2sin (example 3.3)

Percentage Accurate: 62.6% → 99.7%

Time: 14.9s

Alternatives: 13

Speedup: 207.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.6% 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: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right), \cos x, \varepsilon \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.041666666666666664, -0.5\right)\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   (fma eps (* eps -0.16666666666666666) 1.0)
   (cos x)
   (* eps (* (sin x) (fma eps (* eps 0.041666666666666664) -0.5))))))

C

double code(double x, double eps) {
	return eps * fma(fma(eps, (eps * -0.16666666666666666), 1.0), cos(x), (eps * (sin(x) * fma(eps, (eps * 0.041666666666666664), -0.5))));
}

Julia

function code(x, eps)
	return Float64(eps * fma(fma(eps, Float64(eps * -0.16666666666666666), 1.0), cos(x), Float64(eps * Float64(sin(x) * fma(eps, Float64(eps * 0.041666666666666664), -0.5)))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(eps * N[(eps * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(eps * N[(N[Sin[x], $MachinePrecision] * N[(eps * N[(eps * 0.041666666666666664), $MachinePrecision] + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \frac{-1}{2} \cdot \sin x\right)}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\color{blue}{\left(\left(\frac{-1}{6} \cdot \cos x\right) \cdot \varepsilon + \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon\right)} + \frac{-1}{2} \cdot \sin x\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \cos x\right) \cdot \varepsilon + \left(\left(\frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \frac{-1}{2} \cdot \sin x\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\left(\frac{-1}{6} \cdot \cos x\right) \cdot \varepsilon + \left(\color{blue}{\varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)} + \frac{-1}{2} \cdot \sin x\right)\right)\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \color{blue}{\left(\left(\left(\frac{-1}{6} \cdot \cos x\right) \cdot \varepsilon\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \frac{-1}{2} \cdot \sin x\right) \cdot \varepsilon\right)}\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right), \cos x, \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.041666666666666664, -0.5\right)\right) \cdot \varepsilon\right)} \]

6. Final simplification 99.9%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right), \cos x, \varepsilon \cdot \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.041666666666666664, -0.5\right)\right)\right) \]
7. Add Preprocessing

Alternative 2: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \sin x \cdot -0.5, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   eps
   (* (sin x) -0.5)
   (* (fma eps (* eps -0.16666666666666666) 1.0) (cos x)))))

C

double code(double x, double eps) {
	return eps * fma(eps, (sin(x) * -0.5), (fma(eps, (eps * -0.16666666666666666), 1.0) * cos(x)));
}

Julia

function code(x, eps)
	return Float64(eps * fma(eps, Float64(sin(x) * -0.5), Float64(fma(eps, Float64(eps * -0.16666666666666666), 1.0) * cos(x))))
end

Wolfram

code[x_, eps_] := N[(eps * N[(eps * N[(N[Sin[x], $MachinePrecision] * -0.5), $MachinePrecision] + N[(N[(eps * N[(eps * -0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \color{blue}{\sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   8. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right)} + \cos x\right) \]

   9. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x} + \cos x\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 99.6

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\cos x}\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right)} \]

6. Final simplification 99.6%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \sin x \cdot -0.5, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right) \]
7. Add Preprocessing

Alternative 3: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Final simplification 99.0%

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

Alternative 4: 99.0% 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 63.8%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 98.4

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

5. Simplified 98.4%

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

Alternative 5: 98.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.041666666666666664, \varepsilon \cdot \varepsilon, -0.5\right)\\ \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot t\_0, x \cdot \mathsf{fma}\left(-0.16666666666666666, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot t\_0\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.08333333333333333, -0.5\right)\right)\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (fma 0.041666666666666664 (* eps eps) -0.5)))
   (fma
    x
    (fma
     eps
     (* eps t_0)
     (*
      x
      (fma
       -0.16666666666666666
       (* (* eps eps) (* x t_0))
       (* eps (fma (* eps eps) 0.08333333333333333 -0.5)))))
    (fma -0.16666666666666666 (* eps (* eps eps)) eps))))

C

double code(double x, double eps) {
	double t_0 = fma(0.041666666666666664, (eps * eps), -0.5);
	return fma(x, fma(eps, (eps * t_0), (x * fma(-0.16666666666666666, ((eps * eps) * (x * t_0)), (eps * fma((eps * eps), 0.08333333333333333, -0.5))))), fma(-0.16666666666666666, (eps * (eps * eps)), eps));
}

