2sin (example 3.3)

Percentage Accurate: 41.9% → 99.4%

Time: 14.6s

Alternatives: 12

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% 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: 99.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.0%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+r+ 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-def 99.4%

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

   6. neg-mul-1 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-out 99.4%

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

   9. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. add-log-exp_binary64 53.0%

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

7. Applied rewrite-once 53.0%

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

   1. rem-log-exp 99.4%

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

   2. fma-udef 99.4%

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

   3. *-commutative 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. flip-+ 99.2%

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

   7. div-inv 99.3%

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

   8. associate-*l* 99.3%

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

   9. fma-def 99.3%

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

9. Applied egg-rr 99.4%

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

    1. fma-udef 99.5%

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

    2. +-commutative 99.5%

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

    3. distribute-lft-neg-out 99.5%

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

    4. unsub-neg 99.5%

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

    5. *-commutative 99.5%

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

    6. associate-*r/ 99.5%

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

    7. *-commutative 99.5%

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

    8. neg-mul-1 99.5%

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

11. Simplified 99.5%

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

    1. associate-*r/ 99.5%

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

    2. unpow2 99.5%

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

    3. associate-*l* 99.4%

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

    4. associate-/l* 99.5%

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

13. Applied egg-rr 99.5%

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

14. Final simplification 99.5%

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

Alternative 2: 76.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (<= t_0 -0.01)
     t_0
     (if (<= t_0 0.0) (* (cos x) (* 2.0 (sin (* eps 0.5)))) (sin eps)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if (t_0 <= (-0.01d0)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if t_0 <= -0.01:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	else
		tmp = sin(eps);
	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[LessEqual[t$95$0, -0.01], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0100000000000000002

   1. Initial program 69.2%

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

   if -0.0100000000000000002 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 0.0

   1. Initial program 16.7%

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

      1. diff-sin 16.7%

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

      2. div-inv 16.7%

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

      3. +-commutative 16.7%

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

      4. associate--l+ 82.3%

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

      5. +-inverses 82.3%

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

      6. metadata-eval 82.3%

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

      7. div-inv 82.3%

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

      8. +-commutative 82.3%

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

      9. metadata-eval 82.3%

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

   3. Applied egg-rr 82.3%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \cos \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 82.3%

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

      2. *-commutative 82.3%

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

      3. *-commutative 82.3%

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

      4. associate-+r+ 82.3%

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

      5. +-commutative 82.3%

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

      6. +-rgt-identity 82.3%

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

   5. Simplified 82.3%

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

   6. Taylor expanded in eps around 0 82.3%

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

   if 0.0 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 85.3%

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

   2. Taylor expanded in x around 0 85.5%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -0.01:\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{elif}\;\sin \left(\varepsilon + x\right) - \sin x \leq 0:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \cdot \cos x + {\sin \varepsilon}^{2} \cdot \frac{\sin x}{-1 - \cos \varepsilon} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (* (sin eps) (cos x))
  (* (pow (sin eps) 2.0) (/ (sin x) (- -1.0 (cos eps))))))

C

double code(double x, double eps) {
	return (sin(eps) * cos(x)) + (pow(sin(eps), 2.0) * (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(eps) ** 2.0d0) * (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.pow(Math.sin(eps), 2.0) * (Math.sin(x) / (-1.0 - Math.cos(eps))));
}

Python

def code(x, eps):
	return (math.sin(eps) * math.cos(x)) + (math.pow(math.sin(eps), 2.0) * (math.sin(x) / (-1.0 - math.cos(eps))))

Julia

function code(x, eps)
	return Float64(Float64(sin(eps) * cos(x)) + Float64((sin(eps) ^ 2.0) * Float64(sin(x) / Float64(-1.0 - cos(eps)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (sin(eps) * cos(x)) + ((sin(eps) ^ 2.0) * (sin(x) / (-1.0 - cos(eps))));
end

Wolfram

code[x_, eps_] := N[(N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.0%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+r+ 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-def 99.4%

