2cos (problem 3.3.5)

Percentage Accurate: 52.2% → 99.8%

Time: 18.7s

Alternatives: 10

Speedup: 51.3×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

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: 52.2% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \mathsf{fma}\left(t\_0, \cos x, \cos \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right) \cdot \left(t\_0 \cdot -2\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. diff-cos 81.7%

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

   2. div-inv 81.7%

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

   3. associate--l+ 81.6%

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

   4. metadata-eval 81.6%

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

   5. div-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. metadata-eval 81.6%

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

4. Applied egg-rr 81.6%

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

   1. associate-*r* 81.7%

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

   2. *-commutative 81.7%

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

   3. *-commutative 81.7%

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

   4. associate-+r+ 81.7%

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

   5. +-commutative 81.7%

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

   6. count-2 81.7%

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

   7. *-commutative 81.7%

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

   8. associate-+r- 81.7%

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

   9. +-commutative 81.7%

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

   10. associate--l+ 99.7%

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

   11. +-inverses 99.7%

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

   12. +-commutative 99.7%

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

   13. *-lft-identity 99.7%

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

   14. metadata-eval 99.7%

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

   15. cancel-sign-sub-inv 99.7%

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

   16. neg-sub0 99.7%

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

   17. mul-1-neg 99.7%

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

   18. remove-double-neg 99.7%

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

6. Simplified 99.7%

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

   1. distribute-lft-in 99.7%

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

   2. *-commutative 99.7%

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

   3. sin-sum 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r* 99.8%

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

   7. metadata-eval 99.8%

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

   8. *-un-lft-identity 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*r* 99.8%

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

   12. metadata-eval 99.8%

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

   13. *-un-lft-identity 99.8%

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

8. Applied egg-rr 99.8%

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

   1. fma-define 99.8%

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

10. Applied egg-rr 99.8%

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

11. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \left(t\_0 \cdot -2\right) \cdot \left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \sin x + t\_0 \cdot \cos x\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. diff-cos 81.7%

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

   2. div-inv 81.7%

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

   3. associate--l+ 81.6%

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

   4. metadata-eval 81.6%

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

   5. div-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. metadata-eval 81.6%

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

4. Applied egg-rr 81.6%

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

   1. associate-*r* 81.7%

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

   2. *-commutative 81.7%

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

   3. *-commutative 81.7%

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

   4. associate-+r+ 81.7%

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

   5. +-commutative 81.7%

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

   6. count-2 81.7%

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

   7. *-commutative 81.7%

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

   8. associate-+r- 81.7%

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

   9. +-commutative 81.7%

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

   10. associate--l+ 99.7%

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

   11. +-inverses 99.7%

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

   12. +-commutative 99.7%

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

   13. *-lft-identity 99.7%

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

   14. metadata-eval 99.7%

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

   15. cancel-sign-sub-inv 99.7%

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

   16. neg-sub0 99.7%

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

   17. mul-1-neg 99.7%

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

   18. remove-double-neg 99.7%

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

6. Simplified 99.7%

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

   1. distribute-lft-in 99.7%

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

   2. *-commutative 99.7%

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

   3. sin-sum 99.8%

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

   4. *-commutative 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*r* 99.8%

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

   7. metadata-eval 99.8%

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

   8. *-un-lft-identity 99.8%

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

   9. *-commutative 99.8%

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

   10. *-commutative 99.8%

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

   11. associate-*r* 99.8%

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

   12. metadata-eval 99.8%

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

   13. *-un-lft-identity 99.8%

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

8. Applied egg-rr 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

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

   1. diff-cos 81.7%

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

   2. div-inv 81.7%

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

   3. associate--l+ 81.6%

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

   4. metadata-eval 81.6%

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

   5. div-inv 81.6%

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

   6. +-commutative 81.6%

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

   7. metadata-eval 81.6%

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

4. Applied egg-rr 81.6%

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

   1. associate-*r* 81.7%

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

   2. *-commutative 81.7%

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

   3. *-commutative 81.7%

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

   4. associate-+r+ 81.7%

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

   5. +-commutative 81.7%

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

   6. count-2 81.7%

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

   7. *-commutative 81.7%

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

   8. associate-+r- 81.7%

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

   9. +-commutative 81.7%

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

   10. associate--l+ 99.7%

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

   11. +-inverses 99.7%

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

   12. +-commutative 99.7%

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

   13. *-lft-identity 99.7%

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

   14. metadata-eval 99.7%

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

   15. cancel-sign-sub-inv 99.7%

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

   16. neg-sub0 99.7%

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

   17. mul-1-neg 99.7%

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

   18. remove-double-neg 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

5. Simplified 98.9%

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

Alternative 5: 98.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.5%

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

   1. *-commutative 98.5%

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

8. Simplified 98.5%

   \[\leadsto \varepsilon \cdot \left(\color{blue}{\varepsilon \cdot -0.5} - \sin x\right) \]
9. Add Preprocessing

