2cos (problem 3.3.5)

Percentage Accurate: 37.6% → 99.4%

Time: 22.0s

Alternatives: 12

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 37.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. diff-cos 49.9%

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

   2. div-inv 49.9%

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

   3. metadata-eval 49.9%

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

   4. div-inv 49.9%

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

   5. +-commutative 49.9%

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

   6. metadata-eval 49.9%

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

3. Applied egg-rr 49.9%

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

   1. *-commutative 49.9%

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

   2. +-commutative 49.9%

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

   3. associate--l+ 80.8%

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

   4. +-inverses 80.8%

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

   5. distribute-lft-in 80.8%

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

   6. metadata-eval 80.8%

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

   7. *-commutative 80.8%

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

   8. associate-+r+ 80.8%

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

   9. +-commutative 80.8%

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

5. Simplified 80.8%

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

6. Taylor expanded in eps around 0 81.2%

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

   1. sin-sum 99.5%

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

8. Applied egg-rr 99.5%

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

   1. fma-def 99.5%

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

   2. *-commutative 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in eps around inf 99.5%

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

    1. +-commutative 99.5%

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

    2. *-commutative 99.5%

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

    3. *-commutative 99.5%

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

    4. fma-def 99.5%

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

    5. *-commutative 99.5%

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

13. Simplified 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. diff-cos 49.9%

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

   2. div-inv 49.9%

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

   3. metadata-eval 49.9%

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

   4. div-inv 49.9%

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

   5. +-commutative 49.9%

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

   6. metadata-eval 49.9%

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

3. Applied egg-rr 49.9%

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

   1. *-commutative 49.9%

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

   2. +-commutative 49.9%

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

   3. associate--l+ 80.8%

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

   4. +-inverses 80.8%

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

   5. distribute-lft-in 80.8%

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

   6. metadata-eval 80.8%

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

   7. *-commutative 80.8%

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

   8. associate-+r+ 80.8%

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

   9. +-commutative 80.8%

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

5. Simplified 80.8%

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

6. Taylor expanded in eps around 0 81.2%

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

   1. sin-sum 99.5%

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

8. Applied egg-rr 99.5%

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

   1. fma-def 99.5%

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

   2. *-commutative 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in eps around inf 99.5%

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

12. Final simplification 99.5%

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

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0036 \lor \neg \left(\varepsilon \leq 0.0037\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.125\right) + 1\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0036) (not (<= eps 0.0037)))
   (- (- (* (cos x) (cos eps)) (* (sin x) (sin eps))) (cos x))
   (*
    -2.0
    (*
     (sin (* 0.5 eps))
     (+
      (* (cos x) (+ (* 0.5 eps) (* -0.020833333333333332 (pow eps 3.0))))
      (* (sin x) (+ (* eps (* eps -0.125)) 1.0)))))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0036) || !(eps <= 0.0037)) {
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	} else {
		tmp = -2.0 * (sin((0.5 * eps)) * ((cos(x) * ((0.5 * eps) + (-0.020833333333333332 * pow(eps, 3.0)))) + (sin(x) * ((eps * (eps * -0.125)) + 1.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.0036d0)) .or. (.not. (eps <= 0.0037d0))) then
        tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x)
    else
        tmp = (-2.0d0) * (sin((0.5d0 * eps)) * ((cos(x) * ((0.5d0 * eps) + ((-0.020833333333333332d0) * (eps ** 3.0d0)))) + (sin(x) * ((eps * (eps * (-0.125d0))) + 1.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0036) || !(eps <= 0.0037)) {
		tmp = ((Math.cos(x) * Math.cos(eps)) - (Math.sin(x) * Math.sin(eps))) - Math.cos(x);
	} else {
		tmp = -2.0 * (Math.sin((0.5 * eps)) * ((Math.cos(x) * ((0.5 * eps) + (-0.020833333333333332 * Math.pow(eps, 3.0)))) + (Math.sin(x) * ((eps * (eps * -0.125)) + 1.0))));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0036) or not (eps <= 0.0037):
		tmp = ((math.cos(x) * math.cos(eps)) - (math.sin(x) * math.sin(eps))) - math.cos(x)
	else:
		tmp = -2.0 * (math.sin((0.5 * eps)) * ((math.cos(x) * ((0.5 * eps) + (-0.020833333333333332 * math.pow(eps, 3.0)))) + (math.sin(x) * ((eps * (eps * -0.125)) + 1.0))))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0036) || !(eps <= 0.0037))
		tmp = Float64(Float64(Float64(cos(x) * cos(eps)) - Float64(sin(x) * sin(eps))) - cos(x));
	else
		tmp = Float64(-2.0 * Float64(sin(Float64(0.5 * eps)) * Float64(Float64(cos(x) * Float64(Float64(0.5 * eps) + Float64(-0.020833333333333332 * (eps ^ 3.0)))) + Float64(sin(x) * Float64(Float64(eps * Float64(eps * -0.125)) + 1.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0036) || ~((eps <= 0.0037)))
		tmp = ((cos(x) * cos(eps)) - (sin(x) * sin(eps))) - cos(x);
	else
		tmp = -2.0 * (sin((0.5 * eps)) * ((cos(x) * ((0.5 * eps) + (-0.020833333333333332 * (eps ^ 3.0)))) + (sin(x) * ((eps * (eps * -0.125)) + 1.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0036], N[Not[LessEqual[eps, 0.0037]], $MachinePrecision]], N[(N[(N[(N[Cos[x], $MachinePrecision] * N[Cos[eps], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Cos[x], $MachinePrecision] * N[(N[(0.5 * eps), $MachinePrecision] + N[(-0.020833333333333332 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[(eps * N[(eps * -0.125), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0035999999999999999 or 0.0037000000000000002 < eps

