2sin (example 3.3)

Percentage Accurate: 62.8% → 100.0%

Time: 13.1s

Alternatives: 7

Speedup: 205.0×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

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: 62.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \varepsilon\right)\\ \left(t\_0 \cdot \sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - t\_0 \cdot \sin x\right)}^{3}}\right) \cdot 2 \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 eps))))
   (*
    (* t_0 (cbrt (pow (- (* (cos (* 0.5 eps)) (cos x)) (* t_0 (sin x))) 3.0)))
    2.0)))

C

double code(double x, double eps) {
	double t_0 = sin((0.5 * eps));
	return (t_0 * cbrt(pow(((cos((0.5 * eps)) * cos(x)) - (t_0 * sin(x))), 3.0))) * 2.0;
}

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((0.5 * eps));
	return (t_0 * Math.cbrt(Math.pow(((Math.cos((0.5 * eps)) * Math.cos(x)) - (t_0 * Math.sin(x))), 3.0))) * 2.0;
}

Julia

function code(x, eps)
	t_0 = sin(Float64(0.5 * eps))
	return Float64(Float64(t_0 * cbrt((Float64(Float64(cos(Float64(0.5 * eps)) * cos(x)) - Float64(t_0 * sin(x))) ^ 3.0))) * 2.0)
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]}, N[(N[(t$95$0 * N[Power[N[Power[N[(N[(N[Cos[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]]

Derivation

1. Initial program 62.0%

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

   1. diff-sin 62.1%

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

   2. *-commutative 62.1%

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

   3. div-inv 62.1%

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

   4. associate--l+ 62.1%

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

   5. metadata-eval 62.1%

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

   6. div-inv 62.1%

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

   7. +-commutative 62.1%

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

   8. associate-+l+ 62.1%

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

   9. metadata-eval 62.1%

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

4. Applied egg-rr 62.1%

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

5. Taylor expanded in x around -inf 99.9%

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

   1. add-cbrt-cube 99.9%

      \[\leadsto \left(\color{blue}{\sqrt[3]{\left(\cos \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right) \cdot \cos \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)\right) \cdot \cos \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   2. pow3 99.9%

      \[\leadsto \left(\sqrt[3]{\color{blue}{{\cos \left(0.5 \cdot \left(\varepsilon - -2 \cdot x\right)\right)}^{3}}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   3. sub-neg 99.9%

      \[\leadsto \left(\sqrt[3]{{\cos \left(0.5 \cdot \color{blue}{\left(\varepsilon + \left(--2 \cdot x\right)\right)}\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   4. *-commutative 99.9%

      \[\leadsto \left(\sqrt[3]{{\cos \left(0.5 \cdot \left(\varepsilon + \left(-\color{blue}{x \cdot -2}\right)\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   5. distribute-rgt-neg-in 99.9%

      \[\leadsto \left(\sqrt[3]{{\cos \left(0.5 \cdot \left(\varepsilon + \color{blue}{x \cdot \left(--2\right)}\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   6. metadata-eval 99.9%

      \[\leadsto \left(\sqrt[3]{{\cos \left(0.5 \cdot \left(\varepsilon + x \cdot \color{blue}{2}\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

7. Applied egg-rr 99.9%

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

   1. distribute-lft-in 99.9%

      \[\leadsto \left(\sqrt[3]{{\cos \color{blue}{\left(0.5 \cdot \varepsilon + 0.5 \cdot \left(x \cdot 2\right)\right)}}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   2. cos-sum 100.0%

      \[\leadsto \left(\sqrt[3]{{\color{blue}{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \left(x \cdot 2\right)\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)}}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   3. *-commutative 100.0%

      \[\leadsto \left(\sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(0.5 \cdot \color{blue}{\left(2 \cdot x\right)}\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   4. associate-*r* 100.0%

      \[\leadsto \left(\sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \color{blue}{\left(\left(0.5 \cdot 2\right) \cdot x\right)} - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   5. metadata-eval 100.0%

      \[\leadsto \left(\sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \left(\color{blue}{1} \cdot x\right) - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   6. *-un-lft-identity 100.0%

      \[\leadsto \left(\sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos \color{blue}{x} - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \left(x \cdot 2\right)\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   7. *-commutative 100.0%

      \[\leadsto \left(\sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(0.5 \cdot \color{blue}{\left(2 \cdot x\right)}\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   8. associate-*r* 100.0%

      \[\leadsto \left(\sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \color{blue}{\left(\left(0.5 \cdot 2\right) \cdot x\right)}\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   9. metadata-eval 100.0%

      \[\leadsto \left(\sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \left(\color{blue}{1} \cdot x\right)\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