Alternative 6: 98.2% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (* eps -0.5)
   (* x (+ (* x (+ (* x 0.16666666666666666) (* eps 0.25))) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((x * ((x * 0.16666666666666666d0) + (eps * 0.25d0))) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(eps * 0.25))) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((x * ((x * 0.16666666666666666) + (eps * 0.25))) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(eps * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

7. Final simplification 98.3%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666 + \varepsilon \cdot 0.25\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 7: 98.1% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (* eps -0.5) (* x (+ (* x (* x 0.16666666666666666)) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * (x * 0.16666666666666666)) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((x * (x * 0.16666666666666666d0)) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((x * (x * 0.16666666666666666)) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((x * (x * 0.16666666666666666)) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(x * Float64(x * 0.16666666666666666)) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((x * (x * 0.16666666666666666)) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 98.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.16666666666666666 \cdot x + 0.25 \cdot \varepsilon\right) - 1\right)\right)} \]

7. Taylor expanded in x around inf 98.2%

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

   1. *-commutative 98.2%

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

9. Simplified 98.2%

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

10. Final simplification 98.2%

    \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right) + -1\right)\right) \]
11. Add Preprocessing

Alternative 8: 97.7% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) + -1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (+ (* eps -0.5) (* x (+ (* 0.25 (* eps x)) -1.0)))))

C

double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * ((eps * (-0.5d0)) + (x * ((0.25d0 * (eps * x)) + (-1.0d0))))
end function

Java

public static double code(double x, double eps) {
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
}

Python

def code(x, eps):
	return eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)))

Julia

function code(x, eps)
	return Float64(eps * Float64(Float64(eps * -0.5) + Float64(x * Float64(Float64(0.25 * Float64(eps * x)) + -1.0))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * ((eps * -0.5) + (x * ((0.25 * (eps * x)) + -1.0)));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * -0.5), $MachinePrecision] + N[(x * N[(N[(0.25 * N[(eps * x), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 97.9%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(-0.5 \cdot \varepsilon + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) - 1\right)\right)} \]

7. Final simplification 97.9%

   \[\leadsto \varepsilon \cdot \left(\varepsilon \cdot -0.5 + x \cdot \left(0.25 \cdot \left(\varepsilon \cdot x\right) + -1\right)\right) \]
8. Add Preprocessing

Alternative 9: 97.7% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 97.9%

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

   1. +-commutative 97.9%

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

   2. mul-1-neg 97.9%

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

   3. unsub-neg 97.9%

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

   4. *-commutative 97.9%

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

8. Simplified 97.9%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot -0.5 - x\right)} \]
9. Add Preprocessing

Alternative 10: 78.6% accurate, 51.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.7%

   \[\cos \left(x + \varepsilon\right) - \cos x \]
2. Add Preprocessing

3. Taylor expanded in eps around 0 98.9%

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

   1. *-commutative 98.9%

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

   2. *-commutative 98.9%

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

   3. associate-*l* 98.9%

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

5. Simplified 98.9%

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

6. Taylor expanded in x around 0 97.9%

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

   1. +-commutative 97.9%

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

   2. mul-1-neg 97.9%

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

   3. unsub-neg 97.9%

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

   4. *-commutative 97.9%

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

8. Simplified 97.9%

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

9. Taylor expanded in eps around 0 77.2%

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

    1. mul-1-neg 77.2%

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

    2. distribute-rgt-neg-out 77.2%

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

11. Simplified 77.2%

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

12. Final simplification 77.2%

    \[\leadsto -\varepsilon \cdot x \]
13. Add Preprocessing

Developer target: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (* (* -2.0 (sin (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

  (- (cos (+ x eps)) (cos x)))