2sin (example 3.3)

Percentage Accurate: 42.2% → 99.7%

Time: 14.0s

Alternatives: 12

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: 42.2% 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: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (* (sin x) (sin eps)) (- (tan (/ eps 2.0))) (* (sin eps) (cos x))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. sin-sum 63.9%

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

   2. associate--l+ 63.9%

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

3. Applied egg-rr 63.9%

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

   1. +-commutative 63.9%

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

   2. sub-neg 63.9%

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

   3. associate-+l+ 99.1%

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

   4. *-commutative 99.1%

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

   5. neg-mul-1 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-out 99.1%

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

   8. +-commutative 99.1%

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

5. Simplified 99.1%

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

   1. flip-+ 99.0%

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

   2. frac-2neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. sub-1-cos 99.4%

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

   5. pow2 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

   1. remove-double-neg 99.4%

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

   2. neg-sub0 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--r+ 99.4%

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

   5. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. *-commutative 99.4%

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

    3. neg-mul-1 99.4%

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

    4. distribute-lft-neg-in 99.4%

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

    5. associate-/l* 99.4%

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

    6. associate-/r/ 99.4%

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

    7. distribute-neg-frac 99.4%

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

    8. neg-mul-1 99.4%

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

    9. metadata-eval 99.4%

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

    10. times-frac 99.4%

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

    11. distribute-lft-in 99.4%

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

    12. neg-mul-1 99.4%

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

    13. metadata-eval 99.4%

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

    14. +-commutative 99.4%

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

    15. sub-neg 99.4%

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

    16. *-lft-identity 99.4%

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

12. Simplified 99.5%

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

    1. +-commutative 99.5%

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

    2. associate-*r* 99.6%

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

    3. fma-def 99.6%

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

    4. distribute-frac-neg 99.6%

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

    5. tan-neg 99.6%

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

14. Applied egg-rr 99.6%

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

15. Final simplification 99.6%

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

Alternative 2: 76.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-7} \lor \neg \left(t_0 \leq 2 \cdot 10^{-299}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ x eps)) (sin x))))
   (if (or (<= t_0 -1e-7) (not (<= t_0 2e-299))) t_0 (* eps (cos x)))))

C

double code(double x, double eps) {
	double t_0 = sin((x + eps)) - sin(x);
	double tmp;
	if ((t_0 <= -1e-7) || !(t_0 <= 2e-299)) {
		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((x + eps)) - sin(x)
    if ((t_0 <= (-1d-7)) .or. (.not. (t_0 <= 2d-299))) 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((x + eps)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -1e-7) || !(t_0 <= 2e-299)) {
		tmp = t_0;
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((x + eps)) - math.sin(x)
	tmp = 0
	if (t_0 <= -1e-7) or not (t_0 <= 2e-299):
		tmp = t_0
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(x + eps)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -1e-7) || !(t_0 <= 2e-299))
		tmp = t_0;
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((x + eps)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -1e-7) || ~((t_0 <= 2e-299)))
		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[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-7], N[Not[LessEqual[t$95$0, 2e-299]], $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)) < -9.9999999999999995e-8 or 1.99999999999999998e-299 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 73.2%

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

   if -9.9999999999999995e-8 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1.99999999999999998e-299

   1. Initial program 16.7%

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

   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 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(x + \varepsilon\right) - \sin x \leq -1 \cdot 10^{-7} \lor \neg \left(\sin \left(x + \varepsilon\right) - \sin x \leq 2 \cdot 10^{-299}\right):\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (sin eps) (cos x) (* (* (sin x) (sin eps)) (- (tan (/ eps 2.0))))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. sin-sum 63.9%

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

   2. associate--l+ 63.9%

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

3. Applied egg-rr 63.9%

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

   1. +-commutative 63.9%

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

   2. sub-neg 63.9%

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

   3. associate-+l+ 99.1%

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

   4. *-commutative 99.1%

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

   5. neg-mul-1 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-out 99.1%

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

   8. +-commutative 99.1%

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

5. Simplified 99.1%

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

   1. flip-+ 99.0%

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

   2. frac-2neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. sub-1-cos 99.4%

