2sin (example 3.3)

Percentage Accurate: 62.6% → 100.0%

Time: 14.7s

Alternatives: 14

Speedup: 12.2×

Specification

\[\left(\left(-10000 \leq x \land x \leq 10000\right) \land 10^{-16} \cdot \left|x\right| < \varepsilon\right) \land \varepsilon < \left|x\right|\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.6% 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: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ t_1 := t\_0 \cdot \sin x\\ t_2 := \cos \left(\varepsilon \cdot 0.5\right) \cdot \cos x\\ 2 \cdot \left(t\_0 \cdot \frac{{t\_2}^{3} - {t\_1}^{3}}{\mathsf{fma}\left(t\_2, t\_2, \mathsf{fma}\left(t\_1, t\_1, t\_2 \cdot t\_1\right)\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5)))
        (t_1 (* t_0 (sin x)))
        (t_2 (* (cos (* eps 0.5)) (cos x))))
   (*
    2.0
    (*
     t_0
     (/
      (- (pow t_2 3.0) (pow t_1 3.0))
      (fma t_2 t_2 (fma t_1 t_1 (* t_2 t_1))))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	double t_1 = t_0 * sin(x);
	double t_2 = cos((eps * 0.5)) * cos(x);
	return 2.0 * (t_0 * ((pow(t_2, 3.0) - pow(t_1, 3.0)) / fma(t_2, t_2, fma(t_1, t_1, (t_2 * t_1)))));
}

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	t_1 = Float64(t_0 * sin(x))
	t_2 = Float64(cos(Float64(eps * 0.5)) * cos(x))
	return Float64(2.0 * Float64(t_0 * Float64(Float64((t_2 ^ 3.0) - (t_1 ^ 3.0)) / fma(t_2, t_2, fma(t_1, t_1, Float64(t_2 * t_1))))))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]}, N[(2.0 * N[(t$95$0 * N[(N[(N[Power[t$95$2, 3.0], $MachinePrecision] - N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / N[(t$95$2 * t$95$2 + N[(t$95$1 * t$95$1 + N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot 2 \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower-fma.f64 99.8

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 100.0%

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

10. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\varepsilon \cdot 0.5\right)\\ 2 \cdot \left(t\_0 \cdot \left(\cos \left(\varepsilon \cdot 0.5\right) \cdot \cos x - t\_0 \cdot \sin x\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (sin (* eps 0.5))))
   (* 2.0 (* t_0 (- (* (cos (* eps 0.5)) (cos x)) (* t_0 (sin x)))))))

C

double code(double x, double eps) {
	double t_0 = sin((eps * 0.5));
	return 2.0 * (t_0 * ((cos((eps * 0.5)) * cos(x)) - (t_0 * sin(x))));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: t_0
    t_0 = sin((eps * 0.5d0))
    code = 2.0d0 * (t_0 * ((cos((eps * 0.5d0)) * cos(x)) - (t_0 * sin(x))))
end function

Java

public static double code(double x, double eps) {
	double t_0 = Math.sin((eps * 0.5));
	return 2.0 * (t_0 * ((Math.cos((eps * 0.5)) * Math.cos(x)) - (t_0 * Math.sin(x))));
}

Python

def code(x, eps):
	t_0 = math.sin((eps * 0.5))
	return 2.0 * (t_0 * ((math.cos((eps * 0.5)) * math.cos(x)) - (t_0 * math.sin(x))))

Julia

function code(x, eps)
	t_0 = sin(Float64(eps * 0.5))
	return Float64(2.0 * Float64(t_0 * Float64(Float64(cos(Float64(eps * 0.5)) * cos(x)) - Float64(t_0 * sin(x)))))
end

MATLAB

function tmp = code(x, eps)
	t_0 = sin((eps * 0.5));
	tmp = 2.0 * (t_0 * ((cos((eps * 0.5)) * cos(x)) - (t_0 * sin(x))));
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot 2 \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower-fma.f64 99.8

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 100.0%

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

10. Final simplification 100.0%

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

Alternative 3: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot 2 \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower-fma.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

   \[\leadsto 2 \cdot \left(\sin \left(\varepsilon \cdot 0.5\right) \cdot \cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)\right) \]
9. Add Preprocessing

Alternative 4: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (cos (fma 0.5 eps x))
   (*
    eps
    (fma
     (* eps eps)
     (fma
      (* eps eps)
      (fma (* eps eps) -1.5500992063492063e-6 0.00026041666666666666)
      -0.020833333333333332)
     0.5)))))