   1. Initial program 61.3%

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

      1. cos-sum 98.7%

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

   3. Applied egg-rr 98.7%

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

   if -0.0035999999999999999 < eps < 0.0037000000000000002

   1. Initial program 21.0%

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

      1. diff-cos 39.2%

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

      2. div-inv 39.2%

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

      3. metadata-eval 39.2%

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

      4. div-inv 39.2%

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

      5. +-commutative 39.2%

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

      6. metadata-eval 39.2%

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

   3. Applied egg-rr 39.2%

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

      1. *-commutative 39.2%

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

      2. +-commutative 39.2%

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

      3. associate--l+ 98.4%

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

      4. +-inverses 98.4%

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

      5. distribute-lft-in 98.4%

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

      6. metadata-eval 98.4%

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

      7. *-commutative 98.4%

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

      8. associate-+r+ 98.4%

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

      9. +-commutative 98.4%

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

   5. Simplified 98.4%

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

   6. Taylor expanded in eps around 0 99.8%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\left(\sin x + \left(-0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)\right)}\right) \]
   7. Step-by-step derivation

      1. associate-+r+ 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\left(\left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right) + \left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right)\right)}\right) \]

      2. +-commutative 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\left(\left(-0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + 0.5 \cdot \left(\varepsilon \cdot \cos x\right)\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)}\right) \]

      3. +-commutative 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\color{blue}{\left(0.5 \cdot \left(\varepsilon \cdot \cos x\right) + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right)} + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)\right) \]

      4. associate-*r* 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\left(\color{blue}{\left(0.5 \cdot \varepsilon\right) \cdot \cos x} + -0.020833333333333332 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)\right) \]

      5. associate-*r* 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\left(\left(0.5 \cdot \varepsilon\right) \cdot \cos x + \color{blue}{\left(-0.020833333333333332 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right) + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)\right) \]

      6. distribute-rgt-out 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\color{blue}{\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right)} + \left(\sin x + -0.125 \cdot \left({\varepsilon}^{2} \cdot \sin x\right)\right)\right)\right) \]

      7. associate-*r* 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \left(\sin x + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2}\right) \cdot \sin x}\right)\right)\right) \]

      8. distribute-rgt1-in 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \color{blue}{\left(-0.125 \cdot {\varepsilon}^{2} + 1\right) \cdot \sin x}\right)\right) \]

      9. unpow2 99.8%

         \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \left(-0.125 \cdot \color{blue}{\left(\varepsilon \cdot \varepsilon\right)} + 1\right) \cdot \sin x\right)\right) \]

      10. associate-*r* 99.8%

          \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \left(\color{blue}{\left(-0.125 \cdot \varepsilon\right) \cdot \varepsilon} + 1\right) \cdot \sin x\right)\right) \]

