2sin (example 3.3)

Percentage Accurate: 42.7% → 99.5%

Time: 15.9s

Alternatives: 9

Speedup: 2.0×

Specification

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: 42.7% 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: 99.5% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

   1. diff-sin 41.0%

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

   2. div-inv 41.0%

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

   3. metadata-eval 41.0%

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

   4. div-inv 41.0%

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

   5. +-commutative 41.0%

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

   6. metadata-eval 41.0%

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

3. Applied egg-rr 41.0%

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

   1. associate-*r* 41.0%

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

   2. *-commutative 41.0%

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

   3. associate-*l* 41.0%

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

   4. +-commutative 41.0%

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

   5. associate--l+ 75.2%

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

   6. +-inverses 75.2%

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

   7. *-commutative 75.2%

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

   8. associate-+r+ 75.2%

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

   9. +-commutative 75.2%

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

5. Simplified 75.2%

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

   1. distribute-lft-in 75.2%

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

   2. *-commutative 75.2%

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

   3. +-rgt-identity 75.2%

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

   4. cos-sum 99.5%

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

   5. +-rgt-identity 99.5%

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

   6. *-commutative 99.5%

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

   7. +-rgt-identity 99.5%

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

   8. *-commutative 99.5%

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

7. Applied egg-rr 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. count-2 99.5%

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

   4. *-commutative 99.5%

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

   5. count-2 99.5%

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

   6. count-2 99.5%

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

   7. *-commutative 99.5%

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

   8. count-2 99.5%

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

   9. distribute-lft-neg-in 99.5%

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

   10. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.5%

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

    1. associate-*r* 99.5%

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

    2. mul-1-neg 99.5%

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

    3. distribute-rgt-neg-in 99.5%

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

    4. *-commutative 99.5%

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

    5. fma-def 99.5%

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

    6. *-commutative 99.5%

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

    7. *-commutative 99.5%

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

12. Simplified 99.5%

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

    1. cancel-sign-sub-inv 99.5%

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

    2. *-commutative 99.5%

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

    3. fma-def 99.5%

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

14. Applied egg-rr 99.5%

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

15. Final simplification 99.5%

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

Alternative 2: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-7} \lor \neg \left(t_0 \leq 10^{-10}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -2e-7) (not (<= t_0 1e-10)))
     t_0
     (* (cos x) (* (sin (* eps 0.5)) 2.0)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -2e-7) || !(t_0 <= 1e-10)) {
		tmp = t_0;
	} else {
		tmp = cos(x) * (sin((eps * 0.5)) * 2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-2d-7)) .or. (.not. (t_0 <= 1d-10))) then
        tmp = t_0
    else
        tmp = cos(x) * (sin((eps * 0.5d0)) * 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -2e-7) || !(t_0 <= 1e-10)) {
		tmp = t_0;
	} else {
		tmp = Math.cos(x) * (Math.sin((eps * 0.5)) * 2.0);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -2e-7) or not (t_0 <= 1e-10):
		tmp = t_0
	else:
		tmp = math.cos(x) * (math.sin((eps * 0.5)) * 2.0)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -2e-7) || !(t_0 <= 1e-10))
		tmp = t_0;
	else
		tmp = Float64(cos(x) * Float64(sin(Float64(eps * 0.5)) * 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -2e-7) || ~((t_0 <= 1e-10)))
		tmp = t_0;
	else
		tmp = cos(x) * (sin((eps * 0.5)) * 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-7], N[Not[LessEqual[t$95$0, 1e-10]], $MachinePrecision]], t$95$0, N[(N[Cos[x], $MachinePrecision] * N[(N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -1.9999999999999999e-7 or 1.00000000000000004e-10 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 68.0%

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

   if -1.9999999999999999e-7 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1.00000000000000004e-10

