2sin (example 3.3)

Percentage Accurate: 41.9% → 99.7%

Time: 14.7s

Alternatives: 11

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 99.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.0%

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

   1. sin-sum 65.9%

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

   2. associate--l+ 65.9%

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

3. Applied egg-rr 65.9%

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

   1. +-commutative 65.9%

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

   2. associate-+l- 99.7%

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

   3. *-commutative 99.7%

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

   4. *-rgt-identity 99.7%

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

   5. distribute-lft-out-- 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in eps around inf 99.7%

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

   1. fma-neg 99.7%

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

   2. distribute-rgt-neg-in 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. distribute-neg-in 99.7%

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

   6. remove-double-neg 99.7%

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

   7. metadata-eval 99.7%

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

8. Simplified 99.7%

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

   1. flip-+ 99.4%

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

   2. associate-*r/ 99.4%

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

   3. metadata-eval 99.4%

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

   4. sub-1-cos 99.4%

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

   5. unpow2 99.4%

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

   6. sub-neg 99.4%

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

   7. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

    1. *-commutative 99.4%

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

    2. associate-/l* 99.4%

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

    3. associate-/r/ 99.4%

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

    4. distribute-frac-neg 99.4%

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

    5. neg-mul-1 99.4%

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

    6. metadata-eval 99.4%

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

    7. times-frac 99.4%

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

    8. *-lft-identity 99.4%

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

    9. unpow2 99.4%

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

    10. times-frac 99.4%

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

    11. +-commutative 99.4%

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

    12. hang-0p-tan 99.8%

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

12. Simplified 99.8%

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

    1. fma-udef 99.7%

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

    2. +-commutative 99.7%

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

    3. associate-*l* 99.7%

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

    4. frac-2neg 99.7%

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

    5. metadata-eval 99.7%

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

    6. /-rgt-identity 99.7%

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

    7. div-inv 99.7%

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

    8. metadata-eval 99.7%

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

    9. *-commutative 99.7%

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

14. Applied egg-rr 99.7%

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

    1. +-commutative 99.7%

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

    2. fma-def 99.8%

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

    3. *-commutative 99.8%

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

16. Simplified 99.8%

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

17. Final simplification 99.8%

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

Alternative 2: 76.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-8} \lor \neg \left(t_0 \leq 10^{-262}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -2e-8) (not (<= t_0 1e-262)))
     t_0
     (+ (* -0.5 (* (sin x) (* eps eps))) (* eps (cos x))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -2e-8) || !(t_0 <= 1e-262)) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * (sin(x) * (eps * eps))) + (eps * cos(x));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-2d-8)) .or. (.not. (t_0 <= 1d-262))) then
        tmp = t_0
    else
        tmp = ((-0.5d0) * (sin(x) * (eps * eps))) + (eps * cos(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -2e-8) || !(t_0 <= 1e-262)) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * (Math.sin(x) * (eps * eps))) + (eps * Math.cos(x));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -2e-8) or not (t_0 <= 1e-262):
		tmp = t_0
	else:
		tmp = (-0.5 * (math.sin(x) * (eps * eps))) + (eps * math.cos(x))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -2e-8) || !(t_0 <= 1e-262))
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * Float64(eps * eps))) + Float64(eps * cos(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -2e-8) || ~((t_0 <= 1e-262)))
		tmp = t_0;
	else
		tmp = (-0.5 * (sin(x) * (eps * eps))) + (eps * cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-8], N[Not[LessEqual[t$95$0, 1e-262]], $MachinePrecision]], t$95$0, N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -2e-8 or 1.00000000000000001e-262 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 73.4%

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

   if -2e-8 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1.00000000000000001e-262

   1. Initial program 18.6%

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

   2. Taylor expanded in eps around 0 79.7%

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

      1. unpow2 79.7%

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

   4. Applied egg-rr 79.7%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -2 \cdot 10^{-8} \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-262}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot \cos x\\ \end{array} \]

Alternative 3: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-8} \lor \neg \left(t_0 \leq 10^{-262}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ eps x)) (sin x))))
   (if (or (<= t_0 -2e-8) (not (<= t_0 1e-262))) t_0 (* eps (cos x)))))

