2sin (example 3.3)

Percentage Accurate: 42.3% → 99.7%

Time: 14.9s

Alternatives: 8

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: 42.3% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

   1. sin-sum 59.6%

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

   2. associate--l+ 59.6%

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

4. Applied egg-rr 59.6%

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

   1. +-commutative 59.6%

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

   2. associate-+l- 99.1%

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

   3. *-commutative 99.1%

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

   4. *-rgt-identity 99.1%

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

   5. distribute-lft-out-- 99.2%

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

6. Simplified 99.2%

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

   1. flip-- 99.1%

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

   2. associate-*r/ 99.0%

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

   3. metadata-eval 99.0%

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

   4. 1-sub-cos 99.5%

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

   5. pow2 99.5%

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

8. Applied egg-rr 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-/l* 99.5%

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

10. Simplified 99.5%

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

11. Taylor expanded in eps around inf 99.5%

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

    1. associate-/l* 99.5%

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

    2. unpow2 99.5%

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

    3. *-rgt-identity 99.5%

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

    4. associate-*r/ 99.5%

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

    5. times-frac 99.5%

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

    6. hang-0p-tan 99.7%

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

    7. associate-/r/ 99.7%

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

    8. /-rgt-identity 99.7%

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

13. Simplified 99.7%

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

    1. sub-neg 99.7%

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

    2. *-commutative 99.7%

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

    3. distribute-rgt-neg-in 99.7%

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

    4. div-inv 99.7%

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

    5. metadata-eval 99.7%

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

15. Applied egg-rr 99.7%

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

    1. associate-*l* 99.7%

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

    2. distribute-lft-out 99.7%

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

    3. neg-mul-1 99.7%

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

    4. associate-*r* 99.7%

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

    5. *-commutative 99.7%

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

    6. neg-mul-1 99.7%

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

17. Simplified 99.7%

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

18. Final simplification 99.7%

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

Alternative 2: 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 39.8%

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

   1. sin-sum 59.6%

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

   2. associate--l+ 59.6%

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

4. Applied egg-rr 59.6%

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

   1. +-commutative 59.6%

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

   2. associate-+l- 99.1%

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

   3. *-commutative 99.1%

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

   4. *-rgt-identity 99.1%

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

   5. distribute-lft-out-- 99.2%

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

6. Simplified 99.2%

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

7. Final simplification 99.2%

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

Alternative 3: 77.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0013 \lor \neg \left(\varepsilon \leq 0.0007\right):\\ \;\;\;\;\sin \left(\varepsilon - x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0013) (not (<= eps 0.0007)))
   (- (sin (- eps x)) (sin x))
   (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0013) || !(eps <= 0.0007)) {
		tmp = sin((eps - x)) - sin(x);
	} 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) :: tmp
    if ((eps <= (-0.0013d0)) .or. (.not. (eps <= 0.0007d0))) then
        tmp = sin((eps - x)) - sin(x)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0013) || !(eps <= 0.0007)) {
		tmp = Math.sin((eps - x)) - Math.sin(x);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0013) or not (eps <= 0.0007):
		tmp = math.sin((eps - x)) - math.sin(x)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0013) || !(eps <= 0.0007))
		tmp = Float64(sin(Float64(eps - x)) - sin(x));
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0013) || ~((eps <= 0.0007)))
		tmp = sin((eps - x)) - sin(x);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0013], N[Not[LessEqual[eps, 0.0007]], $MachinePrecision]], N[(N[Sin[N[(eps - x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0012999999999999999 or 6.99999999999999993e-4 < eps

   1. Initial program 57.1%

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

      1. sin-sum 99.5%

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

      2. associate--l+ 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. +-commutative 99.5%

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

      2. associate-+l- 99.5%

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

      3. *-commutative 99.5%

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

      4. *-rgt-identity 99.5%

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

      5. distribute-lft-out-- 99.6%

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

   6. Simplified 99.6%

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

      1. sub-neg 99.6%

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

      2. distribute-lft-in 99.5%

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

      3. *-commutative 99.5%

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

      4. *-un-lft-identity 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. *-un-lft-identity 99.5%