   1. Initial program 24.6%

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

      1. diff-sin 24.6%

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

      2. div-inv 24.6%

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

      3. metadata-eval 24.6%

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

      4. div-inv 24.6%

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

      5. +-commutative 24.6%

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

      6. metadata-eval 24.6%

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

   3. Applied egg-rr 24.6%

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

      1. associate-*r* 24.6%

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

      2. *-commutative 24.6%

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

      3. associate-*l* 24.6%

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

      4. +-commutative 24.6%

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

      5. associate--l+ 80.6%

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

      6. +-inverses 80.6%

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

      7. *-commutative 80.6%

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

      8. associate-+r+ 80.6%

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

      9. +-commutative 80.6%

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

   5. Simplified 80.6%

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

      1. distribute-lft-in 80.6%

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

      2. *-commutative 80.6%

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

      3. +-rgt-identity 80.6%

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

      4. cos-sum 99.6%

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

      5. +-rgt-identity 99.6%

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

      6. *-commutative 99.6%

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

      7. +-rgt-identity 99.6%

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

      8. *-commutative 99.6%

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

   7. Applied egg-rr 99.6%

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

      1. cancel-sign-sub-inv 99.6%

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

      2. *-commutative 99.6%

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

      3. count-2 99.6%

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

      4. *-commutative 99.6%

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

      5. count-2 99.6%

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

      6. count-2 99.6%

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

      7. *-commutative 99.6%

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

      8. count-2 99.6%

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

      9. distribute-lft-neg-in 99.6%

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

      10. fma-def 99.6%

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

   9. Simplified 99.6%

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

   10. Taylor expanded in eps around inf 99.6%

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

       1. associate-*r* 99.6%

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

       2. mul-1-neg 99.6%

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

       3. distribute-rgt-neg-in 99.6%

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

       4. *-commutative 99.6%

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

       5. fma-def 99.6%

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

       6. *-commutative 99.6%

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

       7. *-commutative 99.6%

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

   12. Simplified 99.6%

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

   13. Taylor expanded in eps around 0 80.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -2 \cdot 10^{-7} \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-10}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot 2\right)\\ \end{array} \]

Alternative 3: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-7} \lor \neg \left(t_0 \leq 10^{-10}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -2e-7) (not (<= t_0 1e-10))) t_0 (* eps (cos x)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -2e-7) || !(t_0 <= 1e-10)) {
		tmp = t_0;
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-2d-7)) .or. (.not. (t_0 <= 1d-10))) then
        tmp = t_0
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -2e-7) || !(t_0 <= 1e-10)) {
		tmp = t_0;
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -2e-7) or not (t_0 <= 1e-10):
		tmp = t_0
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -2e-7) || !(t_0 <= 1e-10))
		tmp = t_0;
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -2e-7) || ~((t_0 <= 1e-10)))
		tmp = t_0;
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-7], N[Not[LessEqual[t$95$0, 1e-10]], $MachinePrecision]], t$95$0, N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -1.9999999999999999e-7 or 1.00000000000000004e-10 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 68.0%

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

   if -1.9999999999999999e-7 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1.00000000000000004e-10

   1. Initial program 24.6%

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

   2. Taylor expanded in eps around 0 78.9%

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -2 \cdot 10^{-7} \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-10}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 4: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

   1. diff-sin 41.0%

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

   2. div-inv 41.0%

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

   3. metadata-eval 41.0%

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

   4. div-inv 41.0%

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

   5. +-commutative 41.0%

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

   6. metadata-eval 41.0%

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

3. Applied egg-rr 41.0%

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

   1. associate-*r* 41.0%

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

   2. *-commutative 41.0%

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

   3. associate-*l* 41.0%

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

   4. +-commutative 41.0%

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

   5. associate--l+ 75.2%

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

   6. +-inverses 75.2%

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

   7. *-commutative 75.2%

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

   8. associate-+r+ 75.2%

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

   9. +-commutative 75.2%

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

5. Simplified 75.2%

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

   1. distribute-lft-in 75.2%

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

   2. *-commutative 75.2%

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

   3. +-rgt-identity 75.2%

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

   4. cos-sum 99.5%

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

   5. +-rgt-identity 99.5%

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

   6. *-commutative 99.5%

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

   7. +-rgt-identity 99.5%

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

   8. *-commutative 99.5%

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

7. Applied egg-rr 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. count-2 99.5%

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

   4. *-commutative 99.5%

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

   5. count-2 99.5%

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

   6. count-2 99.5%

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

   7. *-commutative 99.5%

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

   8. count-2 99.5%

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

   9. distribute-lft-neg-in 99.5%

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

   10. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.5%

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

    1. associate-*r* 99.5%

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

    2. mul-1-neg 99.5%

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

    3. distribute-rgt-neg-in 99.5%

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

    4. *-commutative 99.5%

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

    5. fma-def 99.5%

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

    6. *-commutative 99.5%

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

    7. *-commutative 99.5%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(-1 + \cos \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (cos x) (sin eps) (* (sin x) (+ -1.0 (cos eps)))))

C

double code(double x, double eps) {
	return fma(cos(x), sin(eps), (sin(x) * (-1.0 + cos(eps))));
}

Julia

function code(x, eps)
	return fma(cos(x), sin(eps), Float64(sin(x) * Float64(-1.0 + cos(eps))))
end