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

   5. pow2 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

   1. remove-double-neg 99.4%

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

   2. neg-sub0 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--r+ 99.4%

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

   5. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. *-commutative 99.4%

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

    3. neg-mul-1 99.4%

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

    4. distribute-lft-neg-in 99.4%

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

    5. associate-/l* 99.4%

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

    6. associate-/r/ 99.4%

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

    7. distribute-neg-frac 99.4%

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

    8. neg-mul-1 99.4%

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

    9. metadata-eval 99.4%

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

    10. times-frac 99.4%

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

    11. distribute-lft-in 99.4%

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

    12. neg-mul-1 99.4%

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

    13. metadata-eval 99.4%

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

    14. +-commutative 99.4%

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

    15. sub-neg 99.4%

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

    16. *-lft-identity 99.4%

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

12. Simplified 99.5%

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

    1. fma-def 99.6%

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

    2. associate-*r* 99.6%

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

    3. distribute-frac-neg 99.6%

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

    4. tan-neg 99.6%

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

14. Applied egg-rr 99.6%

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

15. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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 / 2.0d0)))
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 / 2.0)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. sin-sum 63.9%

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

   2. associate--l+ 63.9%

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

3. Applied egg-rr 63.9%

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

   1. +-commutative 63.9%

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

   2. sub-neg 63.9%

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

   3. associate-+l+ 99.1%

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

   4. *-commutative 99.1%

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

   5. neg-mul-1 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-out 99.1%

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

   8. +-commutative 99.1%

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

5. Simplified 99.1%

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

   1. flip-+ 99.0%

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

   2. frac-2neg 99.0%

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

   3. metadata-eval 99.0%

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

   4. sub-1-cos 99.4%

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

   5. pow2 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

7. Applied egg-rr 99.4%

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

   1. remove-double-neg 99.4%

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

   2. neg-sub0 99.4%

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

   3. +-commutative 99.4%

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

   4. associate--r+ 99.4%

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

   5. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

    1. associate-*r/ 99.4%

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

    2. *-commutative 99.4%

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

    3. neg-mul-1 99.4%

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

    4. distribute-lft-neg-in 99.4%

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

    5. associate-/l* 99.4%

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

    6. associate-/r/ 99.4%

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

    7. distribute-neg-frac 99.4%

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

    8. neg-mul-1 99.4%

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

    9. metadata-eval 99.4%

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

    10. times-frac 99.4%

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

    11. distribute-lft-in 99.4%

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

    12. neg-mul-1 99.4%

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

    13. metadata-eval 99.4%

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

    14. +-commutative 99.4%

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

    15. sub-neg 99.4%

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

    16. *-lft-identity 99.4%

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

12. Simplified 99.5%

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

    1. expm1-log1p-u 93.3%

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

    2. expm1-udef 47.4%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\sin \varepsilon \cdot \tan \left(\frac{-\varepsilon}{2}\right)\right)\right)} - 1} \]

    3. +-commutative 47.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\sin x \cdot \left(\sin \varepsilon \cdot \tan \left(\frac{-\varepsilon}{2}\right)\right) + \sin \varepsilon \cdot \cos x}\right)} - 1 \]

    4. associate-*r* 47.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(\sin x \cdot \sin \varepsilon\right) \cdot \tan \left(\frac{-\varepsilon}{2}\right)} + \sin \varepsilon \cdot \cos x\right)} - 1 \]

    5. fma-def 47.4%

       \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, \tan \left(\frac{-\varepsilon}{2}\right), \sin \varepsilon \cdot \cos x\right)}\right)} - 1 \]

    6. distribute-frac-neg 47.4%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, \tan \color{blue}{\left(-\frac{\varepsilon}{2}\right)}, \sin \varepsilon \cdot \cos x\right)\right)} - 1 \]

    7. tan-neg 47.4%

       \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, \color{blue}{-\tan \left(\frac{\varepsilon}{2}\right)}, \sin \varepsilon \cdot \cos x\right)\right)} - 1 \]

14. Applied egg-rr 47.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, -\tan \left(\frac{\varepsilon}{2}\right), \sin \varepsilon \cdot \cos x\right)\right)} - 1} \]
15. Step-by-step derivation

