array} \]

Alternative 5: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.000145) (not (<= eps 0.00015)))
   (fma (sin x) (- (sin eps)) (* (cos x) (+ -1.0 (cos eps))))
   (+
    (* -0.5 (* eps (* eps (cos x))))
    (* (sin x) (- (* (pow eps 3.0) 0.16666666666666666) eps)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.000145) || !(eps <= 0.00015)) {
		tmp = fma(sin(x), -sin(eps), (cos(x) * (-1.0 + cos(eps))));
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) + (sin(x) * ((pow(eps, 3.0) * 0.16666666666666666) - eps));
	}
	return tmp;
}

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.000145) || !(eps <= 0.00015))
		tmp = fma(sin(x), Float64(-sin(eps)), Float64(cos(x) * Float64(-1.0 + cos(eps))));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) + Float64(sin(x) * Float64(Float64((eps ^ 3.0) * 0.16666666666666666) - eps)));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.000145], N[Not[LessEqual[eps, 0.00015]], $MachinePrecision]], N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision]) + N[(N[Cos[x], $MachinePrecision] * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(N[Power[eps, 3.0], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] - eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.45e-4 or 1.49999999999999987e-4 < eps

   1. Initial program 47.4%

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

      1. cos-sum 98.6%

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

      2. cancel-sign-sub-inv 98.6%

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

      3. fma-def 98.6%

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

   3. Applied egg-rr 98.6%

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

   4. Taylor expanded in x around inf 98.6%

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

      1. neg-mul-1 98.6%

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

      2. associate--l+ 98.8%

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

      3. *-commutative 98.8%

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

      4. distribute-rgt-neg-in 98.8%

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

      5. fma-def 98.7%

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

      6. *-rgt-identity 98.7%

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

      7. distribute-lft-out-- 98.7%

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

      8. sub-neg 98.7%

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

      9. metadata-eval 98.7%

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

      10. +-commutative 98.7%

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

   6. Simplified 98.7%

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

   if -1.45e-4 < eps < 1.49999999999999987e-4

   1. Initial program 26.2%

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

   2. Taylor expanded in eps around 0 99.8%

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

      1. +-commutative 99.8%

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

      2. associate-+l+ 99.8%

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

      3. unpow2 99.8%

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

      4. associate-*l* 99.8%

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

      5. associate-*r* 99.8%

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

      6. associate-*r* 99.8%

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

      7. distribute-rgt-out 99.8%

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

      8. mul-1-neg 99.8%

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

   4. Simplified 99.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000145 \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;\mathsf{fma}\left(\sin x, -\sin \varepsilon, \cos x \cdot \left(-1 + \cos \varepsilon\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) + \sin x \cdot \left({\varepsilon}^{3} \cdot 0.16666666666666666 - \varepsilon\right)\\ \end{array} \]

Alternative 6: 75.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -1e-14)
   (* -2.0 (pow (sin (* 0.5 eps)) 2.0))
   (- (* -0.5 (* eps (* eps (cos x)))) (* eps (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -1e-14) {
		tmp = -2.0 * pow(sin((0.5 * eps)), 2.0);
	} else {
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (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)) <= (-1d-14)) then
        tmp = (-2.0d0) * (sin((0.5d0 * eps)) ** 2.0d0)
    else
        tmp = ((-0.5d0) * (eps * (eps * cos(x)))) - (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)) <= -1e-14) {
		tmp = -2.0 * Math.pow(Math.sin((0.5 * eps)), 2.0);
	} else {
		tmp = (-0.5 * (eps * (eps * Math.cos(x)))) - (eps * Math.sin(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -1e-14:
		tmp = -2.0 * math.pow(math.sin((0.5 * eps)), 2.0)
	else:
		tmp = (-0.5 * (eps * (eps * math.cos(x)))) - (eps * math.sin(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -1e-14)
		tmp = Float64(-2.0 * (sin(Float64(0.5 * eps)) ^ 2.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(eps * Float64(eps * cos(x)))) - Float64(eps * sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -1e-14)
		tmp = -2.0 * (sin((0.5 * eps)) ^ 2.0);
	else
		tmp = (-0.5 * (eps * (eps * cos(x)))) - (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], -1e-14], N[(-2.0 * N[Power[N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(eps * N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 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)) < -9.99999999999999999e-15

   1. Initial program 69.1%

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

      1. diff-cos 70.8%

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

      2. div-inv 70.8%

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

      3. metadata-eval 70.8%

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

      4. div-inv 70.8%

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

      5. +-commutative 70.8%

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

      6. metadata-eval 70.8%

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

   3. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. +-commutative 70.8%

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

      3. associate--l+ 70.8%

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

      4. *-commutative 70.8%

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

      5. associate-+r+ 70.8%

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

      6. +-commutative 70.8%

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

   5. Simplified 70.8%

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

   6. Taylor expanded in x around 0 70.8%

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

   if -9.99999999999999999e-15 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 21.8%

