2sin (example 3.3)

Percentage Accurate: 42.2% → 99.3%

Time: 13.5s

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: 42.2% 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: 99.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \sin \varepsilon, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin x \cdot \left(\cos \varepsilon + -1\right)\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma (cos x) (sin eps) (log1p (expm1 (* (sin x) (+ (cos eps) -1.0))))))

C

double code(double x, double eps) {
	return fma(cos(x), sin(eps), log1p(expm1((sin(x) * (cos(eps) + -1.0)))));
}

Julia

function code(x, eps)
	return fma(cos(x), sin(eps), log1p(expm1(Float64(sin(x) * Float64(cos(eps) + -1.0)))))
end

Wolfram

code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[Log[1 + N[(Exp[N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.8%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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+ 69.3%

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

3. Applied egg-rr 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around inf 99.4%

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

   1. fma-neg 99.4%

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

   2. distribute-rgt-neg-in 99.4%

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

   3. neg-sub0 99.4%

      \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \color{blue}{\left(0 - \left(1 - \cos \varepsilon\right)\right)}\right) \]

   4. associate--r- 99.4%

      \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \color{blue}{\left(\left(0 - 1\right) + \cos \varepsilon\right)}\right) \]

   5. metadata-eval 99.4%

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

   6. +-commutative 99.4%

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

8. Simplified 99.4%

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

   1. log1p-expm1-u 99.4%

      \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin x \cdot \left(\cos \varepsilon + -1\right)\right)\right)}\right) \]

   2. *-commutative 99.4%

      \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\left(\cos \varepsilon + -1\right) \cdot \sin x}\right)\right)\right) \]

10. Applied egg-rr 99.4%

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(\cos \varepsilon + -1\right) \cdot \sin x\right)\right)}\right) \]

11. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \mathsf{log1p}\left(\mathsf{expm1}\left(\sin x \cdot \left(\cos \varepsilon + -1\right)\right)\right)\right) \]

Alternative 2: 76.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.01:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \varepsilon\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ x eps)) (sin x))))
   (if (<= t_0 -0.01)
     t_0
     (if (<= t_0 0.0)
       (* (cos x) (* 2.0 (sin (* eps 0.5))))
       (fabs (sin eps))))))

C

double code(double x, double eps) {
	double t_0 = sin((x + eps)) - sin(x);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = fabs(sin(eps));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((x + eps)) - sin(x)
    if (t_0 <= (-0.01d0)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    else
        tmp = abs(sin(eps))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((x + eps)) - Math.sin(x);
	double tmp;
	if (t_0 <= -0.01) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.abs(Math.sin(eps));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((x + eps)) - math.sin(x)
	tmp = 0
	if t_0 <= -0.01:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	else:
		tmp = math.fabs(math.sin(eps))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(x + eps)) - sin(x))
	tmp = 0.0
	if (t_0 <= -0.01)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = abs(sin(eps));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((x + eps)) - sin(x);
	tmp = 0.0;
	if (t_0 <= -0.01)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	else
		tmp = abs(sin(eps));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.01], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Abs[N[Sin[eps], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.0100000000000000002

   1. Initial program 64.6%

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

   if -0.0100000000000000002 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 0.0

   1. Initial program 19.3%

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

      1. diff-sin 19.3%

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

      2. div-inv 19.3%

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

      3. associate--l+ 19.3%

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

      4. metadata-eval 19.3%

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

      5. div-inv 19.3%

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

      6. +-commutative 19.3%

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

      7. associate-+l+ 19.3%

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

      8. metadata-eval 19.3%

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

   3. Applied egg-rr 19.3%

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

      1. associate-*r* 19.3%

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

      2. *-commutative 19.3%

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

      3. *-commutative 19.3%

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

      4. +-commutative 19.3%

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

      5. count-2 19.3%

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

      6. fma-def 19.3%

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

      7. sub-neg 19.3%

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

      8. mul-1-neg 19.3%

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

      9. +-commutative 19.3%

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

      10. associate-+r+ 78.4%

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

      11. mul-1-neg 78.4%

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

      12. sub-neg 78.4%

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

      13. +-inverses 78.4%

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

      14. remove-double-neg 78.4%

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

      15. mul-1-neg 78.4%

          \[\leadsto \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\left(0 + \left(-\color{blue}{-1 \cdot \varepsilon}\right)\right) \cdot 0.5\right)\right) \]

      16. sub-neg 78.4%

          \[\leadsto \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\color{blue}{\left(0 - -1 \cdot \varepsilon\right)} \cdot 0.5\right)\right) \]

      17. neg-sub0 78.4%

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

      18. mul-1-neg 78.4%

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

      19. remove-double-neg 78.4%

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

   5. Simplified 78.4%

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

   6. Taylor expanded in eps around 0 78.4%

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

   if 0.0 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 71.6%

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

   2. Taylor expanded in x around 0 73.4%

      \[\leadsto \color{blue}{\sin \varepsilon} \]
   3. Step-by-step derivation

      1. add-sqr-sqrt 72.2%

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

      2. sqrt-unprod 67.6%

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

      3. pow2 67.6%

         \[\leadsto \sqrt{\color{blue}{{\sin \varepsilon}^{2}}} \]

   4. Applied egg-rr 67.6%

      \[\leadsto \color{blue}{\sqrt{{\sin \varepsilon}^{2}}} \]
   5. Step-by-step derivation

      1. unpow2 67.6%

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

      2. rem-sqrt-square 74.6%

         \[\leadsto \color{blue}{\left|\sin \varepsilon\right|} \]

   6. Simplified 74.6%

      \[\leadsto \color{blue}{\left|\sin \varepsilon\right|} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(x + \varepsilon\right) - \sin x \leq -0.01:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{elif}\;\sin \left(x + \varepsilon\right) - \sin x \leq 0:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|\sin \varepsilon\right|\\ \end{array} \]