   8. Simplified 99.8%

      \[\leadsto -2 \cdot \left(\sin \left(0.5 \cdot \varepsilon + 0\right) \cdot \color{blue}{\left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \left(\left(-0.125 \cdot \varepsilon\right) \cdot \varepsilon + 1\right) \cdot \sin x\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0036 \lor \neg \left(\varepsilon \leq 0.0037\right):\\ \;\;\;\;\left(\cos x \cdot \cos \varepsilon - \sin x \cdot \sin \varepsilon\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \left(\cos x \cdot \left(0.5 \cdot \varepsilon + -0.020833333333333332 \cdot {\varepsilon}^{3}\right) + \sin x \cdot \left(\varepsilon \cdot \left(\varepsilon \cdot -0.125\right) + 1\right)\right)\right)\\ \end{array} \]

Alternative 4: 66.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{if}\;t_0 \leq -0.002:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (cos (+ eps x)) (cos x))))
   (if (<= t_0 -0.002) t_0 (expm1 (* eps (- (sin x)))))))

C

double code(double x, double eps) {
	double t_0 = cos((eps + x)) - cos(x);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_0;
	} else {
		tmp = expm1((eps * -sin(x)));
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos((eps + x)) - Math.cos(x);
	double tmp;
	if (t_0 <= -0.002) {
		tmp = t_0;
	} else {
		tmp = Math.expm1((eps * -Math.sin(x)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos((eps + x)) - math.cos(x)
	tmp = 0
	if t_0 <= -0.002:
		tmp = t_0
	else:
		tmp = math.expm1((eps * -math.sin(x)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(Float64(eps + x)) - cos(x))
	tmp = 0.0
	if (t_0 <= -0.002)
		tmp = t_0;
	else
		tmp = expm1(Float64(eps * Float64(-sin(x))));
	end
	return tmp
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.002], t$95$0, N[(Exp[N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x)) < -2e-3

   1. Initial program 85.8%

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

   if -2e-3 < (-.f64 (cos.f64 (+.f64 x eps)) (cos.f64 x))

   1. Initial program 17.7%

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

      1. expm1-log1p-u 17.7%

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

   3. Applied egg-rr 17.7%

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

   4. Taylor expanded in eps around 0 63.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)}\right) \]
   5. Step-by-step derivation

      1. associate-*r* 63.6%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}\right) \]

      2. mul-1-neg 63.6%

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

   6. Simplified 63.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-\varepsilon\right) \cdot \sin x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cos \left(\varepsilon + x\right) - \cos x \leq -0.002:\\ \;\;\;\;\cos \left(\varepsilon + x\right) - \cos x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon \cdot \left(-\sin x\right)\right)\\ \end{array} \]

Alternative 5: 69.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-13} \lor \neg \left(x \leq 5.6 \cdot 10^{-28}\right):\\ \;\;\;\;-2 \cdot \left(t_0 \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {t_0}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (if (or (<= x -1.9e-13) (not (<= x 5.6e-28)))
     (* -2.0 (* t_0 (sin x)))
     (* -2.0 (pow t_0 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	double tmp;
	if ((x <= -1.9e-13) || !(x <= 5.6e-28)) {
		tmp = -2.0 * (t_0 * sin(x));
	} else {
		tmp = -2.0 * pow(t_0, 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((0.5d0 * eps))
    if ((x <= (-1.9d-13)) .or. (.not. (x <= 5.6d-28))) then
        tmp = (-2.0d0) * (t_0 * sin(x))
    else
        tmp = (-2.0d0) * (t_0 ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((0.5 * eps));
	double tmp;
	if ((x <= -1.9e-13) || !(x <= 5.6e-28)) {
		tmp = -2.0 * (t_0 * Math.sin(x));
	} else {
		tmp = -2.0 * Math.pow(t_0, 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((0.5 * eps))
	tmp = 0
	if (x <= -1.9e-13) or not (x <= 5.6e-28):
		tmp = -2.0 * (t_0 * math.sin(x))
	else:
		tmp = -2.0 * math.pow(t_0, 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	tmp = 0.0
	if ((x <= -1.9e-13) || !(x <= 5.6e-28))
		tmp = Float64(-2.0 * Float64(t_0 * sin(x)));
	else
		tmp = Float64(-2.0 * (t_0 ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((0.5 * eps));
	tmp = 0.0;
	if ((x <= -1.9e-13) || ~((x <= 5.6e-28)))
		tmp = -2.0 * (t_0 * sin(x));
	else
		tmp = -2.0 * (t_0 ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[x, -1.9e-13], N[Not[LessEqual[x, 5.6e-28]], $MachinePrecision]], N[(-2.0 * N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.9e-13 or 5.5999999999999996e-28 < x