Wolfram

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

Derivation

1. Initial program 41.5%

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

   1. sin-sum 67.2%

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

   2. associate--l+ 67.1%

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

3. Applied egg-rr 67.1%

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

4. Taylor expanded in x around inf 67.2%

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

   1. associate--l+ 99.2%

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

   2. fma-def 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.2%

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

   6. sub-neg 99.2%

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

   7. metadata-eval 99.2%

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

   8. +-commutative 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 6: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

   1. diff-sin 41.0%

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

   2. div-inv 41.0%

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

   3. metadata-eval 41.0%

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

   4. div-inv 41.0%

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

   5. +-commutative 41.0%

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

   6. metadata-eval 41.0%

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

3. Applied egg-rr 41.0%

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

   1. associate-*r* 41.0%

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

   2. *-commutative 41.0%

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

   3. associate-*l* 41.0%

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

   4. +-commutative 41.0%

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

   5. associate--l+ 75.2%

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

   6. +-inverses 75.2%

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

   7. *-commutative 75.2%

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

   8. associate-+r+ 75.2%

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

   9. +-commutative 75.2%

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

5. Simplified 75.2%

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

   1. distribute-lft-in 75.2%

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

   2. *-commutative 75.2%

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

   3. +-rgt-identity 75.2%

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

   4. cos-sum 99.5%

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

   5. +-rgt-identity 99.5%

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

   6. *-commutative 99.5%

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

   7. +-rgt-identity 99.5%

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

   8. *-commutative 99.5%

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

7. Applied egg-rr 99.5%

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

   1. cancel-sign-sub-inv 99.5%

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

   2. *-commutative 99.5%

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

   3. count-2 99.5%

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

   4. *-commutative 99.5%

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

   5. count-2 99.5%

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

   6. count-2 99.5%

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

   7. *-commutative 99.5%

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

   8. count-2 99.5%

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

   9. distribute-lft-neg-in 99.5%

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

   10. fma-def 99.5%

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

9. Simplified 99.5%

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

10. Taylor expanded in eps around inf 99.5%

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

    1. associate-*r* 99.5%

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

    2. mul-1-neg 99.5%

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

    3. distribute-rgt-neg-in 99.5%

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

    4. *-commutative 99.5%

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

    5. fma-def 99.5%

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

    6. *-commutative 99.5%

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

    7. *-commutative 99.5%

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

12. Simplified 99.5%

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

    1. cos-sum 75.2%

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

14. Applied egg-rr 75.2%

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

15. Final simplification 75.2%

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

Alternative 7: 76.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.66:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.8e-5) (sin eps) (if (<= eps 0.66) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.8e-5) {
		tmp = sin(eps);
	} else if (eps <= 0.66) {
		tmp = eps * cos(x);
	} else {
		tmp = sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-2.8d-5)) then
        tmp = sin(eps)
    else if (eps <= 0.66d0) then
        tmp = eps * cos(x)
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -2.8e-5) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.66) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -2.8e-5:
		tmp = math.sin(eps)
	elif eps <= 0.66:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -2.8e-5)
		tmp = sin(eps);
	elseif (eps <= 0.66)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -2.8e-5)
		tmp = sin(eps);
	elseif (eps <= 0.66)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.8e-5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.66], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.79999999999999996e-5 or 0.660000000000000031 < eps

   1. Initial program 51.9%

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

   2. Taylor expanded in x around 0 51.7%

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

   if -2.79999999999999996e-5 < eps < 0.660000000000000031

   1. Initial program 30.7%

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

   2. Taylor expanded in eps around 0 98.1%

      \[\leadsto \color{blue}{\cos x \cdot \varepsilon} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.8 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.66:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 8: 56.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return sin(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

Derivation

1. Initial program 41.5%

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

2. Taylor expanded in x around 0 52.6%

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

3. Final simplification 52.6%

   \[\leadsto \sin \varepsilon \]

Alternative 9: 30.3% 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 41.5%

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

2. Taylor expanded in eps around 0 50.2%

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

   1. fma-def 50.2%

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

   2. *-commutative 50.2%

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

   3. unpow2 50.2%

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

4. Simplified 50.2%

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

5. Taylor expanded in x around 0 27.8%

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

6. Final simplification 27.8%

   \[\leadsto \varepsilon \]

Developer target: 77.0% accurate, 1.0× speedup

Math

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

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(2.0 * Float64(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[(2.0 * N[(N[Cos[N[(x + N[(eps / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023274 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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