2sin (example 3.3)

Percentage Accurate: 62.3% → 99.8%

Time: 13.4s

Alternatives: 5

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: 62.3% 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: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + {\varepsilon}^{4} \cdot \left(\sin x \cdot 0.041666666666666664\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (+
  (fma
   -0.5
   (* (sin x) (pow eps 2.0))
   (* (cos x) (+ eps (* -0.16666666666666666 (pow eps 3.0)))))
  (* (pow eps 4.0) (* (sin x) 0.041666666666666664))))

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))))) + (pow(eps, 4.0) * (sin(x) * 0.041666666666666664));
}

Julia

function code(x, eps)
	return Float64(fma(-0.5, Float64(sin(x) * (eps ^ 2.0)), Float64(cos(x) * Float64(eps + Float64(-0.16666666666666666 * (eps ^ 3.0))))) + Float64((eps ^ 4.0) * Float64(sin(x) * 0.041666666666666664)))
end

Wolfram

code[x_, eps_] := N[(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] + N[(N[Power[eps, 4.0], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0 100.0%

   \[\leadsto \color{blue}{-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \left(0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) + \varepsilon \cdot \cos x\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 100.0%

      \[\leadsto -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \color{blue}{\left(\varepsilon \cdot \cos x + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right)}\right) \]

   2. associate-+r+ 100.0%

      \[\leadsto -0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \color{blue}{\left(\left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)\right)} \]

   3. associate-+r+ 100.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left({\varepsilon}^{2} \cdot \sin x\right) + \left(-0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right)} \]

   4. fma-def 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, {\varepsilon}^{2} \cdot \sin x, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right)} + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

   5. *-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \color{blue}{\sin x \cdot {\varepsilon}^{2}}, -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right) + \varepsilon \cdot \cos x\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

   6. +-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\varepsilon \cdot \cos x + -0.16666666666666666 \cdot \left({\varepsilon}^{3} \cdot \cos x\right)}\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

   7. associate-*r* 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \varepsilon \cdot \cos x + \color{blue}{\left(-0.16666666666666666 \cdot {\varepsilon}^{3}\right) \cdot \cos x}\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

   8. distribute-rgt-out 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \color{blue}{\cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)}\right) + 0.041666666666666664 \cdot \left({\varepsilon}^{4} \cdot \sin x\right) \]

   9. *-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + \color{blue}{\left({\varepsilon}^{4} \cdot \sin x\right) \cdot 0.041666666666666664} \]

   10. associate-*l* 100.0%

       \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + \color{blue}{{\varepsilon}^{4} \cdot \left(\sin x \cdot 0.041666666666666664\right)} \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + {\varepsilon}^{4} \cdot \left(\sin x \cdot 0.041666666666666664\right)} \]

6. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(-0.5, \sin x \cdot {\varepsilon}^{2}, \cos x \cdot \left(\varepsilon + -0.16666666666666666 \cdot {\varepsilon}^{3}\right)\right) + {\varepsilon}^{4} \cdot \left(\sin x \cdot 0.041666666666666664\right) \]
7. Add Preprocessing

Alternative 2: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + x)))))
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 + x)))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 63.8%

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

   1. diff-sin 63.8%

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

   2. *-commutative 63.8%

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

   3. div-inv 63.8%

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

   4. associate--l+ 63.8%

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

   5. metadata-eval 63.8%

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

   6. div-inv 63.8%

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

   7. +-commutative 63.8%

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

   8. associate-+l+ 63.8%

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

   9. metadata-eval 63.8%

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

4. Applied egg-rr 63.8%

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

5. Taylor expanded in x around 0 99.9%

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

6. Final simplification 99.9%

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

Alternative 3: 99.3% 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 63.8%

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

3. Taylor expanded in eps around 0 99.4%

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

4. Final simplification 99.4%

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

Alternative 4: 98.1% 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 63.8%

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

3. Taylor expanded in x around 0 98.1%

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

4. Final simplification 98.1%

   \[\leadsto \sin \varepsilon \]
5. Add Preprocessing

Alternative 5: 98.1% 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 63.8%

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

3. Taylor expanded in eps around 0 99.8%

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

4. Taylor expanded in x around 0 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in x around inf 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. unpow2 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

   7. distribute-rgt-out 99.3%

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

9. Simplified 99.3%

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

10. Taylor expanded in x around 0 98.1%

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

11. Final simplification 98.1%

    \[\leadsto \varepsilon \]
12. 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 2024031 
(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)))

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

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