   1. Initial program 8.0%

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

      1. diff-cos 7.3%

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

      2. div-inv 7.3%

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

      3. metadata-eval 7.3%

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

      4. div-inv 7.3%

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

      5. +-commutative 7.3%

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

      6. metadata-eval 7.3%

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

   3. Applied egg-rr 7.3%

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

      1. *-commutative 7.3%

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

      2. +-commutative 7.3%

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

      3. associate--l+ 61.0%

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

      4. +-inverses 61.0%

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

      5. distribute-lft-in 61.0%

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

      6. metadata-eval 61.0%

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

      7. *-commutative 61.0%

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

      8. associate-+r+ 61.1%

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

      9. +-commutative 61.1%

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

   5. Simplified 61.1%

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

   6. Taylor expanded in eps around 0 60.2%

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

   if -1.9e-13 < x < 5.5999999999999996e-28

   1. Initial program 71.7%

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

      1. diff-cos 89.9%

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

      2. div-inv 89.9%

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

      3. metadata-eval 89.9%

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

      4. div-inv 89.9%

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

      5. +-commutative 89.9%

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

      6. metadata-eval 89.9%

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

   3. Applied egg-rr 89.9%

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

      1. *-commutative 89.9%

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

      2. +-commutative 89.9%

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

      3. associate--l+ 99.4%

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

      4. +-inverses 99.4%

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

      5. distribute-lft-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-+r+ 99.4%

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

      9. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 89.8%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-13} \lor \neg \left(x \leq 5.6 \cdot 10^{-28}\right):\\ \;\;\;\;-2 \cdot \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \end{array} \]

Alternative 6: 76.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. diff-cos 49.9%

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

   2. div-inv 49.9%

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

   3. metadata-eval 49.9%

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

   4. div-inv 49.9%

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

   5. +-commutative 49.9%

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

   6. metadata-eval 49.9%

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

3. Applied egg-rr 49.9%

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

   1. *-commutative 49.9%

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

   2. +-commutative 49.9%

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

   3. associate--l+ 80.8%

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

   4. +-inverses 80.8%

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

   5. distribute-lft-in 80.8%

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

   6. metadata-eval 80.8%

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

   7. *-commutative 80.8%

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

   8. associate-+r+ 80.8%

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

   9. +-commutative 80.8%

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

5. Simplified 80.8%

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

6. Taylor expanded in eps around 0 81.2%

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

7. Taylor expanded in eps around inf 81.2%

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

8. Final simplification 81.2%

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

Alternative 7: 68.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-12} \lor \neg \left(x \leq 4.1 \cdot 10^{-28}\right):\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -4.2e-12) (not (<= x 4.1e-28)))
   (expm1 (* eps (- (sin x))))
   (* -2.0 (pow (sin (* 0.5 eps)) 2.0))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -4.2e-12) || !(x <= 4.1e-28)) {
		tmp = expm1((eps * -sin(x)));
	} else {
		tmp = -2.0 * pow(sin((0.5 * eps)), 2.0);
	}
	return tmp;
}

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -4.2e-12) || !(x <= 4.1e-28)) {
		tmp = Math.expm1((eps * -Math.sin(x)));
	} else {
		tmp = -2.0 * Math.pow(Math.sin((0.5 * eps)), 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -4.2e-12) or not (x <= 4.1e-28):
		tmp = math.expm1((eps * -math.sin(x)))
	else:
		tmp = -2.0 * math.pow(math.sin((0.5 * eps)), 2.0)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -4.2e-12) || !(x <= 4.1e-28))
		tmp = expm1(Float64(eps * Float64(-sin(x))));
	else
		tmp = Float64(-2.0 * (sin(Float64(0.5 * eps)) ^ 2.0));
	end
	return tmp
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -4.2e-12], N[Not[LessEqual[x, 4.1e-28]], $MachinePrecision]], N[(Exp[N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]] - 1), $MachinePrecision], N[(-2.0 * N[Power[N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.19999999999999988e-12 or 4.1000000000000002e-28 < x

   1. Initial program 8.0%

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

      1. expm1-log1p-u 6.5%

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

   3. Applied egg-rr 6.5%

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

   4. Taylor expanded in eps around 0 58.6%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \left(\varepsilon \cdot \sin x\right)}\right) \]
   5. Step-by-step derivation