C

double code(double x, double eps) {
	double t_0 = sin((eps + x)) - sin(x);
	double tmp;
	if ((t_0 <= -2e-8) || !(t_0 <= 1e-262)) {
		tmp = t_0;
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((eps + x)) - sin(x)
    if ((t_0 <= (-2d-8)) .or. (.not. (t_0 <= 1d-262))) then
        tmp = t_0
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps + x)) - Math.sin(x);
	double tmp;
	if ((t_0 <= -2e-8) || !(t_0 <= 1e-262)) {
		tmp = t_0;
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((eps + x)) - math.sin(x)
	tmp = 0
	if (t_0 <= -2e-8) or not (t_0 <= 1e-262):
		tmp = t_0
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(eps + x)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -2e-8) || !(t_0 <= 1e-262))
		tmp = t_0;
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((eps + x)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -2e-8) || ~((t_0 <= 1e-262)))
		tmp = t_0;
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-8], N[Not[LessEqual[t$95$0, 1e-262]], $MachinePrecision]], t$95$0, N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -2e-8 or 1.00000000000000001e-262 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 73.4%

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

   if -2e-8 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1.00000000000000001e-262

   1. Initial program 18.6%

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

   2. Taylor expanded in eps around 0 79.6%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\varepsilon + x\right) - \sin x \leq -2 \cdot 10^{-8} \lor \neg \left(\sin \left(\varepsilon + x\right) - \sin x \leq 10^{-262}\right):\\ \;\;\;\;\sin \left(\varepsilon + x\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 43.0%

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

   1. sin-sum 65.9%

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

   2. associate--l+ 65.9%

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

3. Applied egg-rr 65.9%

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

   1. +-commutative 65.9%

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

   2. associate-+l- 99.7%

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

   3. *-commutative 99.7%

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

   4. *-rgt-identity 99.7%

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

   5. distribute-lft-out-- 99.7%

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

5. Simplified 99.7%

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

6. Taylor expanded in eps around inf 99.7%

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

   1. fma-neg 99.7%

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

   2. distribute-rgt-neg-in 99.7%

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

   3. sub-neg 99.7%

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

   4. +-commutative 99.7%

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

   5. distribute-neg-in 99.7%

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

   6. remove-double-neg 99.7%

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

   7. metadata-eval 99.7%

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

8. Simplified 99.7%

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

9. Taylor expanded in x around inf 99.7%

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

    1. +-commutative 99.7%

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

    2. sub-neg 99.7%

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

    3. metadata-eval 99.7%

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

    4. *-commutative 99.7%

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

    5. fma-def 99.7%

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

    6. *-commutative 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.0%

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

   1. sin-sum 65.9%

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

   2. associate--l+ 65.9%

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

3. Applied egg-rr 65.9%

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

   1. +-commutative 65.9%

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

   2. associate-+l- 99.7%

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

   3. *-commutative 99.7%

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

   4. *-rgt-identity 99.7%

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

   5. distribute-lft-out-- 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 6: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.0%

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

   1. add-sqr-sqrt 20.7%

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

   2. sqrt-unprod 19.8%

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

   3. pow2 19.8%

      \[\leadsto \sqrt{\color{blue}{{\left(\sin \left(x + \varepsilon\right) - \sin x\right)}^{2}}} \]