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

   6. neg-mul-1 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-out 99.4%

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

   9. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. add-log-exp_binary64 53.0%

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

7. Applied rewrite-once 53.0%

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

   1. rem-log-exp 99.4%

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

   2. fma-udef 99.4%

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

   3. *-commutative 99.4%

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

   4. +-commutative 99.4%

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

   5. *-commutative 99.4%

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

   6. flip-+ 99.2%

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

   7. div-inv 99.3%

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

   8. associate-*l* 99.3%

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

   9. fma-def 99.3%

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

9. Applied egg-rr 99.4%

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

    1. fma-udef 99.5%

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

    2. +-commutative 99.5%

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

    3. distribute-lft-neg-out 99.5%

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

    4. unsub-neg 99.5%

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

    5. *-commutative 99.5%

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

    6. associate-*r/ 99.5%

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

    7. *-commutative 99.5%

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

    8. neg-mul-1 99.5%

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

11. Simplified 99.5%

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

12. Final simplification 99.5%

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

Alternative 4: 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 45.0%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+r+ 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-def 99.4%

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

   6. neg-mul-1 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-out 99.4%

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

   9. +-commutative 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 45.0%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+r+ 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-def 99.4%

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

   6. neg-mul-1 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-out 99.4%

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

   9. +-commutative 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around inf 99.4%

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

   1. +-commutative 99.4%

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

   2. sub-neg 99.4%

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

   3. metadata-eval 99.4%

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

   4. fma-def 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 6: 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 45.0%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+r+ 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-def 99.4%

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

   6. neg-mul-1 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-out 99.4%

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

   9. +-commutative 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around inf 99.4%

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

7. Final simplification 99.4%

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

Alternative 7: 77.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot 0\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (sin eps) (cos x) (* (sin x) 0.0)))

C

double code(double x, double eps) {
	return fma(sin(eps), cos(x), (sin(x) * 0.0));
}

Julia

function code(x, eps)
	return fma(sin(eps), cos(x), Float64(sin(x) * 0.0))
end

Wolfram

code[x_, eps_] := N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 45.0%

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

   1. sin-sum 65.2%

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

   2. associate--l+ 65.2%

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

3. Applied egg-rr 65.2%

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

   1. +-commutative 65.2%

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

   2. sub-neg 65.2%

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

   3. associate-+r+ 99.4%

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

   4. *-commutative 99.4%

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

   5. fma-def 99.4%

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

   6. neg-mul-1 99.4%

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

   7. *-commutative 99.4%

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

   8. distribute-rgt-out 99.4%

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

   9. +-commutative 99.4%

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

5. Simplified 99.4%

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

7. Applied egg-rr 81.4%

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

8. Final simplification 81.4%

   \[\leadsto \mathsf{fma}\left(\sin \varepsilon, \cos x, \sin x \cdot 0\right) \]

Alternative 8: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double eps) {
	return cos((0.5 * (eps + (x + 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 = cos((0.5d0 * (eps + (x + x)))) * (2.0d0 * sin((eps * 0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 45.0%

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

   1. diff-sin 44.6%

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

   2. div-inv 44.6%

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

   3. +-commutative 44.6%

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

   4. associate--l+ 79.8%

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

   5. +-inverses 79.8%

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

   6. metadata-eval 79.8%

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

   7. div-inv 79.8%

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

   8. +-commutative 79.8%

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

   9. metadata-eval 79.8%

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

3. Applied egg-rr 79.8%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \cos \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
4. Step-by-step derivation