    1. expm1-def 93.3%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\sin x \cdot \sin \varepsilon, -\tan \left(\frac{\varepsilon}{2}\right), \sin \varepsilon \cdot \cos x\right)\right)\right)} \]

    2. expm1-log1p 99.6%

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

    3. fma-udef 99.6%

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

    4. +-commutative 99.6%

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

    5. distribute-rgt-neg-out 99.6%

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

    6. unsub-neg 99.6%

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

    7. *-commutative 99.6%

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

    8. *-commutative 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 5: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. sin-sum 63.9%

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

   2. associate--l+ 63.9%

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

3. Applied egg-rr 63.9%

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

   1. +-commutative 63.9%

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

   2. sub-neg 63.9%

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

   3. associate-+l+ 99.1%

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

   4. *-commutative 99.1%

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

   5. neg-mul-1 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-out 99.1%

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

   8. +-commutative 99.1%

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

5. Simplified 99.1%

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

   1. fma-def 99.1%

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

   2. *-commutative 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 6: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. sin-sum 63.9%

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

   2. associate--l+ 63.9%

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

3. Applied egg-rr 63.9%

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

   1. +-commutative 63.9%

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

   2. sub-neg 63.9%

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

   3. associate-+l+ 99.1%

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

   4. *-commutative 99.1%

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

   5. neg-mul-1 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-out 99.1%

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

   8. +-commutative 99.1%

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

5. Simplified 99.1%

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

   1. +-commutative 99.1%

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

   2. *-commutative 99.1%

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

   3. *-commutative 99.1%

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

   4. fma-def 99.1%

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

   5. *-commutative 99.1%

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

7. Applied egg-rr 99.1%

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

8. Final simplification 99.1%

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

Alternative 7: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return (Math.sin(eps) * Math.cos(x)) + (Math.sin(x) * (Math.cos(eps) + -1.0));
}

Python

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) + (math.sin(x) * (math.cos(eps) + -1.0))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. sin-sum 63.9%

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

   2. associate--l+ 63.9%

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

3. Applied egg-rr 63.9%

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

   1. +-commutative 63.9%

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

   2. sub-neg 63.9%

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

   3. associate-+l+ 99.1%

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

   4. *-commutative 99.1%

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

   5. neg-mul-1 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-out 99.1%

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

   8. +-commutative 99.1%

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

5. Simplified 99.1%

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

6. Final simplification 99.1%

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

Alternative 8: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00074:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00074)
   (sin eps)
   (if (<= eps 1.7e-7)
     (* eps (+ (cos x) (* (sin x) (* eps -0.5))))
     (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00074) {
		tmp = sin(eps);
	} else if (eps <= 1.7e-7) {
		tmp = eps * (cos(x) + (sin(x) * (eps * -0.5)));
	} 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 <= (-0.00074d0)) then
        tmp = sin(eps)
    else if (eps <= 1.7d-7) then
        tmp = eps * (cos(x) + (sin(x) * (eps * (-0.5d0))))
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00074) {
		tmp = Math.sin(eps);
	} else if (eps <= 1.7e-7) {
		tmp = eps * (Math.cos(x) + (Math.sin(x) * (eps * -0.5)));
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00074:
		tmp = math.sin(eps)
	elif eps <= 1.7e-7:
		tmp = eps * (math.cos(x) + (math.sin(x) * (eps * -0.5)))
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00074)
		tmp = sin(eps);
	elseif (eps <= 1.7e-7)
		tmp = Float64(eps * Float64(cos(x) + Float64(sin(x) * Float64(eps * -0.5))));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00074)
		tmp = sin(eps);
	elseif (eps <= 1.7e-7)
		tmp = eps * (cos(x) + (sin(x) * (eps * -0.5)));
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00074], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 1.7e-7], N[(eps * N[(N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(eps * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.3999999999999999e-4 or 1.69999999999999987e-7 < eps