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

   2. Taylor expanded in eps around 0 77.5%

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

      1. mul-1-neg 77.5%

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

      2. unsub-neg 77.5%

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

      3. unpow2 77.5%

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

      4. associate-*l* 77.5%

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

   4. Simplified 77.5%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot \cos x\right)\right) - \varepsilon \cdot \sin x\\ \end{array} \]

Alternative 7: 67.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -1e-14)
   (* -2.0 (pow (sin (* 0.5 eps)) 2.0))
   (* (sin x) (* -2.0 (sin (/ eps 2.0))))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -1e-14) {
		tmp = -2.0 * pow(sin((0.5 * eps)), 2.0);
	} else {
		tmp = sin(x) * (-2.0 * sin((eps / 2.0)));
	}
	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)) <= (-1d-14)) then
        tmp = (-2.0d0) * (sin((0.5d0 * eps)) ** 2.0d0)
    else
        tmp = sin(x) * ((-2.0d0) * sin((eps / 2.0d0)))
    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)) <= -1e-14) {
		tmp = -2.0 * Math.pow(Math.sin((0.5 * eps)), 2.0);
	} else {
		tmp = Math.sin(x) * (-2.0 * Math.sin((eps / 2.0)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -1e-14:
		tmp = -2.0 * math.pow(math.sin((0.5 * eps)), 2.0)
	else:
		tmp = math.sin(x) * (-2.0 * math.sin((eps / 2.0)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -1e-14)
		tmp = Float64(-2.0 * (sin(Float64(0.5 * eps)) ^ 2.0));
	else
		tmp = Float64(sin(x) * Float64(-2.0 * sin(Float64(eps / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -1e-14)
		tmp = -2.0 * (sin((0.5 * eps)) ^ 2.0);
	else
		tmp = sin(x) * (-2.0 * sin((eps / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], -1e-14], N[(-2.0 * N[Power[N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[x], $MachinePrecision] * N[(-2.0 * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.99999999999999999e-15

   1. Initial program 69.1%

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

      1. diff-cos 70.8%

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

      2. div-inv 70.8%

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

      3. metadata-eval 70.8%

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

      4. div-inv 70.8%

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

      5. +-commutative 70.8%

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

      6. metadata-eval 70.8%

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

   3. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. +-commutative 70.8%

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

      3. associate--l+ 70.8%

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

      4. *-commutative 70.8%

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

      5. associate-+r+ 70.8%

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

      6. +-commutative 70.8%

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

   5. Simplified 70.8%

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

   6. Taylor expanded in x around 0 70.8%

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

   if -9.99999999999999999e-15 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 21.8%

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

      1. expm1-log1p-u 21.8%

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

   3. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right) - \cos x\right)\right)} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 21.8%

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

      2. diff-cos 33.4%

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

      3. +-commutative 33.4%

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

      4. associate-+r- 78.4%

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

      5. +-inverses 78.4%

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

      6. +-commutative 78.4%

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

      7. associate-+r+ 78.6%

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

   5. Applied egg-rr 78.6%

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

      1. associate-*r* 78.6%

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

      2. +-rgt-identity 78.6%

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

   7. Simplified 78.6%

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

   8. Taylor expanded in eps around 0 67.6%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\sin x \cdot \left(-2 \cdot \sin \left(\frac{\varepsilon}{2}\right)\right)\\ \end{array} \]

Alternative 8: 66.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -1e-14)
   (* -2.0 (pow (sin (* 0.5 eps)) 2.0))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -1e-14) {
		tmp = -2.0 * pow(sin((0.5 * eps)), 2.0);
	} 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 ((cos((eps + x)) - cos(x)) <= (-1d-14)) then
        tmp = (-2.0d0) * (sin((0.5d0 * eps)) ** 2.0d0)
    else
        tmp = 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)) <= -1e-14) {
		tmp = -2.0 * Math.pow(Math.sin((0.5 * eps)), 2.0);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -1e-14:
		tmp = -2.0 * math.pow(math.sin((0.5 * eps)), 2.0)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -1e-14)
		tmp = Float64(-2.0 * (sin(Float64(0.5 * eps)) ^ 2.0));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -1e-14)
		tmp = -2.0 * (sin((0.5 * eps)) ^ 2.0);
	else
		tmp = 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], -1e-14], N[(-2.0 * N[Power[N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.99999999999999999e-15