   1. associate-*r* 79.8%

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

   2. *-commutative 79.8%

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

   3. *-commutative 79.8%

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

   4. associate-+r+ 80.0%

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

   5. +-commutative 80.0%

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

   6. +-rgt-identity 80.0%

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

5. Simplified 80.0%

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

6. Final simplification 80.0%

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

Alternative 9: 76.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.052 \lor \neg \left(x \leq 0.00039\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(-1 + \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -0.052) (not (<= x 0.00039)))
   (* (cos x) (* 2.0 (sin (* eps 0.5))))
   (+ (sin eps) (* x (+ -1.0 (cos eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -0.052) || !(x <= 0.00039)) {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = sin(eps) + (x * (-1.0 + cos(eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-0.052d0)) .or. (.not. (x <= 0.00039d0))) then
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    else
        tmp = sin(eps) + (x * ((-1.0d0) + cos(eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -0.052) || !(x <= 0.00039)) {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps) + (x * (-1.0 + Math.cos(eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -0.052) or not (x <= 0.00039):
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps) + (x * (-1.0 + math.cos(eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -0.052) || !(x <= 0.00039))
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(sin(eps) + Float64(x * Float64(-1.0 + cos(eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -0.052) || ~((x <= 0.00039)))
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	else
		tmp = sin(eps) + (x * (-1.0 + cos(eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -0.052], N[Not[LessEqual[x, 0.00039]], $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[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0519999999999999976 or 3.89999999999999993e-4 < x

   1. Initial program 6.5%

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

      1. diff-sin 5.9%

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

      2. div-inv 5.9%

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

      3. +-commutative 5.9%

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

      4. associate--l+ 59.2%

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

      5. +-inverses 59.2%

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

      6. metadata-eval 59.2%

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

      7. div-inv 59.2%

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

      8. +-commutative 59.2%

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

      9. metadata-eval 59.2%

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

   3. Applied egg-rr 59.2%

      \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\left(\varepsilon + 0\right) \cdot 0.5\right) \cdot \cos \left(\left(x + \left(x + \varepsilon\right)\right) \cdot 0.5\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* 59.2%

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

      2. *-commutative 59.2%

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

      3. *-commutative 59.2%

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

      4. associate-+r+ 59.5%

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

      5. +-commutative 59.5%

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

      6. +-rgt-identity 59.5%

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

   5. Simplified 59.5%

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

   6. Taylor expanded in eps around 0 59.9%

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

   if -0.0519999999999999976 < x < 3.89999999999999993e-4

   1. Initial program 79.6%

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

   2. Taylor expanded in x around 0 98.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.052 \lor \neg \left(x \leq 0.00039\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(-1 + \cos \varepsilon\right)\\ \end{array} \]

Alternative 10: 75.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.8 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0106:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -8.8e-6)
   (sin eps)
   (if (<= eps 0.0106) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -8.8e-6) {
		tmp = sin(eps);
	} else if (eps <= 0.0106) {
		tmp = eps * cos(x);
	} else {
		tmp = sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-8.8d-6)) then
        tmp = sin(eps)
    else if (eps <= 0.0106d0) then
        tmp = eps * cos(x)
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -8.8e-6) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.0106) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -8.8e-6:
		tmp = math.sin(eps)
	elif eps <= 0.0106:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -8.8e-6)
		tmp = sin(eps);
	elseif (eps <= 0.0106)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -8.8e-6)
		tmp = sin(eps);
	elseif (eps <= 0.0106)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -8.8e-6], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.0106], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -8.8000000000000004e-6 or 0.0106 < eps

   1. Initial program 60.2%

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

   2. Taylor expanded in x around 0 61.2%

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

   if -8.8000000000000004e-6 < eps < 0.0106

   1. Initial program 29.8%

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

   2. Taylor expanded in eps around 0 99.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8.8 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0106:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 11: 55.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sin \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, eps)
	return sin(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

Derivation

1. Initial program 45.0%

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

2. Taylor expanded in x around 0 57.2%

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

3. Final simplification 57.2%

   \[\leadsto \sin \varepsilon \]

Alternative 12: 28.8% 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 45.0%

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

2. Taylor expanded in eps around 0 51.1%

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

3. Taylor expanded in x around 0 28.1%

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

4. Final simplification 28.1%

   \[\leadsto \varepsilon \]

Developer target: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023297 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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