2sin (example 3.3)

Percentage Accurate: 41.7% → 99.4%

Time: 14.5s

Alternatives: 11

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: 41.7% 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: 99.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - \cos \varepsilon\\ \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot \left(\cos \varepsilon + -1\right)\right) + \mathsf{fma}\left(-\sin x, t_0, t_0 \cdot \sin x\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- 1.0 (cos eps))))
   (+
    (fma (cos x) (sin eps) (* (sin x) (+ (cos eps) -1.0)))
    (fma (- (sin x)) t_0 (* t_0 (sin x))))))

C

double code(double x, double eps) {
	double t_0 = 1.0 - cos(eps);
	return fma(cos(x), sin(eps), (sin(x) * (cos(eps) + -1.0))) + fma(-sin(x), t_0, (t_0 * sin(x)));
}

Julia

function code(x, eps)
	t_0 = Float64(1.0 - cos(eps))
	return Float64(fma(cos(x), sin(eps), Float64(sin(x) * Float64(cos(eps) + -1.0))) + fma(Float64(-sin(x)), t_0, Float64(t_0 * sin(x))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * N[(N[Cos[eps], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-N[Sin[x], $MachinePrecision]) * t$95$0 + N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 42.4%

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

   1. sin-sum 65.9%

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

   2. associate--l+ 65.9%

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

4. Applied egg-rr 65.9%

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

   1. +-commutative 65.9%

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

   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-- 99.3%

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

6. Simplified 99.3%

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

   1. prod-diff 99.4%

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

   2. *-commutative 99.4%

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

   3. fma-neg 99.4%

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

   4. *-commutative 99.4%

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

8. Applied egg-rr 99.4%

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

   1. *-commutative 99.4%

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

   2. fma-neg 99.4%

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

   3. distribute-rgt-neg-in 99.4%

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

   4. fma-udef 99.4%

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

   5. distribute-lft-neg-in 99.4%

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

   6. distribute-rgt-neg-in 99.4%

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

   7. *-commutative 99.4%

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

   8. fma-udef 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 75.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{if}\;t_0 \leq -0.05 \lor \neg \left(t_0 \leq 10^{-286}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (- (sin (+ x eps)) (sin x))))
   (if (or (<= t_0 -0.05) (not (<= t_0 1e-286)))
     t_0
     (* (cos x) (* 2.0 (sin (* eps 0.5)))))))

C

double code(double x, double eps) {
	double t_0 = sin((x + eps)) - sin(x);
	double tmp;
	if ((t_0 <= -0.05) || !(t_0 <= 1e-286)) {
		tmp = t_0;
	} else {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	}
	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 <= (-0.05d0)) .or. (.not. (t_0 <= 1d-286))) then
        tmp = t_0
    else
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    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 <= -0.05) || !(t_0 <= 1e-286)) {
		tmp = t_0;
	} else {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	}
	return tmp;
}

Python

def code(x, eps):
	t_0 = math.sin((x + eps)) - math.sin(x)
	tmp = 0
	if (t_0 <= -0.05) or not (t_0 <= 1e-286):
		tmp = t_0
	else:
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	return tmp

Julia

function code(x, eps)
	t_0 = Float64(sin(Float64(x + eps)) - sin(x))
	tmp = 0.0
	if ((t_0 <= -0.05) || !(t_0 <= 1e-286))
		tmp = t_0;
	else
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	t_0 = sin((x + eps)) - sin(x);
	tmp = 0.0;
	if ((t_0 <= -0.05) || ~((t_0 <= 1e-286)))
		tmp = t_0;
	else
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	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[Or[LessEqual[t$95$0, -0.05], N[Not[LessEqual[t$95$0, 1e-286]], $MachinePrecision]], t$95$0, N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < -0.050000000000000003 or 1.00000000000000005e-286 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x))

   1. Initial program 73.3%

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

   if -0.050000000000000003 < (-.f64 (sin.f64 (+.f64 x eps)) (sin.f64 x)) < 1.00000000000000005e-286