   10. *-un-lft-identity 100.0%

       \[\leadsto \left(\sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin \color{blue}{x}\right)}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

9. Applied egg-rr 100.0%

   \[\leadsto \left(\sqrt[3]{{\color{blue}{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)}}^{3}} \cdot \sin \left(0.5 \cdot \varepsilon\right)\right) \cdot 2 \]

10. Final simplification 100.0%

    \[\leadsto \left(\sin \left(0.5 \cdot \varepsilon\right) \cdot \sqrt[3]{{\left(\cos \left(0.5 \cdot \varepsilon\right) \cdot \cos x - \sin \left(0.5 \cdot \varepsilon\right) \cdot \sin x\right)}^{3}}\right) \cdot 2 \]
11. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

   1. diff-sin 62.1%

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

   2. *-commutative 62.1%

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

   3. div-inv 62.1%

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

   4. associate--l+ 62.1%

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

   5. metadata-eval 62.1%

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

   6. div-inv 62.1%

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

   7. +-commutative 62.1%

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

   8. associate-+l+ 62.1%

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

   9. metadata-eval 62.1%

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

4. Applied egg-rr 62.1%

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

5. Taylor expanded in x around -inf 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 99.3%

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

   1. associate-*r* 99.3%

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

5. Simplified 99.3%

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

   1. +-commutative 99.3%

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

   2. distribute-lft-in 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-*l* 99.3%

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

7. Applied egg-rr 99.3%

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

8. Final simplification 99.3%

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

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 99.3%

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

   1. associate-*r* 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 5: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (cos x)))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps * math.cos(x)

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = eps * cos(x);
end

Wolfram

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

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 98.9%

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

4. Final simplification 98.9%

   \[\leadsto \varepsilon \cdot \cos x \]
5. Add Preprocessing

Alternative 6: 98.2% accurate, 18.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \left(1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + x\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (+ 1.0 (* x (* -0.5 (+ eps x))))))

C

double code(double x, double eps) {
	return eps * (1.0 + (x * (-0.5 * (eps + x))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = eps * (1.0d0 + (x * ((-0.5d0) * (eps + x))))
end function

Java

public static double code(double x, double eps) {
	return eps * (1.0 + (x * (-0.5 * (eps + x))));
}

Python

def code(x, eps):
	return eps * (1.0 + (x * (-0.5 * (eps + x))))

Julia

function code(x, eps)
	return Float64(eps * Float64(1.0 + Float64(x * Float64(-0.5 * Float64(eps + x)))))
end

MATLAB

function tmp = code(x, eps)
	tmp = eps * (1.0 + (x * (-0.5 * (eps + x))));
end

Wolfram

code[x_, eps_] := N[(eps * N[(1.0 + N[(x * N[(-0.5 * N[(eps + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.0%

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

3. Taylor expanded in eps around 0 99.3%

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

   1. associate-*r* 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in x around 0 98.7%

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

8. Simplified 98.7%

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

9. Taylor expanded in x around 0 98.0%

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + x \cdot \left(-0.5 \cdot \varepsilon + -0.5 \cdot x\right)\right)} \]
10. Step-by-step derivation

    1. distribute-lft-out 98.0%

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

11. Simplified 98.0%

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

12. Final simplification 98.0%

    \[\leadsto \varepsilon \cdot \left(1 + x \cdot \left(-0.5 \cdot \left(\varepsilon + x\right)\right)\right) \]
13. Add Preprocessing

Alternative 7: 97.7% 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 62.0%

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

3. Taylor expanded in eps around 0 98.9%

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

4. Taylor expanded in x around 0 97.8%

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

5. Final simplification 97.8%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * N[Cos[N[(x + N[(eps / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(eps / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024080 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (* (* 2.0 (cos (+ x (/ eps 2.0)))) (sin (/ eps 2.0)))

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