2sin (example 3.3)

Percentage Accurate: 41.8% → 99.7%

Time: 13.0s

Alternatives: 10

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: 41.8% 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

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

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. sin-sum 63.7%

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

   2. associate--l+ 63.6%

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

3. Applied egg-rr 63.6%

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

   1. +-commutative 63.6%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in eps around inf 99.3%

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

   1. sub-neg 99.3%

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

   2. distribute-rgt-neg-out 99.3%

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

   3. *-commutative 99.3%

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

   4. +-commutative 99.3%

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

   5. fma-def 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-neg-in 99.4%

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

   9. remove-double-neg 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

8. Simplified 99.4%

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

   1. +-commutative 99.4%

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

   2. flip-+ 99.3%

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

   3. metadata-eval 99.3%

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

   4. 1-sub-cos 99.5%

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

   5. unpow2 99.5%

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

10. Applied egg-rr 99.5%

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

11. Taylor expanded in eps around inf 99.5%

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

    1. metadata-eval 99.5%

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

    2. times-frac 99.5%

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

    3. distribute-lft-in 99.5%

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

    4. metadata-eval 99.5%

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

    5. neg-mul-1 99.5%

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

    6. sub-neg 99.5%

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

    7. associate-*l/ 99.5%

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

    8. *-commutative 99.5%

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

    9. unpow2 99.5%

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

    10. associate-*r* 99.5%

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

    11. associate-*r/ 99.5%

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

    12. sub-neg 99.5%

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

    13. metadata-eval 99.5%

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

    14. neg-mul-1 99.5%

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

    15. distribute-lft-in 99.5%

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

    16. *-commutative 99.5%

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

    17. times-frac 99.5%

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

    18. hang-0p-tan 99.6%

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

    19. metadata-eval 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 2: 76.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.02 \lor \neg \left(t_0 \leq 5 \cdot 10^{-141}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ x eps)) (sin x))))
   (if (or (<= t_0 -0.02) (not (<= t_0 5e-141)))
     t_0
     (* (cos x) (* 2.0 (sin (* eps 0.5)))))))

C

double code(double x, double eps) {
	double t_0 = sin((x + eps)) - sin(x);
	double tmp;
	if ((t_0 <= -0.02) || !(t_0 <= 5e-141)) {
		tmp = t_0;
	} else {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	}
	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 <= (-0.02d0)) .or. (.not. (t_0 <= 5d-141))) then
        tmp = t_0
    else
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    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 <= -0.02) || !(t_0 <= 5e-141)) {
		tmp = t_0;
	} else {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((x + eps)) - math.sin(x)
	tmp = 0
	if (t_0 <= -0.02) or not (t_0 <= 5e-141):
		tmp = t_0
	else:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(x + eps)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -0.02) || !(t_0 <= 5e-141))
		tmp = t_0;
	else
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((x + eps)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -0.02) || ~((t_0 <= 5e-141)))
		tmp = t_0;
	else
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	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, -0.02], N[Not[LessEqual[t$95$0, 5e-141]], $MachinePrecision]], t$95$0, N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0200000000000000004 or 4.9999999999999999e-141 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 67.4%

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

   if -0.0200000000000000004 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 4.9999999999999999e-141

   1. Initial program 20.0%

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

      1. diff-sin 20.0%

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

      2. div-inv 20.0%

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

      3. associate--l+ 20.0%

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

      4. metadata-eval 20.0%

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

      5. div-inv 20.0%

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

      6. +-commutative 20.0%

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

      7. associate-+l+ 20.0%

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

      8. metadata-eval 20.0%

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

   3. Applied egg-rr 20.0%

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

      1. associate-*r* 20.0%

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

      2. *-commutative 20.0%

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

      3. *-commutative 20.0%

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

      4. +-commutative 20.0%

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

      5. count-2 20.0%

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

      6. fma-def 20.0%

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

      7. sub-neg 20.0%

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

      8. mul-1-neg 20.0%

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

      9. +-commutative 20.0%

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

      10. associate-+r+ 84.3%

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

      11. mul-1-neg 84.3%

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

      12. sub-neg 84.3%

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

      13. +-inverses 84.3%

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

      14. remove-double-neg 84.3%

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

      15. mul-1-neg 84.3%

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

      16. sub-neg 84.3%

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

      17. neg-sub0 84.3%

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

      18. mul-1-neg 84.3%

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

      19. remove-double-neg 84.3%

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

   5. Simplified 84.3%

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

   6. Taylor expanded in eps around 0 84.3%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(x + \varepsilon\right) - \sin x \leq -0.02 \lor \neg \left(\sin \left(x + \varepsilon\right) - \sin x \leq 5 \cdot 10^{-141}\right):\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. sin-sum 63.7%

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

   2. associate--l+ 63.6%

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

3. Applied egg-rr 63.6%

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

   1. +-commutative 63.6%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in eps around inf 99.3%

