2sin (example 3.3)

Percentage Accurate: 42.4% → 99.3%

Time: 11.8s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.4% 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: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

   1. sin-sum 66.8%

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

   2. associate--l+ 66.8%

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

4. Applied egg-rr 66.8%

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

   1. +-commutative 66.8%

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

   2. sub-neg 66.8%

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

   3. associate-+l+ 99.5%

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

   4. *-commutative 99.5%

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

   5. neg-mul-1 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-out 99.5%

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

   8. +-commutative 99.5%

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

6. Simplified 99.5%

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

   1. fma-def 99.5%

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

   2. *-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Final simplification 99.5%

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

Alternative 2: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{-6} \lor \neg \left(t_0 \leq 2 \cdot 10^{-104}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -1e-6) (not (<= t_0 2e-104)))
     t_0
     (* (cos x) (* 2.0 (sin (* eps 0.5)))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -1e-6) || !(t_0 <= 2e-104)) {
		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((eps + x)) - sin(x)
    if ((t_0 <= (-1d-6)) .or. (.not. (t_0 <= 2d-104))) 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((eps + x)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -1e-6) || !(t_0 <= 2e-104)) {
		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((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -1e-6) or not (t_0 <= 2e-104):
		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(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -1e-6) || !(t_0 <= 2e-104))
		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((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -1e-6) || ~((t_0 <= 2e-104)))
		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[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -1e-6], N[Not[LessEqual[t$95$0, 2e-104]], $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)) < -9.99999999999999955e-7 or 1.99999999999999985e-104 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 70.0%

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

   if -9.99999999999999955e-7 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1.99999999999999985e-104

   1. Initial program 21.8%

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

      1. diff-sin 21.8%

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

      2. div-inv 21.8%

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

      3. associate--l+ 21.8%

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

      4. metadata-eval 21.8%

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

      5. div-inv 21.8%

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

      6. +-commutative 21.8%

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

      7. associate-+l+ 21.8%

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

      8. metadata-eval 21.8%

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

   4. Applied egg-rr 21.8%

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

      1. associate-*r* 21.8%

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

      2. *-commutative 21.8%

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

      3. *-commutative 21.8%

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

      4. +-commutative 21.8%

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

      5. count-2 21.8%

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

      6. fma-def 21.8%

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

      7. sub-neg 21.8%

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

      8. mul-1-neg 21.8%

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

      9. +-commutative 21.8%

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

      10. associate-+r+ 80.5%

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

      11. mul-1-neg 80.5%

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

      12. sub-neg 80.5%

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

      13. +-inverses 80.5%

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

      14. remove-double-neg 80.5%

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

      15. mul-1-neg 80.5%

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

      16. sub-neg 80.5%

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

      17. neg-sub0 80.5%

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

      18. mul-1-neg 80.5%

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

      19. remove-double-neg 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in eps around 0 80.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -1 \cdot 10^{-6} \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 2 \cdot 10^{-104}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

   1. sin-sum 66.8%

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

   2. associate--l+ 66.8%

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

4. Applied egg-rr 66.8%

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

   1. +-commutative 66.8%

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

   2. sub-neg 66.8%

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

   3. associate-+l+ 99.5%

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

   4. *-commutative 99.5%

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

   5. neg-mul-1 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-out 99.5%

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

   8. +-commutative 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 76.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00065 \lor \neg \left(x \leq 0.0004\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -0.00065) (not (<= x 0.0004)))
   (* (cos x) (* 2.0 (sin (* eps 0.5))))
   (+ (sin eps) (* x (+ (cos eps) -1.0)))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -0.00065) || !(x <= 0.0004)) {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = sin(eps) + (x * (cos(eps) + -1.0));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-0.00065d0)) .or. (.not. (x <= 0.0004d0))) then
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    else
        tmp = sin(eps) + (x * (cos(eps) + (-1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -0.00065) || !(x <= 0.0004)) {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps) + (x * (Math.cos(eps) + -1.0));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -0.00065) or not (x <= 0.0004):
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps) + (x * (math.cos(eps) + -1.0))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -0.00065) || !(x <= 0.0004))
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(sin(eps) + Float64(x * Float64(cos(eps) + -1.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -0.00065) || ~((x <= 0.0004)))
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	else
		tmp = sin(eps) + (x * (cos(eps) + -1.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -0.00065], N[Not[LessEqual[x, 0.0004]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] + N[(x * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.4999999999999997e-4 or 4.00000000000000019e-4 < x