   1. Initial program 69.1%

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

      1. diff-cos 70.8%

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

      2. div-inv 70.8%

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

      3. metadata-eval 70.8%

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

      4. div-inv 70.8%

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

      5. +-commutative 70.8%

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

      6. metadata-eval 70.8%

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

   3. Applied egg-rr 70.8%

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

      1. *-commutative 70.8%

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

      2. +-commutative 70.8%

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

      3. associate--l+ 70.8%

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

      4. *-commutative 70.8%

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

      5. associate-+r+ 70.8%

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

      6. +-commutative 70.8%

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

   5. Simplified 70.8%

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

   6. Taylor expanded in x around 0 70.8%

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

   if -9.99999999999999999e-15 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 21.8%

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

   2. Taylor expanded in eps around 0 66.1%

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

      1. associate-*r* 65.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}}\right)}^{3} \]

      2. mul-1-neg 65.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-\varepsilon\right)} \cdot \sin x}\right)}^{3} \]

   4. Simplified 66.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 9: 66.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-1 + \cos x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -1e-14)
   (+ -1.0 (* (cos x) (cos eps)))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -1e-14) {
		tmp = -1.0 + (cos(x) * 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 ((cos((eps + x)) - cos(x)) <= (-1d-14)) then
        tmp = (-1.0d0) + (cos(x) * 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 ((Math.cos((eps + x)) - Math.cos(x)) <= -1e-14) {
		tmp = -1.0 + (Math.cos(x) * Math.cos(eps));
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -1e-14:
		tmp = -1.0 + (math.cos(x) * math.cos(eps))
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -1e-14)
		tmp = Float64(-1.0 + Float64(cos(x) * cos(eps)));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -1e-14)
		tmp = -1.0 + (cos(x) * cos(eps));
	else
		tmp = 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], -1e-14], N[(-1.0 + N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.99999999999999999e-15

   1. Initial program 69.1%

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

      1. cos-sum 97.6%

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

      2. associate--l- 97.7%

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

      3. fma-neg 97.7%

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

      4. fma-def 97.7%

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

   3. Applied egg-rr 97.7%

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

   4. Taylor expanded in x around 0 69.2%

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

   5. Taylor expanded in x around -inf 69.2%

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

   if -9.99999999999999999e-15 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 21.8%

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

   2. Taylor expanded in eps around 0 66.1%

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

      1. associate-*r* 65.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}}\right)}^{3} \]

      2. mul-1-neg 65.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-\varepsilon\right)} \cdot \sin x}\right)}^{3} \]

   4. Simplified 66.1%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-1 + \cos x \cdot \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 10: 66.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= (- (cos (+ eps x)) (cos x)) -1e-14)
   (+ -1.0 (cos eps))
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((cos((eps + x)) - cos(x)) <= -1e-14) {
		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 ((cos((eps + x)) - cos(x)) <= (-1d-14)) 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 ((Math.cos((eps + x)) - Math.cos(x)) <= -1e-14) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (math.cos((eps + x)) - math.cos(x)) <= -1e-14:
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (Float64(cos(Float64(eps + x)) - cos(x)) <= -1e-14)
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((cos((eps + x)) - cos(x)) <= -1e-14)
		tmp = -1.0 + cos(eps);
	else
		tmp = 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], -1e-14], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -9.99999999999999999e-15

   1. Initial program 69.1%

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

   2. Taylor expanded in x around 0 69.2%

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

   if -9.99999999999999999e-15 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 21.8%

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

   2. Taylor expanded in eps around 0 66.1%

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

      1. associate-*r* 65.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}}\right)}^{3} \]

      2. mul-1-neg 65.2%

         \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-\varepsilon\right)} \cdot \sin x}\right)}^{3} \]

   4. Simplified 66.1%

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

4. Final simplification 67.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -1 \cdot 10^{-14}:\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 11: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

   1. expm1-log1p-u 26.3%

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

3. Applied egg-rr 26.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \left(x + \varepsilon\right) - \cos x\right)\right)} \]
4. Step-by-step derivation