      1. associate-*r* 58.6%

         \[\leadsto \mathsf{expm1}\left(\color{blue}{\left(-1 \cdot \varepsilon\right) \cdot \sin x}\right) \]

      2. mul-1-neg 58.6%

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

   6. Simplified 58.6%

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

   if -4.19999999999999988e-12 < x < 4.1000000000000002e-28

   1. Initial program 71.7%

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

      1. diff-cos 89.9%

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

      2. div-inv 89.9%

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

      3. metadata-eval 89.9%

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

      4. div-inv 89.9%

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

      5. +-commutative 89.9%

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

      6. metadata-eval 89.9%

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

   3. Applied egg-rr 89.9%

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

      1. *-commutative 89.9%

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

      2. +-commutative 89.9%

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

      3. associate--l+ 99.4%

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

      4. +-inverses 99.4%

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

      5. distribute-lft-in 99.4%

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

      6. metadata-eval 99.4%

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

      7. *-commutative 99.4%

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

      8. associate-+r+ 99.4%

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

      9. +-commutative 99.4%

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

   5. Simplified 99.4%

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

   6. Taylor expanded in x around 0 89.8%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.2 \cdot 10^{-12} \lor \neg \left(x \leq 4.1 \cdot 10^{-28}\right):\\ \;\;\;\;\mathsf{expm1}\left(\varepsilon \cdot \left(-\sin x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot {\sin \left(0.5 \cdot \varepsilon\right)}^{2}\\ \end{array} \]

Alternative 8: 67.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -4.2e-5)
   (- (cos eps) (cos x))
   (if (<= eps 4.8e-20) (* eps (- (sin x))) (+ (cos eps) -1.0))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -4.2e-5) {
		tmp = cos(eps) - cos(x);
	} else if (eps <= 4.8e-20) {
		tmp = eps * -sin(x);
	} else {
		tmp = cos(eps) + -1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-4.2d-5)) then
        tmp = cos(eps) - cos(x)
    else if (eps <= 4.8d-20) then
        tmp = eps * -sin(x)
    else
        tmp = cos(eps) + (-1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -4.2e-5) {
		tmp = Math.cos(eps) - Math.cos(x);
	} else if (eps <= 4.8e-20) {
		tmp = eps * -Math.sin(x);
	} else {
		tmp = Math.cos(eps) + -1.0;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -4.2e-5:
		tmp = math.cos(eps) - math.cos(x)
	elif eps <= 4.8e-20:
		tmp = eps * -math.sin(x)
	else:
		tmp = math.cos(eps) + -1.0
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -4.2e-5)
		tmp = Float64(cos(eps) - cos(x));
	elseif (eps <= 4.8e-20)
		tmp = Float64(eps * Float64(-sin(x)));
	else
		tmp = Float64(cos(eps) + -1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -4.2e-5)
		tmp = cos(eps) - cos(x);
	elseif (eps <= 4.8e-20)
		tmp = eps * -sin(x);
	else
		tmp = cos(eps) + -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -4.2e-5], N[(N[Cos[eps], $MachinePrecision] - N[Cos[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 4.8e-20], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4.19999999999999977e-5

   1. Initial program 62.6%

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

   2. Taylor expanded in x around 0 64.5%

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

   if -4.19999999999999977e-5 < eps < 4.79999999999999986e-20

   1. Initial program 21.1%

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

   2. Taylor expanded in eps around 0 81.5%

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

      1. mul-1-neg 81.5%

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

      2. *-commutative 81.5%

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

      3. distribute-rgt-neg-in 81.5%

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

   4. Simplified 81.5%

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

   if 4.79999999999999986e-20 < eps

   1. Initial program 58.9%

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

   2. Taylor expanded in x around 0 60.0%

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -4.2 \cdot 10^{-5}:\\ \;\;\;\;\cos \varepsilon - \cos x\\ \mathbf{elif}\;\varepsilon \leq 4.8 \cdot 10^{-20}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 9: 48.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \varepsilon + -1\\ t_1 := \left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;t_0\\ \mathbf{elif}\;\varepsilon \leq -5 \cdot 10^{-138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-122}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000175:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (+ (cos eps) -1.0)) (t_1 (* (* eps eps) -0.5)))
   (if (<= eps -0.00017)
     t_0
     (if (<= eps -5e-138)
       t_1
       (if (<= eps 6.5e-122) (* eps (- x)) (if (<= eps 0.000175) t_1 t_0))))))

