2sin (example 3.3)

Percentage Accurate: 62.9% → 99.9%

Time: 8.3s

Alternatives: 6

Speedup: 205.0×

Specification

\[\left(\left(0 \leq x \land x \leq 2 \cdot \pi\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

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: 62.9% 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: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  -0.5
  (* (sin x) (pow eps 2.0))
  (* (cos x) (+ eps (* -0.16666666666666666 (pow eps 3.0))))))

C

double code(double x, double eps) {
	return fma(-0.5, (sin(x) * pow(eps, 2.0)), (cos(x) * (eps + (-0.16666666666666666 * pow(eps, 3.0)))));
}

Julia

function code(x, eps)
	return fma(-0.5, Float64(sin(x) * (eps ^ 2.0)), Float64(cos(x) * Float64(eps + Float64(-0.16666666666666666 * (eps ^ 3.0)))))
end

Wolfram

code[x_, eps_] := N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[Power[eps, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[Cos[x], $MachinePrecision] * N[(eps + N[(-0.16666666666666666 * N[Power[eps, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

2. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} \]
3. Step-by-step derivation

   1. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\varepsilon}^{2} \cdot \sin x, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} \]

   2. *-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\sin x \cdot {\varepsilon}^{2}}, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right) \]

   3. +-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\varepsilon \cdot \cos x + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)}\right) \]

   4. associate-*r* 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \varepsilon \cdot \cos x + \color{blue}{\left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right) \]

   5. distribute-rgt-out 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) \]

4. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)} \]

5. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

   1. diff-sin 61.5%

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

   2. *-commutative 61.5%

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

   3. div-inv 61.5%

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

   4. associate--l+ 61.5%

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

   5. metadata-eval 61.5%

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

   6. div-inv 61.5%

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

   7. +-commutative 61.5%

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

   8. associate-+l+ 61.5%

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

   9. metadata-eval 61.5%

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

3. Applied egg-rr 61.5%

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

4. Taylor expanded in x around -inf 100.0%

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

5. Final simplification 100.0%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

2. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} \]
3. Step-by-step derivation

   1. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\varepsilon}^{2} \cdot \sin x, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} \]

   2. *-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\sin x \cdot {\varepsilon}^{2}}, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right) \]

   3. +-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\varepsilon \cdot \cos x + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)}\right) \]

   4. associate-*r* 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \varepsilon \cdot \cos x + \color{blue}{\left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right) \]

   5. distribute-rgt-out 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) \]

4. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right)} \]

5. Taylor expanded in x around 0 99.6%

   \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{{\varepsilon}^{2} \cdot x}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) \]

6. Taylor expanded in eps around 0 99.3%

   \[\leadsto \mathsf{fma}\left(-0.5, {\varepsilon}^{2} \cdot x, \color{blue}{\varepsilon \cdot \cos x}\right) \]
7. Step-by-step derivation

   1. fma-udef 99.3%

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

   2. associate-*r* 99.3%

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

8. Applied egg-rr 99.3%

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

9. Final simplification 99.3%

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

Alternative 4: 99.6% 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 61.5%

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

2. Taylor expanded in eps around 0 99.1%

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

3. Final simplification 99.1%

   \[\leadsto \varepsilon \cdot \cos x \]

Alternative 5: 98.8% 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 61.5%

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

2. Taylor expanded in x around 0 97.9%

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

3. Final simplification 97.9%

   \[\leadsto \sin \varepsilon \]

Alternative 6: 98.8% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 61.5%

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

2. Taylor expanded in eps around 0 99.1%

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

3. Taylor expanded in x around 0 97.9%

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

4. Final simplification 97.9%

   \[\leadsto \varepsilon \]

Developer target: 100.0% 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 2023310 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= 0.0 x) (<= x (* 2.0 PI))) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

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

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