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

   1. sub-neg 99.3%

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

   2. distribute-rgt-neg-out 99.3%

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

   3. *-commutative 99.3%

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

   4. +-commutative 99.3%

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

   5. fma-def 99.4%

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

   6. sub-neg 99.4%

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

   7. +-commutative 99.4%

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

   8. distribute-neg-in 99.4%

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

   9. remove-double-neg 99.4%

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

   10. metadata-eval 99.4%

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

   11. *-commutative 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in x around inf 99.3%

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

    1. sub-neg 99.3%

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

    2. metadata-eval 99.3%

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

    3. *-commutative 99.3%

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

    4. fma-def 99.4%

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

11. Simplified 99.4%

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

12. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \cdot \cos x - \frac{\sin x}{\frac{1}{1 - \cos \varepsilon}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. sin-sum 63.7%

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

   2. associate--l+ 63.6%

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

3. Applied egg-rr 63.6%

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

   1. +-commutative 63.6%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

   1. flip-- 99.3%

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

   2. associate-*r/ 99.3%

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

   3. metadata-eval 99.3%

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

   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%

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

7. Applied egg-rr 99.5%

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

   1. associate-/l* 99.5%

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

9. Simplified 99.5%

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

    1. expm1-log1p-u 99.5%

       \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\sin x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1 + \cos \varepsilon}{{\sin \varepsilon}^{2}}\right)\right)}} \]

    2. expm1-udef 99.5%

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

    3. clear-num 99.5%

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

    4. unpow2 99.5%

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

    5. 1-sub-cos 99.2%

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

    6. metadata-eval 99.2%

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

    7. flip-- 99.3%

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

11. Applied egg-rr 99.3%

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

    1. expm1-def 99.3%

       \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\sin x}{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{1 - \cos \varepsilon}\right)\right)}} \]

    2. expm1-log1p 99.4%

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

13. Simplified 99.4%

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

14. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. sin-sum 63.7%

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

   2. associate--l+ 63.6%

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

3. Applied egg-rr 63.6%

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

   1. +-commutative 63.6%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 6: 76.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. add-cbrt-cube 32.6%

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

   2. pow3 32.7%

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

3. Applied egg-rr 32.7%

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

   1. rem-cbrt-cube 40.4%

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

   2. diff-sin 39.9%

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

   3. +-commutative 39.9%

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

   4. +-commutative 39.9%

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

5. Applied egg-rr 39.9%

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

   1. associate--l+ 76.7%

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

   2. +-commutative 76.7%

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

7. Simplified 76.7%

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

8. Final simplification 76.7%

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

Alternative 7: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.4%

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

   1. log1p-expm1-u 40.3%

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

3. Applied egg-rr 40.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(x + \varepsilon\right)\right)\right)} - \sin x \]
4. Step-by-step derivation

   1. log1p-expm1-u 40.4%

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

   2. diff-sin 39.9%

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

   3. +-commutative 39.9%

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

   4. +-commutative 39.9%

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

5. Applied egg-rr 39.9%

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

   1. associate-*r* 39.9%

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

   2. *-commutative 39.9%

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

   3. associate-+l+ 39.9%

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

   4. count-2 39.9%

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

   5. associate--l+ 76.6%

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

   6. +-inverses 76.6%

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

7. Simplified 76.6%

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

8. Final simplification 76.6%

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

Alternative 8: 76.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 1.12 \cdot 10^{-7}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -3e-5) (not (<= eps 1.12e-7))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -3e-5) || !(eps <= 1.12e-7)) {
		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 <= (-3d-5)) .or. (.not. (eps <= 1.12d-7))) 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 <= -3e-5) || !(eps <= 1.12e-7)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -3e-5) or not (eps <= 1.12e-7):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -3e-5) || !(eps <= 1.12e-7))
		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 <= -3e-5) || ~((eps <= 1.12e-7)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -3e-5], N[Not[LessEqual[eps, 1.12e-7]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -3.00000000000000008e-5 or 1.12e-7 < eps

   1. Initial program 52.8%

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

   2. Taylor expanded in x around 0 53.4%

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

   if -3.00000000000000008e-5 < eps < 1.12e-7

   1. Initial program 28.6%

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

   2. Taylor expanded in eps around 0 99.4%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -3 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 1.12 \cdot 10^{-7}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9: 55.1% 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 40.4%

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

2. Taylor expanded in x around 0 52.7%

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

3. Final simplification 52.7%

   \[\leadsto \sin \varepsilon \]

Alternative 10: 29.6% 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 40.4%

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

2. Taylor expanded in eps around 0 53.1%

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

3. Taylor expanded in x around 0 28.7%

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

4. Final simplification 28.7%

   \[\leadsto \varepsilon \]

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 2023334 
(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)))