   1. expm1-log1p-u 35.8%

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

   2. diff-cos 44.5%

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

   3. +-commutative 44.5%

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

   4. associate-+r- 76.1%

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

   5. +-inverses 76.1%

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

   6. +-commutative 76.1%

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

   7. associate-+r+ 76.3%

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

5. Applied egg-rr 76.3%

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

   1. associate-*r* 76.3%

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

   2. +-rgt-identity 76.3%

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

7. Simplified 76.3%

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

8. Final simplification 76.3%

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

Alternative 12: 47.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.8e-7) (not (<= eps 0.00015)))
   (+ -1.0 (cos eps))
   (* -0.5 (* eps eps))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.8e-7) || !(eps <= 0.00015)) {
		tmp = -1.0 + cos(eps);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-3.8d-7)) .or. (.not. (eps <= 0.00015d0))) then
        tmp = (-1.0d0) + cos(eps)
    else
        tmp = (-0.5d0) * (eps * eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.8e-7) || !(eps <= 0.00015)) {
		tmp = -1.0 + Math.cos(eps);
	} else {
		tmp = -0.5 * (eps * eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.8e-7) or not (eps <= 0.00015):
		tmp = -1.0 + math.cos(eps)
	else:
		tmp = -0.5 * (eps * eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.8e-7) || !(eps <= 0.00015))
		tmp = Float64(-1.0 + cos(eps));
	else
		tmp = Float64(-0.5 * Float64(eps * eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.8e-7) || ~((eps <= 0.00015)))
		tmp = -1.0 + cos(eps);
	else
		tmp = -0.5 * (eps * eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.8e-7], N[Not[LessEqual[eps, 0.00015]], $MachinePrecision]], N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(eps * eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.80000000000000015e-7 or 1.49999999999999987e-4 < eps

   1. Initial program 46.6%

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

   2. Taylor expanded in x around 0 48.6%

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

   if -3.80000000000000015e-7 < eps < 1.49999999999999987e-4

   1. Initial program 26.6%

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

   2. Taylor expanded in x around 0 26.5%

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

   3. Taylor expanded in eps around 0 39.5%

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

      1. *-commutative 39.5%

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

      2. unpow2 39.5%

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

   5. Simplified 39.5%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-7} \lor \neg \left(\varepsilon \leq 0.00015\right):\\ \;\;\;\;-1 + \cos \varepsilon\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\varepsilon \cdot \varepsilon\right)\\ \end{array} \]

Alternative 13: 21.4% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 35.8%

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

2. Taylor expanded in x around 0 36.7%

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

3. Taylor expanded in eps around 0 23.2%

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

   1. *-commutative 23.2%

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

   2. unpow2 23.2%

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

5. Simplified 23.2%

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

6. Final simplification 23.2%

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

Alternative 14: 12.2% accurate, 68.3× 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(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 35.8%

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

   1. add-cube-cbrt 35.6%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x} \cdot \sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right) \cdot \sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}} \]

   2. pow3 35.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^{3}} \]

3. Applied egg-rr 35.6%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^{3}} \]

4. Taylor expanded in eps around 0 46.9%

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

   1. associate-*r* 46.9%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}}\right)}^{3} \]

   2. mul-1-neg 46.9%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\left(-\varepsilon\right)} \cdot \sin x}\right)}^{3} \]

6. Simplified 46.9%

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

   1. rem-cube-cbrt 47.6%

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

   2. expm1-log1p-u 46.8%

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

   3. expm1-udef 15.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-\varepsilon\right) \cdot \sin x\right)} - 1} \]

   4. rem-cube-cbrt 15.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{\left(-\varepsilon\right) \cdot \sin x}\right)}^{3}}\right)} - 1 \]

   5. rem-cube-cbrt 15.5%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(-\varepsilon\right) \cdot \sin x}\right)} - 1 \]

   6. add-sqr-sqrt 8.0%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{-\varepsilon} \cdot \sqrt{-\varepsilon}\right)} \cdot \sin x\right)} - 1 \]

   7. sqrt-unprod 15.2%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\left(-\varepsilon\right) \cdot \left(-\varepsilon\right)}} \cdot \sin x\right)} - 1 \]

   8. sqr-neg 15.2%

      \[\leadsto e^{\mathsf{log1p}\left(\sqrt{\color{blue}{\varepsilon \cdot \varepsilon}} \cdot \sin x\right)} - 1 \]

   9. sqrt-unprod 7.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sqrt{\varepsilon} \cdot \sqrt{\varepsilon}\right)} \cdot \sin x\right)} - 1 \]

   10. add-sqr-sqrt 14.9%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\varepsilon} \cdot \sin x\right)} - 1 \]

8. Applied egg-rr 14.9%

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

   1. expm1-def 14.6%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\varepsilon \cdot \sin x\right)\right)} \]

   2. expm1-log1p 15.1%

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

   3. *-commutative 15.1%

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

10. Simplified 15.1%

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

11. Taylor expanded in x around 0 15.0%

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

    1. *-commutative 15.0%

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

13. Simplified 15.0%

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

14. Final simplification 15.0%

    \[\leadsto \varepsilon \cdot x \]

Reproduce

herbie shell --seed 2023238 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))