2sin (example 3.3)

Percentage Accurate: 62.0% → 100.0%

Time: 15.7s

Alternatives: 14

Speedup: 205.0×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

   1. sin-sum 60.3%

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

   2. associate--l+ 60.3%

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

4. Applied egg-rr 60.3%

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

   1. +-commutative 60.3%

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

   2. sub-neg 60.3%

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

   3. associate-+l+ 99.5%

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

   4. *-commutative 99.5%

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

   5. neg-mul-1 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-out 99.6%

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

   8. +-commutative 99.6%

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

6. Simplified 99.6%

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

   1. flip-+ 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. sub-1-cos 100.0%

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

   5. pow2 100.0%

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

   6. sub-neg 100.0%

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

   7. metadata-eval 100.0%

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

8. Applied egg-rr 100.0%

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

   1. associate-*r/ 100.0%

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

   2. *-rgt-identity 100.0%

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

   3. distribute-frac-neg 100.0%

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

   4. metadata-eval 100.0%

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

   5. sub-neg 100.0%

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

   6. unpow2 100.0%

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

   7. associate-/l* 100.0%

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

   8. sub-neg 100.0%

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

   9. metadata-eval 100.0%

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

   10. +-commutative 100.0%

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

   11. hang-0p-tan 100.0%

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

10. Simplified 100.0%

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

    1. add-sqr-sqrt 48.4%

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

    2. sqrt-unprod 98.6%

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

    3. sqr-neg 98.6%

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

    4. sqrt-unprod 98.6%

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

    5. add-sqr-sqrt 98.6%

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

    6. pow1 98.6%

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

12. Applied egg-rr 100.0%

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

    1. unpow1 100.0%

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

14. Simplified 100.0%

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

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

   1. diff-sin 60.2%

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

   2. div-inv 60.2%

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

   3. associate--l+ 60.2%

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

   4. metadata-eval 60.2%

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

   5. div-inv 60.2%

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

   6. +-commutative 60.2%

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

   7. associate-+l+ 60.2%

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

   8. metadata-eval 60.2%

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

4. Applied egg-rr 60.2%

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

   1. associate-*r* 60.2%

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

   2. *-commutative 60.2%

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

   3. *-commutative 60.2%

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

   4. +-commutative 60.2%

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

   5. count-2 60.2%

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

   6. fma-define 60.2%

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

   7. associate-+r- 60.2%

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

   8. +-commutative 60.2%

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

   9. associate--l+ 99.9%

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

   10. +-inverses 99.9%

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

   11. +-commutative 99.9%

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

   12. *-lft-identity 99.9%

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

   13. metadata-eval 99.9%

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

   14. cancel-sign-sub-inv 99.9%

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

   15. neg-sub0 99.9%

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

   16. mul-1-neg 99.9%

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

   17. remove-double-neg 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in x around -inf 99.9%

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

8. Final simplification 99.9%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return eps * (Math.cos(x) + (eps * ((Math.sin(x) * -0.5) + (eps * -0.16666666666666666))));
}

Python

def code(x, eps):
	return eps * (math.cos(x) + (eps * ((math.sin(x) * -0.5) + (eps * -0.16666666666666666))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 99.5%

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

   1. *-commutative 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

   1. +-commutative 99.2%

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

   2. distribute-rgt-in 99.2%

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

   3. *-commutative 99.2%

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

   4. associate-*r* 99.2%

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

   5. *-commutative 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 6: 99.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\varepsilon \cdot -0.16666666666666666 + x \cdot \left(x \cdot \left(0.08333333333333333 \cdot \left(\varepsilon + x\right)\right) - 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (+
   (cos x)
   (*
    eps
    (+
     (* eps -0.16666666666666666)
     (* x (- (* x (* 0.08333333333333333 (+ eps x))) 0.5)))))))

C

double code(double x, double eps) {
	return eps * (cos(x) + (eps * ((eps * -0.16666666666666666) + (x * ((x * (0.08333333333333333 * (eps + x))) - 0.5)))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (cos(x) + (eps * ((eps * (-0.16666666666666666d0)) + (x * ((x * (0.08333333333333333d0 * (eps + x))) - 0.5d0)))))
end function