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

      2. +-commutative 99.5%

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

      3. associate--r+ 99.5%

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

      4. *-commutative 99.5%

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

      5. add-sqr-sqrt 45.3%

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

      6. sqrt-unprod 83.5%

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

      7. sqr-neg 83.5%

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

      8. sqrt-unprod 38.2%

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

      9. add-sqr-sqrt 60.9%

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

      10. sin-diff 60.6%

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

   10. Applied egg-rr 60.6%

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

       1. *-lft-identity 60.6%

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

   12. Simplified 60.6%

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

   if -0.0012999999999999999 < eps < 6.99999999999999993e-4

   1. Initial program 26.3%

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

   3. Taylor expanded in eps around 0 98.3%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0013 \lor \neg \left(\varepsilon \leq 0.0007\right):\\ \;\;\;\;\sin \left(\varepsilon - x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -7.4e-6)
   (- (sin (+ eps x)) (sin x))
   (if (<= eps 1.8e-5) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -7.4e-6) {
		tmp = sin((eps + x)) - sin(x);
	} else if (eps <= 1.8e-5) {
		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 <= (-7.4d-6)) then
        tmp = sin((eps + x)) - sin(x)
    else if (eps <= 1.8d-5) 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 <= -7.4e-6) {
		tmp = Math.sin((eps + x)) - Math.sin(x);
	} else if (eps <= 1.8e-5) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -7.4e-6:
		tmp = math.sin((eps + x)) - math.sin(x)
	elif eps <= 1.8e-5:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -7.4e-6)
		tmp = Float64(sin(Float64(eps + x)) - sin(x));
	elseif (eps <= 1.8e-5)
		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 <= -7.4e-6)
		tmp = sin((eps + x)) - sin(x);
	elseif (eps <= 1.8e-5)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -7.4e-6], N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 1.8e-5], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -7.4000000000000003e-6

   1. Initial program 64.1%

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

   if -7.4000000000000003e-6 < eps < 1.80000000000000005e-5

   1. Initial program 26.1%

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

   3. Taylor expanded in eps around 0 98.8%

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

   if 1.80000000000000005e-5 < eps

   1. Initial program 47.8%

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

   3. Taylor expanded in x around 0 51.3%

      \[\leadsto \color{blue}{\sin \varepsilon} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -7.4 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.8%

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

   1. sin-sum 59.6%

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

   2. associate--l+ 59.6%

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

4. Applied egg-rr 59.6%

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

   1. +-commutative 59.6%

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

   2. associate-+l- 99.1%

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

   3. *-commutative 99.1%

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

   4. *-rgt-identity 99.1%

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

   5. distribute-lft-out-- 99.2%

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

6. Simplified 99.2%

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

   1. sub-neg 99.2%

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

   2. distribute-lft-in 99.1%

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

   3. *-commutative 99.1%

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

   4. *-un-lft-identity 99.1%

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

8. Applied egg-rr 99.1%

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

9. Taylor expanded in eps around 0 82.0%

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

    1. distribute-rgt1-in 82.0%

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

    2. metadata-eval 82.0%

       \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{0} \cdot \sin x \]

    3. mul0-lft 82.0%

       \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{0} \]

11. Simplified 82.0%

    \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{0} \]

12. Final simplification 82.0%

    \[\leadsto \sin \varepsilon \cdot \cos x \]
13. Add Preprocessing

Alternative 6: 76.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00125 \lor \neg \left(\varepsilon \leq 9.6 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00125) (not (<= eps 9.6e-5))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00125) || !(eps <= 9.6e-5)) {
		tmp = sin(eps);
	} 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) :: tmp
    if ((eps <= (-0.00125d0)) .or. (.not. (eps <= 9.6d-5))) then
        tmp = sin(eps)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00125) || !(eps <= 9.6e-5)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00125) or not (eps <= 9.6e-5):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00125) || !(eps <= 9.6e-5))
		tmp = sin(eps);
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00125) || ~((eps <= 9.6e-5)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00125], N[Not[LessEqual[eps, 9.6e-5]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00125000000000000003 or 9.6000000000000002e-5 < eps

   1. Initial program 57.1%

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

   3. Taylor expanded in x around 0 58.8%

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

   if -0.00125000000000000003 < eps < 9.6000000000000002e-5

   1. Initial program 26.3%

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

   3. Taylor expanded in eps around 0 98.3%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00125 \lor \neg \left(\varepsilon \leq 9.6 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.7% 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 39.8%

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

3. Taylor expanded in x around 0 55.5%

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

4. Final simplification 55.5%

   \[\leadsto \sin \varepsilon \]
5. Add Preprocessing

Alternative 8: 29.9% 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 39.8%

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

3. Taylor expanded in x around 0 55.5%

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

4. Taylor expanded in eps around 0 31.3%

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

5. Final simplification 31.3%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024013 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x)))

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