2sin (example 3.3)

Percentage Accurate: 62.4% → 99.6%

Time: 12.4s

Alternatives: 4

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: 62.4% 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: 99.6% 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 60.9%

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

3. Taylor expanded in eps around 0 100.0%

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

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

3. Taylor expanded in eps around 0 100.0%

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

4. Final simplification 100.0%

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

Alternative 3: 99.1% 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 60.9%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 97.9% 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 60.9%

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

3. Taylor expanded in eps around 0 99.5%

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

4. Taylor expanded in x around 0 98.1%

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

5. Final simplification 98.1%

   \[\leadsto \varepsilon \]
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 2024026 
(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)))