   1. Initial program 14.3%

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

      1. diff-sin 14.2%

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

      2. div-inv 14.2%

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

      3. associate--l+ 14.2%

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

      4. metadata-eval 14.2%

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

      5. div-inv 14.2%

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

      6. +-commutative 14.2%

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

      7. associate-+l+ 14.2%

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

      8. metadata-eval 14.2%

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

   4. Applied egg-rr 14.2%

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

      1. associate-*r* 14.2%

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

      2. *-commutative 14.2%

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

      3. *-commutative 14.2%

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

      4. +-commutative 14.2%

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

      5. count-2 14.2%

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

      6. fma-def 14.2%

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

      7. sub-neg 14.2%

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

      8. mul-1-neg 14.2%

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

      9. +-commutative 14.2%

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

      10. associate-+r+ 80.5%

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

      11. mul-1-neg 80.5%

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

      12. sub-neg 80.5%

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

      13. +-inverses 80.5%

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

      14. remove-double-neg 80.5%

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

      15. mul-1-neg 80.5%

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

      16. sub-neg 80.5%

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

      17. neg-sub0 80.5%

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

      18. mul-1-neg 80.5%

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

      19. remove-double-neg 80.5%

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

   6. Simplified 80.5%

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

   7. Taylor expanded in eps around 0 80.8%

      \[\leadsto \color{blue}{\cos x} \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(x + \varepsilon\right) - \sin x \leq -0.05 \lor \neg \left(\sin \left(x + \varepsilon\right) - \sin x \leq 10^{-286}\right):\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

   1. sin-sum 65.9%

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

   2. associate--l+ 65.9%

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

4. Applied egg-rr 65.9%

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

   1. +-commutative 65.9%

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

   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-- 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in eps around inf 99.3%

   \[\leadsto \color{blue}{\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)} \]
8. 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. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. remove-double-neg 99.4%

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

   7. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

   1. sin-sum 65.9%

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

   2. associate--l+ 65.9%

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

4. Applied egg-rr 65.9%

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

   1. +-commutative 65.9%

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

   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-- 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in eps around inf 99.3%

   \[\leadsto \color{blue}{\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)} \]
8. 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. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. remove-double-neg 99.4%

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

   7. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around inf 99.3%

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

    1. +-commutative 99.3%

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

    2. sub-neg 99.3%

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

    3. metadata-eval 99.3%

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

    4. *-commutative 99.3%

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

    5. fma-def 99.4%

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

    6. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

   1. sin-sum 65.9%

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

   2. associate--l+ 65.9%

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

4. Applied egg-rr 65.9%

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

   1. +-commutative 65.9%

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

   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-- 99.3%

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

6. Simplified 99.3%

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

7. Final simplification 99.3%

   \[\leadsto \cos x \cdot \sin \varepsilon - \left(1 - \cos \varepsilon\right) \cdot \sin x \]
8. Add Preprocessing

Alternative 6: 76.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot 0\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma (cos x) (sin eps) (* (sin x) 0.0)))

C

double code(double x, double eps) {
	return fma(cos(x), sin(eps), (sin(x) * 0.0));
}

Julia

function code(x, eps)
	return fma(cos(x), sin(eps), Float64(sin(x) * 0.0))
end

Wolfram

code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[(N[Sin[x], $MachinePrecision] * 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.4%

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

   1. sin-sum 65.9%

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

   2. associate--l+ 65.9%

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

4. Applied egg-rr 65.9%

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

   1. +-commutative 65.9%

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

   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-- 99.3%

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

6. Simplified 99.3%

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

7. Taylor expanded in eps around inf 99.3%

   \[\leadsto \color{blue}{\cos x \cdot \sin \varepsilon - \sin x \cdot \left(1 - \cos \varepsilon\right)} \]
8. 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. sub-neg 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. remove-double-neg 99.4%

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

   7. metadata-eval 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in eps around 0 77.6%

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

11. Final simplification 77.6%

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \sin x \cdot 0\right) \]
12. Add Preprocessing