Julia

function code(x, eps)
	t_0 = fma(0.041666666666666664, Float64(eps * eps), -0.5)
	return fma(x, fma(eps, Float64(eps * t_0), Float64(x * fma(-0.16666666666666666, Float64(Float64(eps * eps) * Float64(x * t_0)), Float64(eps * fma(Float64(eps * eps), 0.08333333333333333, -0.5))))), fma(-0.16666666666666666, Float64(eps * Float64(eps * eps)), eps))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(eps * eps), $MachinePrecision] + -0.5), $MachinePrecision]}, N[(x * N[(eps * N[(eps * t$95$0), $MachinePrecision] + N[(x * N[(-0.16666666666666666 * N[(N[(eps * eps), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(eps * N[(N[(eps * eps), $MachinePrecision] * 0.08333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \color{blue}{\varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\frac{-1}{2} \cdot \sin x + \varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \cos x + \frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \frac{-1}{2} \cdot \sin x\right)}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\color{blue}{\left(\left(\frac{-1}{6} \cdot \cos x\right) \cdot \varepsilon + \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon\right)} + \frac{-1}{2} \cdot \sin x\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \cos x\right) \cdot \varepsilon + \left(\left(\frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right) \cdot \varepsilon + \frac{-1}{2} \cdot \sin x\right)\right)}\right) \]

   5. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \varepsilon \cdot \left(\left(\frac{-1}{6} \cdot \cos x\right) \cdot \varepsilon + \left(\color{blue}{\varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right)} + \frac{-1}{2} \cdot \sin x\right)\right)\right) \]

   6. distribute-rgt-in N/A

      \[\leadsto \varepsilon \cdot \left(\cos x + \color{blue}{\left(\left(\left(\frac{-1}{6} \cdot \cos x\right) \cdot \varepsilon\right) \cdot \varepsilon + \left(\varepsilon \cdot \left(\frac{1}{24} \cdot \left(\varepsilon \cdot \sin x\right)\right) + \frac{-1}{2} \cdot \sin x\right) \cdot \varepsilon\right)}\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right), \cos x, \left(\sin x \cdot \mathsf{fma}\left(\varepsilon, \varepsilon \cdot 0.041666666666666664, -0.5\right)\right) \cdot \varepsilon\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + x \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) + \frac{-1}{6} \cdot \left({\varepsilon}^{2} \cdot \left(x \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)\right)\right) + {\varepsilon}^{2} \cdot \left(\frac{1}{24} \cdot {\varepsilon}^{2} - \frac{1}{2}\right)\right)} \]

8. Simplified 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot \mathsf{fma}\left(0.041666666666666664, \varepsilon \cdot \varepsilon, -0.5\right), x \cdot \mathsf{fma}\left(-0.16666666666666666, \left(\varepsilon \cdot \varepsilon\right) \cdot \left(x \cdot \mathsf{fma}\left(0.041666666666666664, \varepsilon \cdot \varepsilon, -0.5\right)\right), \varepsilon \cdot \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.08333333333333333, -0.5\right)\right)\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)\right)} \]
9. Add Preprocessing

Alternative 6: 98.3% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot 0.08333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.08333333333333333, -0.5\right)\right), \varepsilon \cdot -0.5\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   x
   (fma
    x
    (fma
     eps
     (* x 0.08333333333333333)
     (fma (* eps eps) 0.08333333333333333 -0.5))
    (* eps -0.5))
   (fma -0.16666666666666666 (* eps eps) 1.0))))

C

double code(double x, double eps) {
	return eps * fma(x, fma(x, fma(eps, (x * 0.08333333333333333), fma((eps * eps), 0.08333333333333333, -0.5)), (eps * -0.5)), fma(-0.16666666666666666, (eps * eps), 1.0));
}

Julia

function code(x, eps)
	return Float64(eps * fma(x, fma(x, fma(eps, Float64(x * 0.08333333333333333), fma(Float64(eps * eps), 0.08333333333333333, -0.5)), Float64(eps * -0.5)), fma(-0.16666666666666666, Float64(eps * eps), 1.0)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * N[(x * N[(eps * N[(x * 0.08333333333333333), $MachinePrecision] + N[(N[(eps * eps), $MachinePrecision] * 0.08333333333333333 + -0.5), $MachinePrecision]), $MachinePrecision] + N[(eps * -0.5), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \color{blue}{\sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   8. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right)} + \cos x\right) \]