C

double code(double x, double eps) {
	double t_0 = cos(eps) + -1.0;
	double t_1 = (eps * eps) * -0.5;
	double tmp;
	if (eps <= -0.00017) {
		tmp = t_0;
	} else if (eps <= -5e-138) {
		tmp = t_1;
	} else if (eps <= 6.5e-122) {
		tmp = eps * -x;
	} else if (eps <= 0.000175) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(eps) + (-1.0d0)
    t_1 = (eps * eps) * (-0.5d0)
    if (eps <= (-0.00017d0)) then
        tmp = t_0
    else if (eps <= (-5d-138)) then
        tmp = t_1
    else if (eps <= 6.5d-122) then
        tmp = eps * -x
    else if (eps <= 0.000175d0) then
        tmp = t_1
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.cos(eps) + -1.0;
	double t_1 = (eps * eps) * -0.5;
	double tmp;
	if (eps <= -0.00017) {
		tmp = t_0;
	} else if (eps <= -5e-138) {
		tmp = t_1;
	} else if (eps <= 6.5e-122) {
		tmp = eps * -x;
	} else if (eps <= 0.000175) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.cos(eps) + -1.0
	t_1 = (eps * eps) * -0.5
	tmp = 0
	if eps <= -0.00017:
		tmp = t_0
	elif eps <= -5e-138:
		tmp = t_1
	elif eps <= 6.5e-122:
		tmp = eps * -x
	elif eps <= 0.000175:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(cos(eps) + -1.0)
	t_1 = Float64(Float64(eps * eps) * -0.5)
	tmp = 0.0
	if (eps <= -0.00017)
		tmp = t_0;
	elseif (eps <= -5e-138)
		tmp = t_1;
	elseif (eps <= 6.5e-122)
		tmp = Float64(eps * Float64(-x));
	elseif (eps <= 0.000175)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = cos(eps) + -1.0;
	t_1 = (eps * eps) * -0.5;
	tmp = 0.0;
	if (eps <= -0.00017)
		tmp = t_0;
	elseif (eps <= -5e-138)
		tmp = t_1;
	elseif (eps <= 6.5e-122)
		tmp = eps * -x;
	elseif (eps <= 0.000175)
		tmp = t_1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[eps, -0.00017], t$95$0, If[LessEqual[eps, -5e-138], t$95$1, If[LessEqual[eps, 6.5e-122], N[(eps * (-x)), $MachinePrecision], If[LessEqual[eps, 0.000175], t$95$1, t$95$0]]]]]]

Derivation

1. Split input into 3 regimes

2. if eps < -1.7e-4 or 1.74999999999999998e-4 < eps

   1. Initial program 61.3%

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

   2. Taylor expanded in x around 0 61.8%

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

   if -1.7e-4 < eps < -4.99999999999999989e-138 or 6.49999999999999965e-122 < eps < 1.74999999999999998e-4

   1. Initial program 4.1%

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

   2. Taylor expanded in x around 0 4.2%

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

   3. Taylor expanded in eps around 0 41.7%

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

      1. *-commutative 41.7%

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

      2. unpow2 41.7%

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

   5. Simplified 41.7%

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

   if -4.99999999999999989e-138 < eps < 6.49999999999999965e-122

   1. Initial program 36.4%

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

   2. Taylor expanded in eps around 0 98.5%

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

      1. mul-1-neg 98.5%

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

      2. *-commutative 98.5%

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

      3. distribute-rgt-neg-in 98.5%

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

   4. Simplified 98.5%

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

   5. Taylor expanded in x around 0 55.0%

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

      1. associate-*r* 55.0%

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

      2. neg-mul-1 55.0%

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

   7. Simplified 55.0%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00017:\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{elif}\;\varepsilon \leq -5 \cdot 10^{-138}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{elif}\;\varepsilon \leq 6.5 \cdot 10^{-122}:\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{elif}\;\varepsilon \leq 0.000175:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\cos \varepsilon + -1\\ \end{array} \]