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.8%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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+ 69.3%

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

3. Applied egg-rr 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around inf 99.4%

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

   1. fma-neg 99.4%

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

   2. distribute-rgt-neg-in 99.4%

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

   3. neg-sub0 99.4%

      \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \color{blue}{\left(0 - \left(1 - \cos \varepsilon\right)\right)}\right) \]

   4. associate--r- 99.4%

      \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \color{blue}{\left(\left(0 - 1\right) + \cos \varepsilon\right)}\right) \]

   5. metadata-eval 99.4%

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

   6. +-commutative 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.8%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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+ 69.3%

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

3. Applied egg-rr 69.3%

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

   1. +-commutative 69.3%

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

   2. associate-+l- 99.4%

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

   3. *-commutative 99.4%

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

   4. *-rgt-identity 99.4%

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

   5. distribute-lft-out-- 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 5: 76.1% 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 42.8%

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

   1. add-sqr-sqrt 20.5%

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

   2. sqrt-unprod 20.4%

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

   3. pow2 20.4%

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

3. Applied egg-rr 20.4%

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

   1. sqrt-pow1 42.8%

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

   2. metadata-eval 42.8%

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

   3. pow1 42.8%

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

   4. diff-sin 42.0%

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

   5. div-inv 42.0%

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

   6. +-commutative 42.0%

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

   7. associate--l+ 73.5%

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

   8. metadata-eval 73.5%

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

   9. div-inv 73.5%

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

   10. associate-+l+ 73.5%

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

   11. metadata-eval 73.5%

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

5. Applied egg-rr 73.5%

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

   1. associate-*r* 73.5%

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

   2. +-inverses 73.5%

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

   3. +-rgt-identity 73.5%

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

   4. *-commutative 73.5%

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

   5. *-commutative 73.5%

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

   6. +-commutative 73.5%

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

   7. associate-+l+ 73.5%

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

7. Simplified 73.5%

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

8. Final simplification 73.5%

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

Alternative 6: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0044:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.0044)
   (sin eps)
   (if (<= eps 1.8e-5) (* (cos x) eps) (- (sin (+ x eps)) (sin x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0044) {
		tmp = sin(eps);
	} else if (eps <= 1.8e-5) {
		tmp = cos(x) * eps;
	} else {
		tmp = sin((x + eps)) - sin(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.0044d0)) then
        tmp = sin(eps)
    else if (eps <= 1.8d-5) then
        tmp = cos(x) * eps
    else
        tmp = sin((x + eps)) - sin(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.0044) {
		tmp = Math.sin(eps);
	} else if (eps <= 1.8e-5) {
		tmp = Math.cos(x) * eps;
	} else {
		tmp = Math.sin((x + eps)) - Math.sin(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.0044:
		tmp = math.sin(eps)
	elif eps <= 1.8e-5:
		tmp = math.cos(x) * eps
	else:
		tmp = math.sin((x + eps)) - math.sin(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.0044)
		tmp = sin(eps);
	elseif (eps <= 1.8e-5)
		tmp = Float64(cos(x) * eps);
	else
		tmp = Float64(sin(Float64(x + eps)) - sin(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.0044)
		tmp = sin(eps);
	elseif (eps <= 1.8e-5)
		tmp = cos(x) * eps;
	else
		tmp = sin((x + eps)) - sin(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.0044], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 1.8e-5], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision], N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.00440000000000000027

   1. Initial program 45.9%

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

   2. Taylor expanded in x around 0 47.6%

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

   if -0.00440000000000000027 < eps < 1.80000000000000005e-5

   1. Initial program 33.9%

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

   2. Taylor expanded in eps around 0 98.8%

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

   if 1.80000000000000005e-5 < eps

   1. Initial program 55.4%

      \[\sin \left(x + \varepsilon\right) - \sin x \]
3. Recombined 3 regimes into one program.

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0044:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 1.8 \cdot 10^{-5}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \end{array} \]

Alternative 7: 76.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0044 \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0044) (not (<= eps 1.9e-5))) (sin eps) (* (cos x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0044) || !(eps <= 1.9e-5)) {
		tmp = sin(eps);
	} else {
		tmp = cos(x) * 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 <= (-0.0044d0)) .or. (.not. (eps <= 1.9d-5))) then
        tmp = sin(eps)
    else
        tmp = cos(x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0044) || !(eps <= 1.9e-5)) {
		tmp = Math.sin(eps);
	} else {
		tmp = Math.cos(x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0044) or not (eps <= 1.9e-5):
		tmp = math.sin(eps)
	else:
		tmp = math.cos(x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0044) || !(eps <= 1.9e-5))
		tmp = sin(eps);
	else
		tmp = Float64(cos(x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0044) || ~((eps <= 1.9e-5)))
		tmp = sin(eps);
	else
		tmp = cos(x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0044], N[Not[LessEqual[eps, 1.9e-5]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00440000000000000027 or 1.9000000000000001e-5 < eps

   1. Initial program 50.7%

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

   2. Taylor expanded in x around 0 51.5%

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

   if -0.00440000000000000027 < eps < 1.9000000000000001e-5

   1. Initial program 33.9%

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

   2. Taylor expanded in eps around 0 98.8%

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

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0044 \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]

Alternative 8: 54.9% 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 42.8%

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

2. Taylor expanded in x around 0 54.2%

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

3. Final simplification 54.2%

   \[\leadsto \sin \varepsilon \]

Alternative 9: 28.8% 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 42.8%

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

2. Taylor expanded in x around 0 54.2%

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

3. Taylor expanded in eps around 0 29.1%

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

4. Final simplification 29.1%

   \[\leadsto \varepsilon \]

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2023322 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x)))

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