C

double code(double x, double eps) {
	return 2.0 * (cos(fma(0.5, eps, x)) * (eps * fma((eps * eps), fma((eps * eps), fma((eps * eps), -1.5500992063492063e-6, 0.00026041666666666666), -0.020833333333333332), 0.5)));
}

Julia

function code(x, eps)
	return Float64(2.0 * Float64(cos(fma(0.5, eps, x)) * Float64(eps * fma(Float64(eps * eps), fma(Float64(eps * eps), fma(Float64(eps * eps), -1.5500992063492063e-6, 0.00026041666666666666), -0.020833333333333332), 0.5))))
end

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * N[(N[(eps * eps), $MachinePrecision] * -1.5500992063492063e-6 + 0.00026041666666666666), $MachinePrecision] + -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot 2 \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower-fma.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in eps around 0

   \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left({\varepsilon}^{2} \cdot \left(\frac{1}{3840} + \frac{-1}{645120} \cdot {\varepsilon}^{2}\right) - \frac{1}{48}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\right) \cdot 2 \]
9. Step-by-step derivation

10. Applied rewrites 99.5%

    \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)}\right) \cdot 2 \]

11. Final simplification 99.5%

    \[\leadsto 2 \cdot \left(\cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, -1.5500992063492063 \cdot 10^{-6}, 0.00026041666666666666\right), -0.020833333333333332\right), 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 5: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (cos (fma 0.5 eps x))
   (*
    eps
    (fma
     eps
     (* eps (fma (* eps eps) 0.00026041666666666666 -0.020833333333333332))
     0.5)))))

C

double code(double x, double eps) {
	return 2.0 * (cos(fma(0.5, eps, x)) * (eps * fma(eps, (eps * fma((eps * eps), 0.00026041666666666666, -0.020833333333333332)), 0.5)));
}

Julia

function code(x, eps)
	return Float64(2.0 * Float64(cos(fma(0.5, eps, x)) * Float64(eps * fma(eps, Float64(eps * fma(Float64(eps * eps), 0.00026041666666666666, -0.020833333333333332)), 0.5))))
end

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(eps * N[(eps * N[(N[(eps * eps), $MachinePrecision] * 0.00026041666666666666 + -0.020833333333333332), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot 2 \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower-fma.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in eps around 0

   \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + {\varepsilon}^{2} \cdot \left(\frac{1}{3840} \cdot {\varepsilon}^{2} - \frac{1}{48}\right)\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\right) \cdot 2 \]
9. Step-by-step derivation

10. Applied rewrites 99.4%

    \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)}\right) \cdot 2 \]

11. Final simplification 99.4%

    \[\leadsto 2 \cdot \left(\cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.00026041666666666666, -0.020833333333333332\right), 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 6: 99.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  2.0
  (*
   (cos (fma 0.5 eps x))
   (* eps (fma eps (* eps -0.020833333333333332) 0.5)))))

C

double code(double x, double eps) {
	return 2.0 * (cos(fma(0.5, eps, x)) * (eps * fma(eps, (eps * -0.020833333333333332), 0.5)));
}

Julia

function code(x, eps)
	return Float64(2.0 * Float64(cos(fma(0.5, eps, x)) * Float64(eps * fma(eps, Float64(eps * -0.020833333333333332), 0.5))))
end

Wolfram

code[x_, eps_] := N[(2.0 * N[(N[Cos[N[(0.5 * eps + x), $MachinePrecision]], $MachinePrecision] * N[(eps * N[(eps * N[(eps * -0.020833333333333332), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eps around inf

   \[\leadsto \color{blue}{\left(\cos \left(\frac{1}{2} \cdot \left(\varepsilon + 2 \cdot x\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \varepsilon\right)\right)} \cdot 2 \]
6. Step-by-step derivation

   1. *-commutative N/A

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

   2. metadata-eval N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sin.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-cos.f64 N/A

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

   8. cancel-sign-sub-inv N/A

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

   9. metadata-eval N/A

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

   10. distribute-lft-in N/A

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

   11. associate-*r* N/A

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

   12. metadata-eval N/A

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

   13. *-lft-identity N/A

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

   14. lower-fma.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Taylor expanded in eps around 0

   \[\leadsto \left(\left(\varepsilon \cdot \left(\frac{1}{2} + \frac{-1}{48} \cdot {\varepsilon}^{2}\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \varepsilon, x\right)\right)}\right) \cdot 2 \]
9. Step-by-step derivation