Alternative 10: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-20}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3.8e-5) (not (<= eps 4.8e-20)))
   (+ (cos eps) -1.0)
   (* eps (- (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.8e-5) || !(eps <= 4.8e-20)) {
		tmp = cos(eps) + -1.0;
	} 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 ((eps <= (-3.8d-5)) .or. (.not. (eps <= 4.8d-20))) then
        tmp = cos(eps) + (-1.0d0)
    else
        tmp = eps * -sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -3.8e-5) || !(eps <= 4.8e-20)) {
		tmp = Math.cos(eps) + -1.0;
	} else {
		tmp = eps * -Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3.8e-5) or not (eps <= 4.8e-20):
		tmp = math.cos(eps) + -1.0
	else:
		tmp = eps * -math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3.8e-5) || !(eps <= 4.8e-20))
		tmp = Float64(cos(eps) + -1.0);
	else
		tmp = Float64(eps * Float64(-sin(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -3.8e-5) || ~((eps <= 4.8e-20)))
		tmp = cos(eps) + -1.0;
	else
		tmp = eps * -sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3.8e-5], N[Not[LessEqual[eps, 4.8e-20]], $MachinePrecision]], N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision], N[(eps * (-N[Sin[x], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.8000000000000002e-5 or 4.79999999999999986e-20 < eps

   1. Initial program 60.9%

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

   2. Taylor expanded in x around 0 61.3%

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

   if -3.8000000000000002e-5 < eps < 4.79999999999999986e-20

   1. Initial program 21.1%

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

   2. Taylor expanded in eps around 0 81.5%

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

      1. mul-1-neg 81.5%

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

      2. *-commutative 81.5%

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

      3. distribute-rgt-neg-in 81.5%

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

   4. Simplified 81.5%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3.8 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 4.8 \cdot 10^{-20}\right):\\ \;\;\;\;\cos \varepsilon + -1\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \left(-\sin x\right)\\ \end{array} \]

Alternative 11: 24.6% accurate, 22.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-88} \lor \neg \left(x \leq 4.45 \cdot 10^{-100}\right):\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -9.5e-88) (not (<= x 4.45e-100)))
   (* eps (- x))
   (* (* eps eps) -0.5)))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -9.5e-88) || !(x <= 4.45e-100)) {
		tmp = eps * -x;
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-9.5d-88)) .or. (.not. (x <= 4.45d-100))) then
        tmp = eps * -x
    else
        tmp = (eps * eps) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -9.5e-88) || !(x <= 4.45e-100)) {
		tmp = eps * -x;
	} else {
		tmp = (eps * eps) * -0.5;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -9.5e-88) or not (x <= 4.45e-100):
		tmp = eps * -x
	else:
		tmp = (eps * eps) * -0.5
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -9.5e-88) || !(x <= 4.45e-100))
		tmp = Float64(eps * Float64(-x));
	else
		tmp = Float64(Float64(eps * eps) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -9.5e-88) || ~((x <= 4.45e-100)))
		tmp = eps * -x;
	else
		tmp = (eps * eps) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -9.5e-88], N[Not[LessEqual[x, 4.45e-100]], $MachinePrecision]], N[(eps * (-x)), $MachinePrecision], N[(N[(eps * eps), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5e-88 or 4.4500000000000002e-100 < x

   1. Initial program 21.6%

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

   2. Taylor expanded in eps around 0 52.5%

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

      1. mul-1-neg 52.5%

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

      2. *-commutative 52.5%

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

      3. distribute-rgt-neg-in 52.5%

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

   4. Simplified 52.5%

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

   5. Taylor expanded in x around 0 14.8%

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

      1. associate-*r* 14.8%

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

      2. neg-mul-1 14.8%

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

   7. Simplified 14.8%

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

   if -9.5e-88 < x < 4.4500000000000002e-100

   1. Initial program 74.0%

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

   2. Taylor expanded in x around 0 74.0%

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

   3. Taylor expanded in eps around 0 50.4%

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

      1. *-commutative 50.4%

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

      2. unpow2 50.4%

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

   5. Simplified 50.4%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-88} \lor \neg \left(x \leq 4.45 \cdot 10^{-100}\right):\\ \;\;\;\;\varepsilon \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\varepsilon \cdot \varepsilon\right) \cdot -0.5\\ \end{array} \]

Alternative 12: 18.3% accurate, 51.3× speedup

Math

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

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 * Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

2. Taylor expanded in eps around 0 43.1%

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

   1. mul-1-neg 43.1%

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

   2. *-commutative 43.1%

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

   3. distribute-rgt-neg-in 43.1%

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

4. Simplified 43.1%

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

5. Taylor expanded in x around 0 19.2%

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

   1. associate-*r* 19.2%

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

   2. neg-mul-1 19.2%

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

7. Simplified 19.2%

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

8. Final simplification 19.2%

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

Reproduce

herbie shell --seed 2023275 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))