2sin (example 3.3)

Percentage Accurate: 42.6% → 99.7%

Time: 13.6s

Alternatives: 11

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: 42.6% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+l+ 99.3%

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

   4. *-commutative 99.3%

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

   5. neg-mul-1 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-out 99.3%

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

   8. +-commutative 99.3%

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

5. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)} \]
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. metadata-eval 99.0%

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

   3. fma-neg 99.3%

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

   4. metadata-eval 99.3%

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

7. Applied egg-rr 99.3%

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

   1. fma-udef 99.0%

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

   2. -1-add-cos 99.3%

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

   3. unpow2 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

9. Simplified 99.3%

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

10. Taylor expanded in eps around inf 99.3%

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

    1. mul-1-neg 99.3%

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

    2. unpow2 99.3%

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

    3. +-commutative 99.3%

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

    4. associate-*r/ 99.2%

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

    5. distribute-rgt-neg-in 99.2%

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

    6. +-commutative 99.2%

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

    7. hang-0p-tan 99.6%

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

12. Simplified 99.6%

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

    1. distribute-rgt-neg-out 99.6%

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

    2. distribute-rgt-neg-out 99.6%

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

    3. div-inv 99.6%

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

    4. metadata-eval 99.6%

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

14. Applied egg-rr 99.6%

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

    1. associate-*r* 99.6%

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

    2. distribute-rgt-neg-in 99.6%

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

    3. *-commutative 99.6%

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

    4. *-commutative 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.05 \lor \neg \left(t_0 \leq 5 \cdot 10^{-97}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -0.05) (not (<= t_0 5e-97))) 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 <= -0.05) || !(t_0 <= 5e-97)) {
		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 <= (-0.05d0)) .or. (.not. (t_0 <= 5d-97))) 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 <= -0.05) || !(t_0 <= 5e-97)) {
		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 <= -0.05) or not (t_0 <= 5e-97):
		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 <= -0.05) || !(t_0 <= 5e-97))
		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 <= -0.05) || ~((t_0 <= 5e-97)))
		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, -0.05], N[Not[LessEqual[t$95$0, 5e-97]], $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)) < -0.050000000000000003 or 4.9999999999999995e-97 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 71.9%

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

   if -0.050000000000000003 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 4.9999999999999995e-97

   1. Initial program 19.7%

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

   2. Taylor expanded in eps around 0 78.5%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.05 \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 5 \cdot 10^{-97}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 3: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+l+ 99.3%

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

   4. *-commutative 99.3%

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

   5. neg-mul-1 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-out 99.3%

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

   8. +-commutative 99.3%

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

5. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)} \]
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. metadata-eval 99.0%

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

   3. fma-neg 99.3%

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

   4. metadata-eval 99.3%

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

7. Applied egg-rr 99.3%

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

   1. fma-udef 99.0%

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

   2. -1-add-cos 99.3%

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

   3. unpow2 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

9. Simplified 99.3%

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

10. Taylor expanded in eps around inf 99.3%

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

    1. mul-1-neg 99.3%

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

    2. unpow2 99.3%

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

    3. +-commutative 99.3%

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

    4. associate-*r/ 99.2%

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

    5. distribute-rgt-neg-in 99.2%

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

    6. +-commutative 99.2%

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

    7. hang-0p-tan 99.6%

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

12. Simplified 99.6%

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

    1. *-commutative 99.6%

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

    2. add-sqr-sqrt 51.7%

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

    3. sqrt-unprod 87.4%

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

    4. sqr-neg 87.4%

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

    5. sqrt-unprod 35.6%

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

    6. add-sqr-sqrt 75.1%

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

    7. cancel-sign-sub 75.1%

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

    8. distribute-rgt-neg-out 75.1%

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

    9. *-commutative 75.1%

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

14. Applied egg-rr 99.6%

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

    1. associate-*r* 99.6%

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

    2. *-commutative 99.6%

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

    3. associate-*l* 99.6%

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

    4. *-commutative 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 4: 99.6% accurate, 0.4× speedup

Math

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

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(sin(x) * Float64(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[Sin[x], $MachinePrecision] * N[(N[Sin[eps], $MachinePrecision] * N[Tan[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.1%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+l+ 99.3%

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

   4. *-commutative 99.3%

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

   5. neg-mul-1 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-out 99.3%

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

   8. +-commutative 99.3%

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

5. Simplified 99.3%

   \[\leadsto \color{blue}{\sin \varepsilon \cdot \cos x + \sin x \cdot \left(\cos \varepsilon + -1\right)} \]
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. metadata-eval 99.0%