10. Applied rewrites 99.2%

    \[\leadsto \left(\left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.5\right)\right) \cdot \cos \color{blue}{\left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right)}\right) \cdot 2 \]

11. Final simplification 99.2%

    \[\leadsto 2 \cdot \left(\cos \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \left(\varepsilon \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.020833333333333332, 0.5\right)\right)\right) \]
12. Add Preprocessing

Alternative 7: 99.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \cos \left(\mathsf{fma}\left(\varepsilon, 0.5, x\right)\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

   1. lift--.f64 N/A

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

   2. lift-sin.f64 N/A

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

   3. lift-sin.f64 N/A

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

   4. diff-sin N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in eps around 0

   \[\leadsto \left(\color{blue}{\left(\frac{1}{2} \cdot \varepsilon\right)} \cdot \cos \left(\mathsf{fma}\left(x, 2, \varepsilon\right) \cdot \frac{1}{2}\right)\right) \cdot 2 \]
6. Step-by-step derivation

   1. lower-*.f64 99.1

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

7. Applied rewrites 99.1%

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

8. Taylor expanded in x around 0

   \[\leadsto \left(\left(\frac{1}{2} \cdot \varepsilon\right) \cdot \cos \color{blue}{\left(x + \frac{1}{2} \cdot \varepsilon\right)}\right) \cdot 2 \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 99.1

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

10. Applied rewrites 99.1%

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

11. Final simplification 99.1%

    \[\leadsto 2 \cdot \left(\left(\varepsilon \cdot 0.5\right) \cdot \cos \left(\mathsf{fma}\left(\varepsilon, 0.5, x\right)\right)\right) \]
12. Add Preprocessing

Alternative 8: 98.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \cos x \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (cos x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 98.7

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

5. Applied rewrites 98.7%

   \[\leadsto \color{blue}{\varepsilon \cdot \cos x} \]
6. Add Preprocessing

Alternative 9: 98.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(-0.5, x \cdot \left(\varepsilon + x\right), \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, 0.08333333333333333, -0.16666666666666666\right)\right)\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  eps
  (fma
   -0.5
   (* x (+ eps x))
   (* eps (* eps (fma (* x x) 0.08333333333333333 -0.16666666666666666))))
  eps))

C

double code(double x, double eps) {
	return fma(eps, fma(-0.5, (x * (eps + x)), (eps * (eps * fma((x * x), 0.08333333333333333, -0.16666666666666666)))), eps);
}

Julia

function code(x, eps)
	return fma(eps, fma(-0.5, Float64(x * Float64(eps + x)), Float64(eps * Float64(eps * fma(Float64(x * x), 0.08333333333333333, -0.16666666666666666)))), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(-0.5 * N[(x * N[(eps + x), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(eps * N[(N[(x * x), $MachinePrecision] * 0.08333333333333333 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sin \varepsilon + x \cdot \left(\left(\cos \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \sin \varepsilon\right)\right) - 1\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-+r+ N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

5. Applied rewrites 61.2%

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

6. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{-1}{2} \cdot {x}^{2} + \varepsilon \cdot \left(\frac{-1}{2} \cdot x + \frac{-1}{6} \cdot \left(\varepsilon \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right)\right)\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 97.6%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\mathsf{fma}\left(-0.5, x \cdot \left(\varepsilon + x\right), \varepsilon \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x \cdot x, 0.08333333333333333, -0.16666666666666666\right)\right)\right)}, \varepsilon\right) \]
9. Add Preprocessing

Alternative 10: 98.3% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \varepsilon \cdot \left(x \cdot \left(\varepsilon + x\right)\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma -0.5 (* eps (* x (+ eps x))) eps))

C

double code(double x, double eps) {
	return fma(-0.5, (eps * (x * (eps + x))), eps);
}

Julia

function code(x, eps)
	return fma(-0.5, Float64(eps * Float64(x * Float64(eps + x))), eps)
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sin \varepsilon + x \cdot \left(\left(\cos \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \sin \varepsilon\right)\right) - 1\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-+r+ N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

5. Applied rewrites 61.2%

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

6. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot x\right) + \frac{-1}{2} \cdot {x}^{2}\right)\right)} \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

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

9. Final simplification 97.5%

   \[\leadsto \mathsf{fma}\left(-0.5, \varepsilon \cdot \left(x \cdot \left(\varepsilon + x\right)\right), \varepsilon\right) \]
10. Add Preprocessing