   1. Initial program 6.8%

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

      1. diff-sin 6.2%

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

      2. div-inv 6.2%

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

      3. associate--l+ 6.2%

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

      4. metadata-eval 6.2%

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

      5. div-inv 6.2%

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

      6. +-commutative 6.2%

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

      7. associate-+l+ 6.3%

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

      8. metadata-eval 6.3%

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

   4. Applied egg-rr 6.3%

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

      1. associate-*r* 6.3%

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

      2. *-commutative 6.3%

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

      3. *-commutative 6.3%

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

      4. +-commutative 6.3%

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

      5. count-2 6.3%

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

      6. fma-def 6.3%

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

      7. sub-neg 6.3%

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

      8. mul-1-neg 6.3%

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

      9. +-commutative 6.3%

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

      10. associate-+r+ 51.6%

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

      11. mul-1-neg 51.6%

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

      12. sub-neg 51.6%

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

      13. +-inverses 51.6%

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

      14. remove-double-neg 51.6%

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

      15. mul-1-neg 51.6%

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

      16. sub-neg 51.6%

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

      17. neg-sub0 51.6%

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

      18. mul-1-neg 51.6%

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

      19. remove-double-neg 51.6%

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

   6. Simplified 51.6%

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

   7. Taylor expanded in eps around 0 52.0%

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

   if -6.4999999999999997e-4 < x < 4.00000000000000019e-4

   1. Initial program 76.7%

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

   3. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\sin \varepsilon + x \cdot \left(\cos \varepsilon - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00065 \lor \neg \left(x \leq 0.0004\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(\cos \varepsilon + -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -40000000:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -40000000.0)
   (sin eps)
   (if (<= eps 1.15e-5) (* eps (cos x)) (- (sin (+ eps x)) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -40000000.0) {
		tmp = sin(eps);
	} else if (eps <= 1.15e-5) {
		tmp = eps * cos(x);
	} else {
		tmp = sin((eps + x)) - sin(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 <= (-40000000.0d0)) then
        tmp = sin(eps)
    else if (eps <= 1.15d-5) then
        tmp = eps * cos(x)
    else
        tmp = sin((eps + x)) - sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -40000000.0) {
		tmp = Math.sin(eps);
	} else if (eps <= 1.15e-5) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin((eps + x)) - Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -40000000.0:
		tmp = math.sin(eps)
	elif eps <= 1.15e-5:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin((eps + x)) - math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -40000000.0)
		tmp = sin(eps);
	elseif (eps <= 1.15e-5)
		tmp = Float64(eps * cos(x));
	else
		tmp = Float64(sin(Float64(eps + x)) - sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -40000000.0)
		tmp = sin(eps);
	elseif (eps <= 1.15e-5)
		tmp = eps * cos(x);
	else
		tmp = sin((eps + x)) - sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -40000000.0], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 1.15e-5], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -4e7

   1. Initial program 50.7%

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

   3. Taylor expanded in x around 0 51.3%

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

   if -4e7 < eps < 1.15e-5

   1. Initial program 29.2%

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

   3. Taylor expanded in eps around 0 97.4%

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

   if 1.15e-5 < eps

   1. Initial program 58.3%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -40000000:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.15 \cdot 10^{-5}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.3%

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

   1. sin-sum 66.8%

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

   2. associate--l+ 66.8%

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

4. Applied egg-rr 66.8%

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

   1. +-commutative 66.8%

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

   2. sub-neg 66.8%

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

   3. associate-+l+ 99.5%

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

   4. *-commutative 99.5%

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

   5. neg-mul-1 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-out 99.5%

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

   8. +-commutative 99.5%

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

6. Simplified 99.5%

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

   1. distribute-lft-in 99.5%

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

   2. associate-+r+ 66.8%

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

   3. *-commutative 66.8%

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

   4. sin-sum 42.3%

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

   5. +-commutative 42.3%

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

   6. *-commutative 42.3%

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

   7. neg-mul-1 42.3%

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

   8. sub-neg 42.3%

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

   9. diff-sin 41.8%

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

   10. +-commutative 41.8%

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

   11. +-commutative 41.8%

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

8. Applied egg-rr 41.8%

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

9. Taylor expanded in x around -inf 75.5%

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

10. Final simplification 75.5%

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

Alternative 7: 75.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -40000000 \lor \neg \left(\varepsilon \leq 0.00038\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -40000000.0) (not (<= eps 0.00038)))
   (sin eps)
   (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -40000000.0) || !(eps <= 0.00038)) {
		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 <= (-40000000.0d0)) .or. (.not. (eps <= 0.00038d0))) 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 <= -40000000.0) || !(eps <= 0.00038)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -40000000.0) or not (eps <= 0.00038):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -40000000.0) || !(eps <= 0.00038))
		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 <= -40000000.0) || ~((eps <= 0.00038)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -40000000.0], N[Not[LessEqual[eps, 0.00038]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -4e7 or 3.8000000000000002e-4 < eps

   1. Initial program 54.7%

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

   3. Taylor expanded in x around 0 54.9%

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

   if -4e7 < eps < 3.8000000000000002e-4

   1. Initial program 29.2%

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

   3. Taylor expanded in eps around 0 97.4%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -40000000 \lor \neg \left(\varepsilon \leq 0.00038\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.4% 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 42.3%

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

3. Taylor expanded in x around 0 55.0%

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

4. Final simplification 55.0%

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

Alternative 9: 29.4% 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 42.3%

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

3. Taylor expanded in eps around 0 48.8%

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

4. Taylor expanded in x around 0 28.3%

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

5. Final simplification 28.3%

   \[\leadsto \varepsilon \]
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 2024024 
(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)))