Java

public static double code(double x, double eps) {
	return eps * (Math.cos(x) + (eps * ((eps * -0.16666666666666666) + (x * ((x * (0.08333333333333333 * (eps + x))) - 0.5)))));
}

Python

def code(x, eps):
	return eps * (math.cos(x) + (eps * ((eps * -0.16666666666666666) + (x * ((x * (0.08333333333333333 * (eps + x))) - 0.5)))))

Julia

function code(x, eps)
	return Float64(eps * Float64(cos(x) + Float64(eps * Float64(Float64(eps * -0.16666666666666666) + Float64(x * Float64(Float64(x * Float64(0.08333333333333333 * Float64(eps + x))) - 0.5))))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (cos(x) + (eps * ((eps * -0.16666666666666666) + (x * ((x * (0.08333333333333333 * (eps + x))) - 0.5)))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(eps * N[(N[(eps * -0.16666666666666666), $MachinePrecision] + N[(x * N[(N[(x * N[(0.08333333333333333 * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 99.0%

   \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \color{blue}{\left(-0.16666666666666666 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.08333333333333333 \cdot \varepsilon + 0.08333333333333333 \cdot x\right) - 0.5\right)\right)}\right) \]
5. Step-by-step derivation

   1. distribute-lft-out 99.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(-0.16666666666666666 \cdot \varepsilon + x \cdot \left(x \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(\varepsilon + x\right)\right)} - 0.5\right)\right)\right) \]

   2. +-commutative 99.0%

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(-0.16666666666666666 \cdot \varepsilon + x \cdot \left(x \cdot \left(0.08333333333333333 \cdot \color{blue}{\left(x + \varepsilon\right)}\right) - 0.5\right)\right)\right) \]

6. Applied egg-rr 99.0%

   \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(-0.16666666666666666 \cdot \varepsilon + x \cdot \left(x \cdot \color{blue}{\left(0.08333333333333333 \cdot \left(x + \varepsilon\right)\right)} - 0.5\right)\right)\right) \]

7. Final simplification 99.0%

   \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\varepsilon \cdot -0.16666666666666666 + x \cdot \left(x \cdot \left(0.08333333333333333 \cdot \left(\varepsilon + x\right)\right) - 0.5\right)\right)\right) \]
8. Add Preprocessing

Alternative 7: 99.0% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * (math.cos(x) + (eps * ((eps * -0.16666666666666666) + (x * -0.5))))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.6%

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

4. Taylor expanded in x around 0 98.9%

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

5. Final simplification 98.9%

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

Alternative 8: 99.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 98.7%

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

   1. associate-*r* 98.7%

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

   2. *-commutative 98.7%

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

8. Simplified 98.7%

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

9. Final simplification 98.7%

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

Alternative 9: 99.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 98.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
4. Add Preprocessing

Alternative 10: 98.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 97.5%

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

   1. distribute-lft-out 97.5%

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

   2. unpow2 97.5%

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

   3. distribute-lft-out 97.5%

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

   4. +-commutative 97.5%

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

8. Simplified 97.5%

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

Alternative 11: 98.3% accurate, 18.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 97.5%

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

   1. distribute-lft-out 97.5%

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

8. Simplified 97.5%

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

Alternative 12: 98.2% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 97.5%

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

   1. distribute-lft-out 97.5%

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

   2. unpow2 97.5%

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

   3. distribute-lft-out 97.5%

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

   4. +-commutative 97.5%

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

8. Simplified 97.5%

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

9. Taylor expanded in eps around 0 97.5%

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

    1. *-commutative 97.5%

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

    2. associate-*l* 97.5%

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

11. Simplified 97.5%

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

Alternative 13: 97.7% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 96.9%

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

   1. associate-*r* 96.9%

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

   2. *-commutative 96.9%

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

8. Simplified 96.9%

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

9. Final simplification 96.9%

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

Alternative 14: 97.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in eps around 0 99.2%

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

   1. associate-*r* 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 96.8%

   \[\leadsto \color{blue}{\varepsilon} \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return (2.0 * cos((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 * cos((x + (eps / 2.0d0)))) * sin((eps / 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024086 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

  (- (sin (+ x eps)) (sin x)))