2sin (example 3.3)

Percentage Accurate: 43.0% → 99.4%

Time: 12.9s

Alternatives: 7

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: 43.0% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. sin-sum 64.9%

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

   2. associate--l+ 64.9%

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

4. Applied egg-rr 64.9%

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

   1. +-commutative 64.9%

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

   2. associate-+l- 99.5%

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

   3. *-commutative 99.5%

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

   4. *-rgt-identity 99.5%

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

   5. distribute-lft-out-- 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in eps around inf 99.5%

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

   1. fma-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. neg-sub0 99.5%

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

   4. associate--r- 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

9. Simplified 99.5%

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

    1. distribute-rgt-in 99.5%

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

    2. neg-mul-1 99.5%

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

11. Applied egg-rr 99.5%

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

12. Taylor expanded in eps around inf 99.5%

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

13. Final simplification 99.5%

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

Alternative 2: 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 40.5%

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

   1. sin-sum 64.9%

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

   2. associate--l+ 64.9%

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

4. Applied egg-rr 64.9%

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

   1. +-commutative 64.9%

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

   2. associate-+l- 99.5%

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

   3. *-commutative 99.5%

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

   4. *-rgt-identity 99.5%

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

   5. distribute-lft-out-- 99.5%

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

6. Simplified 99.5%

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

7. Taylor expanded in eps around inf 99.5%

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

   1. fma-neg 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. neg-sub0 99.5%

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

   4. associate--r- 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 3: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.5%

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

   1. sin-sum 64.9%

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

   2. associate--l+ 64.9%

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

4. Applied egg-rr 64.9%

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

   1. +-commutative 64.9%

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

   2. associate-+l- 99.5%

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

   3. *-commutative 99.5%

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

   4. *-rgt-identity 99.5%

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

   5. distribute-lft-out-- 99.5%

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

6. Simplified 99.5%

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

7. Final simplification 99.5%

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

Alternative 4: 76.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(x, eps)
	tmp = 2.0 * (sin(((eps + (x - x)) / 2.0)) * cos(((x + (x + eps)) / 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[(x + N[(x + eps), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.5%

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

   1. log1p-expm1-u 40.3%

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

4. Applied egg-rr 40.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(x + \varepsilon\right)\right)\right)} - \sin x \]
5. Step-by-step derivation

   1. log1p-expm1-u 40.5%

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

   2. diff-sin 39.9%

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

   3. +-commutative 39.9%

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

   4. +-commutative 39.9%

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

6. Applied egg-rr 39.9%

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

   1. associate--l+ 75.8%

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

   2. +-commutative 75.8%

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

8. Simplified 75.8%

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

9. Final simplification 75.8%

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

Alternative 5: 76.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00014) (not (<= eps 3.2e-7))) (sin eps) (* (cos x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00014) || !(eps <= 3.2e-7)) {
		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 <= (-0.00014d0)) .or. (.not. (eps <= 3.2d-7))) 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 <= -0.00014) || !(eps <= 3.2e-7)) {
		tmp = Math.sin(eps);
	} else {
		tmp = Math.cos(x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00014) or not (eps <= 3.2e-7):
		tmp = math.sin(eps)
	else:
		tmp = math.cos(x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00014) || !(eps <= 3.2e-7))
		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 <= -0.00014) || ~((eps <= 3.2e-7)))
		tmp = sin(eps);
	else
		tmp = cos(x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00014], N[Not[LessEqual[eps, 3.2e-7]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -1.3999999999999999e-4 or 3.2000000000000001e-7 < eps

   1. Initial program 52.6%

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

   3. Taylor expanded in x around 0 54.3%

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

   if -1.3999999999999999e-4 < eps < 3.2000000000000001e-7

   1. Initial program 28.0%

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

   3. Taylor expanded in eps around 0 99.4%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00014 \lor \neg \left(\varepsilon \leq 3.2 \cdot 10^{-7}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.8% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (sin eps))

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return math.sin(eps)

Julia

function code(x, eps)
	return sin(eps)
end

MATLAB

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

Wolfram

code[x_, eps_] := N[Sin[eps], $MachinePrecision]

Derivation

1. Initial program 40.5%

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

3. Taylor expanded in x around 0 57.4%

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

4. Final simplification 57.4%

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

Alternative 7: 29.4% 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 40.5%

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

3. Taylor expanded in eps around 0 51.7%

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

   1. +-commutative 51.7%

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

   2. fma-def 51.7%

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

   3. *-commutative 51.7%

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

   4. associate-*l* 51.7%

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

5. Simplified 51.7%

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

6. Taylor expanded in x around 0 32.3%

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

7. Final simplification 32.3%

   \[\leadsto \varepsilon \]
8. 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 2024019 
(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)))