3. Applied egg-rr 19.8%

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

   1. sqrt-pow1 43.0%

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

   2. diff-sin 42.6%

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

   3. metadata-eval 42.6%

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

   4. pow1 42.6%

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

   5. div-inv 42.6%

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

   6. +-commutative 42.6%

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

   7. associate--l+ 77.0%

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

   8. metadata-eval 77.0%

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

   9. div-inv 77.0%

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

   10. +-commutative 77.0%

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

   11. associate-+l+ 77.0%

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

   12. metadata-eval 77.0%

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

5. Applied egg-rr 77.0%

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

   1. *-commutative 77.0%

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

   2. +-inverses 77.0%

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

   3. *-commutative 77.0%

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

   4. count-2 77.0%

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

7. Simplified 77.0%

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

8. Final simplification 77.0%

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

Alternative 7: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.0%

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

   1. expm1-log1p-u 42.9%

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

3. Applied egg-rr 42.9%

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

   1. expm1-log1p-u 43.0%

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

   2. diff-sin 42.6%

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

   3. +-commutative 42.6%

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

   4. +-commutative 42.6%

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

5. Applied egg-rr 42.6%

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

   1. *-commutative 42.6%

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

   2. *-commutative 42.6%

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

   3. associate-*l* 42.6%

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

   4. associate-+l+ 42.6%

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

   5. count-2 42.6%

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

   6. associate--l+ 77.0%

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

   7. +-inverses 77.0%

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

7. Simplified 77.0%

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

8. Final simplification 77.0%

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

Alternative 8: 76.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006 \lor \neg \left(\varepsilon \leq 1.45 \cdot 10^{-28}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.006) (not (<= eps 1.45e-28))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.006) || !(eps <= 1.45e-28)) {
		tmp = sin(eps);
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.006d0)) .or. (.not. (eps <= 1.45d-28))) then
        tmp = sin(eps)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.006) || !(eps <= 1.45e-28)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.006) or not (eps <= 1.45e-28):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.006) || !(eps <= 1.45e-28))
		tmp = sin(eps);
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.006) || ~((eps <= 1.45e-28)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.006], N[Not[LessEqual[eps, 1.45e-28]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.0060000000000000001 or 1.45000000000000006e-28 < eps

   1. Initial program 54.5%

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

   2. Taylor expanded in x around 0 54.4%

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

   if -0.0060000000000000001 < eps < 1.45000000000000006e-28

   1. Initial program 31.0%

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

   2. Taylor expanded in eps around 0 99.1%

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

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.006 \lor \neg \left(\varepsilon \leq 1.45 \cdot 10^{-28}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9: 31.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{expm1}\left(\varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (expm1 eps))

C

double code(double x, double eps) {
	return expm1(eps);
}

Java

public static double code(double x, double eps) {
	return Math.expm1(eps);
}

Python

def code(x, eps):
	return math.expm1(eps)

Julia

function code(x, eps)
	return expm1(eps)
end

Wolfram

code[x_, eps_] := N[(Exp[eps] - 1), $MachinePrecision]

Derivation

1. Initial program 43.0%

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

   1. add-sqr-sqrt 20.7%

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

   2. sqrt-unprod 19.8%

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

   3. pow2 19.8%

      \[\leadsto \sqrt{\color{blue}{{\left(\sin \left(x + \varepsilon\right) - \sin x\right)}^{2}}} \]

3. Applied egg-rr 19.8%

   \[\leadsto \color{blue}{\sqrt{{\left(\sin \left(x + \varepsilon\right) - \sin x\right)}^{2}}} \]

4. Taylor expanded in x around 0 22.7%

   \[\leadsto \sqrt{\color{blue}{{\sin \varepsilon}^{2}}} \]
5. Step-by-step derivation

   1. sqrt-pow1 59.4%

      \[\leadsto \color{blue}{{\sin \varepsilon}^{\left(\frac{2}{2}\right)}} \]

   2. metadata-eval 59.4%

      \[\leadsto {\sin \varepsilon}^{\color{blue}{1}} \]

   3. pow1 59.4%

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

   4. expm1-log1p-u 59.3%

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

6. Applied egg-rr 59.3%

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

7. Taylor expanded in eps around 0 36.1%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\varepsilon}\right) \]

8. Final simplification 36.1%

   \[\leadsto \mathsf{expm1}\left(\varepsilon\right) \]

Alternative 10: 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 43.0%

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

2. Taylor expanded in x around 0 59.4%

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

3. Final simplification 59.4%

   \[\leadsto \sin \varepsilon \]

Alternative 11: 29.7% 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 43.0%

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

   1. add-sqr-sqrt 20.7%

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

   2. sqrt-unprod 19.8%

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

   3. pow2 19.8%

      \[\leadsto \sqrt{\color{blue}{{\left(\sin \left(x + \varepsilon\right) - \sin x\right)}^{2}}} \]

3. Applied egg-rr 19.8%

   \[\leadsto \color{blue}{\sqrt{{\left(\sin \left(x + \varepsilon\right) - \sin x\right)}^{2}}} \]

4. Taylor expanded in x around 0 22.7%

   \[\leadsto \sqrt{\color{blue}{{\sin \varepsilon}^{2}}} \]

5. Taylor expanded in eps around 0 35.1%

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

6. Final simplification 35.1%

   \[\leadsto \varepsilon \]

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023333 
(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)))