Alternative 11: 98.2% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot -0.5\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (fma eps (* x (* x -0.5)) eps))

C

double code(double x, double eps) {
	return fma(eps, (x * (x * -0.5)), eps);
}

Julia

function code(x, eps)
	return fma(eps, Float64(x * Float64(x * -0.5)), eps)
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sin \varepsilon + x \cdot \left(\left(\cos \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \sin \varepsilon\right)\right) - 1\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-+r+ N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

5. Applied rewrites 61.2%

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

6. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(x \cdot -0.5\right)}, \varepsilon\right) \]
9. Add Preprocessing

Alternative 12: 97.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \left(\varepsilon \cdot -0.16666666666666666\right), \varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma eps (* eps (* eps -0.16666666666666666)) eps))

C

double code(double x, double eps) {
	return fma(eps, (eps * (eps * -0.16666666666666666)), eps);
}

Julia

function code(x, eps)
	return fma(eps, Float64(eps * Float64(eps * -0.16666666666666666)), eps)
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(eps * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in x around 0

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

   1. lower-sin.f64 97.5

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

5. Applied rewrites 97.5%

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

6. Taylor expanded in eps around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]
7. Step-by-step derivation

8. Applied rewrites 97.5%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \color{blue}{\varepsilon \cdot \left(\varepsilon \cdot -0.16666666666666666\right)}, \varepsilon\right) \]
9. Add Preprocessing

Alternative 13: 61.5% accurate, 29.6× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 (- (+ eps x) x))

C

double code(double x, double eps) {
	return (eps + x) - x;
}

Fortran

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

Java

public static double code(double x, double eps) {
	return (eps + x) - x;
}

Python

def code(x, eps):
	return (eps + x) - x

Julia

function code(x, eps)
	return Float64(Float64(eps + x) - x)
end

MATLAB

function tmp = code(x, eps)
	tmp = (eps + x) - x;
end

Wolfram

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

Derivation

1. Initial program 62.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sin \varepsilon + x \cdot \left(\left(\cos \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \sin \varepsilon\right)\right) - 1\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-lft-in N/A

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

   4. *-commutative N/A

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

   5. associate-+r+ N/A

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

   6. mul-1-neg N/A

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

   7. unsub-neg N/A

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

   8. lower--.f64 N/A

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

5. Applied rewrites 61.2%

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

6. Taylor expanded in eps around 0

   \[\leadsto \left(x + \varepsilon \cdot \left(1 + \frac{-1}{2} \cdot {x}^{2}\right)\right) - x \]
7. Step-by-step derivation

8. Applied rewrites 61.0%

   \[\leadsto \mathsf{fma}\left(\varepsilon, \mathsf{fma}\left(x, x \cdot -0.5, 1\right), x\right) - x \]

9. Taylor expanded in x around 0

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

11. Applied rewrites 61.2%

    \[\leadsto \left(\varepsilon + x\right) - x \]
12. Add Preprocessing

Alternative 14: 5.4% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

double code(double x, double eps) {
	return 0.0;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = 0.0d0
end function

Java

public static double code(double x, double eps) {
	return 0.0;
}

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

function tmp = code(x, eps)
	tmp = 0.0;
end

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 62.4%

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

4. Applied rewrites 7.2%

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

5. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{\frac{-1}{2} \cdot \cos \left(2 \cdot x\right) + \frac{1}{2} \cdot \cos \left(2 \cdot x\right)}{\sin x}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. distribute-rgt-out N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \left(\cos \left(2 \cdot x\right) \cdot \color{blue}{0}\right)}{\sin x} \]

   4. mul0-rgt N/A

      \[\leadsto \frac{\frac{-1}{2} \cdot \color{blue}{0}}{\sin x} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{\sin x} \]

   6. div0 5.5

      \[\leadsto \color{blue}{0} \]

7. Applied rewrites 5.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Developer Target 3: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos((0.5d0 * (eps - ((-2.0d0) * x)))) * sin((0.5d0 * eps))) * 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (* 2 (cos (+ x (/ eps 2))) (sin (/ eps 2))))

  :alt
  (! :herbie-platform default (+ (* (sin x) (- (cos eps) 1)) (* (cos x) (sin eps))))

  :alt
  (! :herbie-platform default (* (cos (* 1/2 (- eps (* -2 x)))) (sin (* 1/2 eps)) 2))

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