   9. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x} + \cos x\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 99.6

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\cos x}\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{\left(1 + \left(\frac{-1}{6} \cdot {\varepsilon}^{2} + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. associate-+r+ N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right)\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + \frac{1}{12} \cdot \left(\varepsilon \cdot x\right)\right), 1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

8. Simplified 97.4%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon, x \cdot 0.08333333333333333, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.08333333333333333, -0.5\right)\right), \varepsilon \cdot -0.5\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right)} \]
9. Add Preprocessing

Alternative 7: 98.3% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, -0.5 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right), \varepsilon\right)\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  x
  (* -0.5 (* eps (fma x (fma -0.16666666666666666 (* eps eps) 1.0) eps)))
  (fma -0.16666666666666666 (* eps (* eps eps)) eps)))

C

double code(double x, double eps) {
	return fma(x, (-0.5 * (eps * fma(x, fma(-0.16666666666666666, (eps * eps), 1.0), eps))), fma(-0.16666666666666666, (eps * (eps * eps)), eps));
}

Julia

function code(x, eps)
	return fma(x, Float64(-0.5 * Float64(eps * fma(x, fma(-0.16666666666666666, Float64(eps * eps), 1.0), eps))), fma(-0.16666666666666666, Float64(eps * Float64(eps * eps)), eps))
end

Wolfram

code[x_, eps_] := N[(x * N[(-0.5 * N[(eps * N[(x * N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(eps * N[(eps * eps), $MachinePrecision]), $MachinePrecision] + eps), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \color{blue}{\sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   8. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right)} + \cos x\right) \]

   9. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x} + \cos x\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 99.6

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\cos x}\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\varepsilon \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{-1}{2} \cdot \left(\varepsilon \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)\right) + \frac{-1}{2} \cdot {\varepsilon}^{2}\right)} \]

8. Simplified 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, -0.5 \cdot \left(\varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right), \varepsilon\right)\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \left(\varepsilon \cdot \varepsilon\right), \varepsilon\right)\right)} \]
9. Add Preprocessing

Alternative 8: 98.3% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.08333333333333333, -0.5\right), \varepsilon \cdot -0.5\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  eps
  (fma
   x
   (fma x (fma (* eps eps) 0.08333333333333333 -0.5) (* eps -0.5))
   (fma -0.16666666666666666 (* eps eps) 1.0))))

C

double code(double x, double eps) {
	return eps * fma(x, fma(x, fma((eps * eps), 0.08333333333333333, -0.5), (eps * -0.5)), fma(-0.16666666666666666, (eps * eps), 1.0));
}

Julia

function code(x, eps)
	return Float64(eps * fma(x, fma(x, fma(Float64(eps * eps), 0.08333333333333333, -0.5), Float64(eps * -0.5)), fma(-0.16666666666666666, Float64(eps * eps), 1.0)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * N[(x * N[(N[(eps * eps), $MachinePrecision] * 0.08333333333333333 + -0.5), $MachinePrecision] + N[(eps * -0.5), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \color{blue}{\sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   8. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right)} + \cos x\right) \]

   9. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x} + \cos x\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 99.6

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\cos x}\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right)} \]

6. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \cdot \frac{-1}{2}}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \color{blue}{x \cdot \left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) \cdot \frac{-1}{2}\right)}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right), 1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

8. Simplified 97.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.08333333333333333, -0.5\right), \varepsilon \cdot -0.5\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right)} \]
9. Add Preprocessing

Alternative 9: 98.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x, -0.5 \cdot \left(\varepsilon + x\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (fma x (* -0.5 (+ eps x)) (fma -0.16666666666666666 (* eps eps) 1.0))))

C

double code(double x, double eps) {
	return eps * fma(x, (-0.5 * (eps + x)), fma(-0.16666666666666666, (eps * eps), 1.0));
}

Julia

function code(x, eps)
	return Float64(eps * fma(x, Float64(-0.5 * Float64(eps + x)), fma(-0.16666666666666666, Float64(eps * eps), 1.0)))
end

Wolfram

code[x_, eps_] := N[(eps * N[(x * N[(-0.5 * N[(eps + x), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \color{blue}{\sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   8. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right)} + \cos x\right) \]

