2sin (example 3.3)

Percentage Accurate: 41.7% → 99.4%

Time: 11.4s

Alternatives: 9

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.7% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.0%

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

   1. sin-sum 69.3%

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

   2. associate--l+ 69.2%

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

3. Applied egg-rr 69.2%

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

   1. +-commutative 69.2%

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

   2. sub-neg 69.2%

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

   3. associate-+l+ 99.5%

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

   4. *-commutative 99.5%

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

   5. neg-mul-1 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-out 99.5%

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

   8. +-commutative 99.5%

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

5. Simplified 99.5%

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

   1. fma-def 99.5%

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

   2. *-commutative 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.0%

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

   1. sin-sum 69.3%

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

   2. associate--l+ 69.2%

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

3. Applied egg-rr 69.2%

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

   1. +-commutative 69.2%

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

   2. sub-neg 69.2%

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

   3. associate-+l+ 99.5%

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

   4. *-commutative 99.5%

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

   5. neg-mul-1 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-out 99.5%

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

   8. +-commutative 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 3: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00044:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.002:\\ \;\;\;\;\varepsilon \cdot \cos x + \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.00044)
   (sin eps)
   (if (<= eps 0.002)
     (+ (* eps (cos x)) (* (* (sin x) (* eps eps)) -0.5))
     (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00044) {
		tmp = sin(eps);
	} else if (eps <= 0.002) {
		tmp = (eps * cos(x)) + ((sin(x) * (eps * 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) :: tmp
    if (eps <= (-0.00044d0)) then
        tmp = sin(eps)
    else if (eps <= 0.002d0) then
        tmp = (eps * cos(x)) + ((sin(x) * (eps * eps)) * (-0.5d0))
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.00044) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.002) {
		tmp = (eps * Math.cos(x)) + ((Math.sin(x) * (eps * eps)) * -0.5);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.00044:
		tmp = math.sin(eps)
	elif eps <= 0.002:
		tmp = (eps * math.cos(x)) + ((math.sin(x) * (eps * eps)) * -0.5)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.00044)
		tmp = sin(eps);
	elseif (eps <= 0.002)
		tmp = Float64(Float64(eps * cos(x)) + Float64(Float64(sin(x) * Float64(eps * eps)) * -0.5));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.00044)
		tmp = sin(eps);
	elseif (eps <= 0.002)
		tmp = (eps * cos(x)) + ((sin(x) * (eps * eps)) * -0.5);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.00044], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.002], N[(N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sin[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -4.40000000000000016e-4 or 2e-3 < eps

   1. Initial program 56.8%

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

   2. Taylor expanded in x around 0 58.5%

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

   if -4.40000000000000016e-4 < eps < 2e-3

   1. Initial program 26.5%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. fma-def 99.2%

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

      2. *-commutative 99.2%

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

      3. unpow2 99.2%

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

   4. Simplified 99.2%

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

      1. fma-udef 99.2%

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

      2. *-commutative 99.2%

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

   6. Applied egg-rr 99.2%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00044:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.002:\\ \;\;\;\;\varepsilon \cdot \cos x + \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) \cdot -0.5\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 4: 76.6% 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 44.0%