Alternative 7: 75.6% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 42.4%

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

   1. add-sqr-sqrt 27.3%

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

   2. sqrt-unprod 27.4%

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

   3. pow2 27.4%

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

4. Applied egg-rr 27.4%

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

   1. sqrt-pow1 42.4%

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

   2. metadata-eval 42.4%

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

   3. pow1 42.4%

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

   4. diff-sin 42.1%

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

   5. +-commutative 42.1%

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

   6. +-commutative 42.1%

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

6. Applied egg-rr 42.1%

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

   1. associate--l+ 76.7%

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

   2. associate-+r+ 76.7%

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

8. Simplified 76.7%

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

9. Final simplification 76.7%

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

Alternative 8: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.00112:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -2.1e-5)
   (- (sin (+ x eps)) (sin x))
   (if (<= eps 0.00112) (* (cos x) eps) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -2.1e-5) {
		tmp = sin((x + eps)) - sin(x);
	} else if (eps <= 0.00112) {
		tmp = cos(x) * eps;
	} else {
		tmp = sin(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 <= (-2.1d-5)) then
        tmp = sin((x + eps)) - sin(x)
    else if (eps <= 0.00112d0) then
        tmp = cos(x) * eps
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -2.1e-5) {
		tmp = Math.sin((x + eps)) - Math.sin(x);
	} else if (eps <= 0.00112) {
		tmp = Math.cos(x) * eps;
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -2.1e-5:
		tmp = math.sin((x + eps)) - math.sin(x)
	elif eps <= 0.00112:
		tmp = math.cos(x) * eps
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -2.1e-5)
		tmp = Float64(sin(Float64(x + eps)) - sin(x));
	elseif (eps <= 0.00112)
		tmp = Float64(cos(x) * eps);
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -2.1e-5)
		tmp = sin((x + eps)) - sin(x);
	elseif (eps <= 0.00112)
		tmp = cos(x) * eps;
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -2.1e-5], N[(N[Sin[N[(x + eps), $MachinePrecision]], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], If[LessEqual[eps, 0.00112], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -2.09999999999999988e-5

   1. Initial program 62.2%

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

   if -2.09999999999999988e-5 < eps < 0.0011199999999999999

   1. Initial program 30.3%

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

   3. Taylor expanded in eps around 0 98.7%

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

   if 0.0011199999999999999 < eps

   1. Initial program 48.8%

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

   3. Taylor expanded in x around 0 50.8%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.1 \cdot 10^{-5}:\\ \;\;\;\;\sin \left(x + \varepsilon\right) - \sin x\\ \mathbf{elif}\;\varepsilon \leq 0.00112:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 75.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.25 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00112\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -2.25e-5) (not (<= eps 0.00112))) (sin eps) (* (cos x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -2.25e-5) || !(eps <= 0.00112)) {
		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 <= (-2.25d-5)) .or. (.not. (eps <= 0.00112d0))) 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 <= -2.25e-5) || !(eps <= 0.00112)) {
		tmp = Math.sin(eps);
	} else {
		tmp = Math.cos(x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -2.25e-5) or not (eps <= 0.00112):
		tmp = math.sin(eps)
	else:
		tmp = math.cos(x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -2.25e-5) || !(eps <= 0.00112))
		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 <= -2.25e-5) || ~((eps <= 0.00112)))
		tmp = sin(eps);
	else
		tmp = cos(x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -2.25e-5], N[Not[LessEqual[eps, 0.00112]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.25000000000000014e-5 or 0.0011199999999999999 < eps

   1. Initial program 54.1%

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

   3. Taylor expanded in x around 0 55.2%

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

   if -2.25000000000000014e-5 < eps < 0.0011199999999999999

   1. Initial program 30.3%

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

   3. Taylor expanded in eps around 0 98.7%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -2.25 \cdot 10^{-5} \lor \neg \left(\varepsilon \leq 0.00112\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 54.0% 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 42.4%

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

3. Taylor expanded in x around 0 53.2%

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

4. Final simplification 53.2%

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

Alternative 11: 28.3% 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 42.4%

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

3. Taylor expanded in x around 0 53.2%

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

4. Taylor expanded in eps around 0 26.9%

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

5. Final simplification 26.9%

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

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 2024018 
(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)))