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

   3. fma-neg 99.3%

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

   4. metadata-eval 99.3%

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

7. Applied egg-rr 99.3%

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

   1. fma-udef 99.0%

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

   2. -1-add-cos 99.3%

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

   3. unpow2 99.3%

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

   4. sub-neg 99.3%

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

   5. metadata-eval 99.3%

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

9. Simplified 99.3%

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

10. Taylor expanded in eps around inf 99.3%

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

    1. mul-1-neg 99.3%

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

    2. unpow2 99.3%

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

    3. +-commutative 99.3%

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

    4. associate-*r/ 99.2%

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

    5. distribute-rgt-neg-in 99.2%

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

    6. +-commutative 99.2%

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

    7. hang-0p-tan 99.6%

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

12. Simplified 99.6%

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

13. Final simplification 99.6%

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

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 41.1%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+l+ 99.3%

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

   4. *-commutative 99.3%

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

   5. neg-mul-1 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-out 99.3%

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

   8. +-commutative 99.3%

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

5. Simplified 99.3%

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

   1. fma-def 99.3%

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

   2. *-commutative 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 6: 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 41.1%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+l+ 99.3%

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

   4. *-commutative 99.3%

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

   5. neg-mul-1 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-out 99.3%

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

   8. +-commutative 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 7: 77.4% 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 41.1%

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

   1. diff-sin 40.2%

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

   2. div-inv 40.2%

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

   3. metadata-eval 40.2%

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

   4. div-inv 40.2%

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

   5. +-commutative 40.2%

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

   6. metadata-eval 40.2%

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

3. Applied egg-rr 40.2%

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

   1. *-commutative 40.2%

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

   2. +-commutative 40.2%

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

   3. associate--l+ 75.8%

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

   4. +-inverses 75.8%

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

   5. distribute-lft-in 75.8%

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

   6. metadata-eval 75.8%

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

   7. *-commutative 75.8%

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

   8. associate-+r+ 75.8%

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

   9. +-commutative 75.8%

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

5. Simplified 75.8%

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

6. Final simplification 75.8%

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

Alternative 8: 77.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000125:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.1e-5)
   (sin eps)
   (if (<= eps 0.000125) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.1e-5) {
		tmp = sin(eps);
	} else if (eps <= 0.000125) {
		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 <= (-1.1d-5)) then
        tmp = sin(eps)
    else if (eps <= 0.000125d0) 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 <= -1.1e-5) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.000125) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.1e-5:
		tmp = math.sin(eps)
	elif eps <= 0.000125:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.1e-5)
		tmp = sin(eps);
	elseif (eps <= 0.000125)
		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 <= -1.1e-5)
		tmp = sin(eps);
	elseif (eps <= 0.000125)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.1e-5], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.000125], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.1e-5 or 1.25e-4 < eps

   1. Initial program 54.2%

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

   2. Taylor expanded in x around 0 54.2%

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

   if -1.1e-5 < eps < 1.25e-4

   1. Initial program 27.2%

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

   2. Taylor expanded in eps around 0 98.6%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.1 \cdot 10^{-5}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.000125:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 9: 55.5% 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 41.1%

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

2. Taylor expanded in x around 0 54.6%

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

3. Final simplification 54.6%

   \[\leadsto \sin \varepsilon \]

Alternative 10: 4.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 41.1%

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

   1. add-cube-cbrt 40.3%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\sin \left(x + \varepsilon\right)} \cdot \sqrt[3]{\sin \left(x + \varepsilon\right)}\right) \cdot \sqrt[3]{\sin \left(x + \varepsilon\right)}} - \sin x \]

   2. pow3 40.3%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\sin \left(x + \varepsilon\right)}\right)}^{3}} - \sin x \]

3. Applied egg-rr 40.3%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\sin \left(x + \varepsilon\right)}\right)}^{3}} - \sin x \]

4. Taylor expanded in eps around 0 4.4%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \sin x - \sin x} \]
5. Step-by-step derivation

   1. pow-base-1 4.4%

      \[\leadsto \color{blue}{1} \cdot \sin x - \sin x \]

   2. *-lft-identity 4.4%

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

   3. +-inverses 4.4%

      \[\leadsto \color{blue}{0} \]

6. Simplified 4.4%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 4.4%

   \[\leadsto 0 \]

Alternative 11: 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 41.1%

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

2. Taylor expanded in eps around 0 49.4%

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

3. Taylor expanded in x around 0 28.3%

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

4. Final simplification 28.3%

   \[\leadsto \varepsilon \]

Developer target: 77.4% 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 2023242 
(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)))