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

   1. sin-sum 69.3%

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

   2. associate--l+ 69.2%

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

3. Applied egg-rr 69.2%

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

   1. +-commutative 69.2%

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

   2. sub-neg 69.2%

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

   3. associate-+l+ 99.5%

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

   4. *-commutative 99.5%

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

   5. neg-mul-1 99.5%

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

   6. *-commutative 99.5%

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

   7. distribute-rgt-out 99.5%

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

   8. +-commutative 99.5%

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

5. Simplified 99.5%

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

   1. distribute-rgt-in 99.5%

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

   2. neg-mul-1 99.5%

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

   3. associate-+r+ 69.3%

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

   4. sin-sum 44.0%

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

   5. +-commutative 44.0%

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

   6. sub-neg 44.0%

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

   7. diff-sin 43.5%

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

   8. div-inv 43.5%

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

   9. +-commutative 43.5%

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

   10. associate--l+ 75.2%

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

   11. metadata-eval 75.2%

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

   12. div-inv 75.2%

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

   13. +-commutative 75.2%

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

   14. associate-+l+ 75.1%

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

   15. metadata-eval 75.1%

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

7. Applied egg-rr 75.1%

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

   1. associate-*r* 75.1%

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

   2. *-commutative 75.1%

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

   3. *-commutative 75.1%

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

   4. +-inverses 75.1%

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

   5. +-rgt-identity 75.1%

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

9. Simplified 75.1%

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

10. Final simplification 75.1%

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

Alternative 5: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.0%

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

   1. add-log-exp 35.2%

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

3. Applied egg-rr 35.2%

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

   1. add-log-exp 44.0%

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

   2. diff-sin 43.5%

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

   3. div-inv 43.5%

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

   4. +-commutative 43.5%

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

   5. associate--l+ 75.2%

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

   6. metadata-eval 75.2%

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

   7. div-inv 75.2%

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

   8. +-commutative 75.2%

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

   9. associate-+l+ 75.1%

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

   10. metadata-eval 75.1%

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

5. Applied egg-rr 75.1%

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

   1. *-commutative 75.1%

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

   2. +-inverses 75.1%

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

   3. *-commutative 75.1%

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

   4. associate-+r+ 75.2%

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

   5. +-commutative 75.2%

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

7. Simplified 75.2%

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

8. Final simplification 75.2%

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

Alternative 6: 76.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00122:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -1.45e-6)
   (sin eps)
   (if (<= eps 0.00122) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -1.45e-6) {
		tmp = sin(eps);
	} else if (eps <= 0.00122) {
		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 <= (-1.45d-6)) then
        tmp = sin(eps)
    else if (eps <= 0.00122d0) 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 <= -1.45e-6) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.00122) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -1.45e-6:
		tmp = math.sin(eps)
	elif eps <= 0.00122:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -1.45e-6)
		tmp = sin(eps);
	elseif (eps <= 0.00122)
		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 <= -1.45e-6)
		tmp = sin(eps);
	elseif (eps <= 0.00122)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -1.45e-6], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.00122], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.4500000000000001e-6 or 0.00121999999999999995 < eps

   1. Initial program 56.8%

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

   2. Taylor expanded in x around 0 58.5%

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

   if -1.4500000000000001e-6 < eps < 0.00121999999999999995

   1. Initial program 26.5%

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

   2. Taylor expanded in eps around 0 98.7%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -1.45 \cdot 10^{-6}:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.00122:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 7: 55.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.0%

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

2. Taylor expanded in x around 0 56.3%

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

3. Final simplification 56.3%

   \[\leadsto \sin \varepsilon \]

Alternative 8: 4.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 44.0%

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

   1. add-cube-cbrt 43.1%

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

   2. pow3 43.0%

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

3. Applied egg-rr 43.0%

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

4. Taylor expanded in eps around 0 4.0%

   \[\leadsto \color{blue}{{1}^{0.3333333333333333} \cdot \sin x - \sin x} \]
5. Step-by-step derivation

   1. pow-base-1 4.0%

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

   2. *-lft-identity 4.0%

      \[\leadsto \color{blue}{\sin x} - \sin x \]

   3. +-inverses 4.0%

      \[\leadsto \color{blue}{0} \]

6. Simplified 4.0%

   \[\leadsto \color{blue}{0} \]

7. Final simplification 4.0%

   \[\leadsto 0 \]

Alternative 9: 29.3% 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 44.0%

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

2. Taylor expanded in eps around 0 43.5%

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

   1. fma-def 43.5%

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

   2. *-commutative 43.5%

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

   3. unpow2 43.5%

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

4. Simplified 43.5%

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

5. Taylor expanded in x around 0 24.0%

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

6. Final simplification 24.0%

   \[\leadsto \varepsilon \]

Developer target: 76.7% 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 2023229 
(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)))