   1. Initial program 54.4%

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

   2. Taylor expanded in x around 0 54.7%

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

   if -7.3999999999999999e-4 < eps < 1.69999999999999987e-7

   1. Initial program 26.9%

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

      1. sin-sum 28.1%

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

      2. associate--l+ 28.1%

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

   3. Applied egg-rr 28.1%

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

      1. +-commutative 28.1%

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

      2. sub-neg 28.1%

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

      3. associate-+l+ 98.9%

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

      4. *-commutative 98.9%

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

      5. neg-mul-1 98.9%

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

      6. *-commutative 98.9%

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

      7. distribute-rgt-out 98.9%

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

      8. +-commutative 98.9%

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

   5. Simplified 98.9%

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

   6. Taylor expanded in eps around 0 99.4%

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

      1. fma-def 99.4%

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

      2. unpow2 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around inf 99.4%

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

       1. *-commutative 99.4%

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

       2. associate-*r* 99.4%

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

       3. unpow2 99.4%

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

       4. associate-*l* 99.4%

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

       5. *-commutative 99.4%

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

       6. associate-*r* 99.4%

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

       7. distribute-rgt-out 99.4%

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

       8. *-commutative 99.4%

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

   11. Simplified 99.4%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00074:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \left(\cos x + \sin x \cdot \left(\varepsilon \cdot -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.8%

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

   1. sin-sum 63.9%

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

   2. associate--l+ 63.9%

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

3. Applied egg-rr 63.9%

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

   1. +-commutative 63.9%

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

   2. sub-neg 63.9%

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

   3. associate-+l+ 99.1%

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

   4. *-commutative 99.1%

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

   5. neg-mul-1 99.1%

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

   6. *-commutative 99.1%

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

   7. distribute-rgt-out 99.1%

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

   8. +-commutative 99.1%

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

5. Simplified 99.1%

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

   1. *-commutative 99.1%

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

   2. distribute-lft-in 99.1%

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

   3. associate-+r+ 63.9%

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

   4. +-commutative 63.9%

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

   5. *-commutative 63.9%

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

   6. neg-mul-1 63.9%

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

   7. sin-sum 40.8%

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

   8. sub-neg 40.8%

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

   9. diff-sin 40.1%

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

   10. div-inv 40.1%

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

   11. +-commutative 40.1%

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

   12. associate--l+ 76.4%

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

   13. metadata-eval 76.4%

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

   14. div-inv 76.4%

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

   15. associate-+l+ 76.4%

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

   16. metadata-eval 76.4%

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

7. Applied egg-rr 76.4%

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

   1. +-inverses 76.4%

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

   2. +-rgt-identity 76.4%

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

   3. *-commutative 76.4%

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

   4. +-commutative 76.4%

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

   5. associate-+l+ 76.4%

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

9. Simplified 76.4%

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

10. Final simplification 76.4%

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

Alternative 10: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00075:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00075)
   (sin eps)
   (if (<= eps 1.7e-7) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00075) {
		tmp = sin(eps);
	} else if (eps <= 1.7e-7) {
		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 <= (-0.00075d0)) then
        tmp = sin(eps)
    else if (eps <= 1.7d-7) 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 <= -0.00075) {
		tmp = Math.sin(eps);
	} else if (eps <= 1.7e-7) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00075:
		tmp = math.sin(eps)
	elif eps <= 1.7e-7:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00075)
		tmp = sin(eps);
	elseif (eps <= 1.7e-7)
		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 <= -0.00075)
		tmp = sin(eps);
	elseif (eps <= 1.7e-7)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00075], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 1.7e-7], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.5000000000000002e-4 or 1.69999999999999987e-7 < eps

   1. Initial program 54.4%

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

   2. Taylor expanded in x around 0 54.7%

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

   if -7.5000000000000002e-4 < eps < 1.69999999999999987e-7

   1. Initial program 26.9%

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

   2. Taylor expanded in eps around 0 98.4%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00075:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.7 \cdot 10^{-7}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 11: 55.0% 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 40.8%

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

2. Taylor expanded in x around 0 50.6%

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

3. Final simplification 50.6%

   \[\leadsto \sin \varepsilon \]

Alternative 12: 29.1% 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 40.8%

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

2. Taylor expanded in eps around 0 51.1%

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

3. Taylor expanded in x around 0 25.4%

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

4. Final simplification 25.4%

   \[\leadsto \varepsilon \]

Developer target: 76.7% 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 2023230 
(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)))