   9. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x} + \cos x\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 99.6

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\cos x}\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right)} \]

6. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \cdot \frac{-1}{2}}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \color{blue}{x \cdot \left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) \cdot \frac{-1}{2}\right)}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right), 1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

8. Simplified 97.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.08333333333333333, -0.5\right), \varepsilon \cdot -0.5\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right)} \]

9. Taylor expanded in eps around 0

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

    1. distribute-lft-out N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{\frac{-1}{2} \cdot \left(\varepsilon + x\right)}, \mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right)\right) \]

    2. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{\frac{-1}{2} \cdot \left(\varepsilon + x\right)}, \mathsf{fma}\left(\frac{-1}{6}, \varepsilon \cdot \varepsilon, 1\right)\right) \]

    3. lower-+.f64 97.3

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, -0.5 \cdot \color{blue}{\left(\varepsilon + x\right)}, \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right) \]

11. Simplified 97.3%

    \[\leadsto \varepsilon \cdot \mathsf{fma}\left(x, \color{blue}{-0.5 \cdot \left(\varepsilon + x\right)}, \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right) \]
12. Add Preprocessing

Alternative 10: 98.2% accurate, 10.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-cos.f64 99.0

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

5. Simplified 99.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}\right)}, \cos x\right) \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \left(\frac{1}{12} \cdot {x}^{2} - \frac{1}{2}\right)}, \cos x\right) \]

   2. sub-neg N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\left(\frac{1}{12} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)}, \cos x\right) \]

   3. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{12}} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \cos x\right) \]

   4. unpow2 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{12} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \cos x\right) \]

   5. associate-*l* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{12}\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \cos x\right) \]

   6. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \color{blue}{\left(\frac{1}{12} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right), \cos x\right) \]

   7. metadata-eval N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \left(x \cdot \left(\frac{1}{12} \cdot x\right) + \color{blue}{\frac{-1}{2}}\right), \cos x\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \color{blue}{\mathsf{fma}\left(x, \frac{1}{12} \cdot x, \frac{-1}{2}\right)}, \cos x\right) \]

   9. *-commutative N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{12}}, \frac{-1}{2}\right), \cos x\right) \]

   10. lower-*.f64 97.8

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.08333333333333333}, -0.5\right), \cos x\right) \]

8. Simplified 97.8%

   \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.08333333333333333, -0.5\right)}, \cos x\right) \]

9. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

11. Simplified 97.3%

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

12. Final simplification 97.3%

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

Alternative 11: 98.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(x \cdot -0.5, \varepsilon + x, 1\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (* eps (fma (* x -0.5) (+ eps x) 1.0)))

C

double code(double x, double eps) {
	return eps * fma((x * -0.5), (eps + x), 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * fma(Float64(x * -0.5), Float64(eps + x), 1.0))
end

Wolfram

code[x_, eps_] := N[(eps * N[(N[(x * -0.5), $MachinePrecision] * N[(eps + x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \color{blue}{\sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   8. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right)} + \cos x\right) \]

   9. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x} + \cos x\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 99.6

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\cos x}\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right)} \]

6. Taylor expanded in x around 0

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

   1. associate-+r+ N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) + x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \frac{-1}{2} \cdot \left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)} \]

   3. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \color{blue}{\left(x \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \cdot \frac{-1}{2}}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   4. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + \color{blue}{x \cdot \left(\left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right) \cdot \frac{-1}{2}\right)}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \varepsilon \cdot \left(x \cdot \left(\frac{-1}{2} \cdot \varepsilon + x \cdot \color{blue}{\left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right)}\right) + \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right) \]

   6. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \frac{-1}{2} \cdot \varepsilon + x \cdot \left(\frac{-1}{2} \cdot \left(1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)\right), 1 + \frac{-1}{6} \cdot {\varepsilon}^{2}\right)} \]

8. Simplified 97.3%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(\varepsilon \cdot \varepsilon, 0.08333333333333333, -0.5\right), \varepsilon \cdot -0.5\right), \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)\right)} \]

9. Taylor expanded in eps around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. associate-*r* N/A

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

    4. unpow2 N/A

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

    5. associate-*r* N/A

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

    6. distribute-lft-out N/A

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

    7. lower-fma.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-+.f64 97.3

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

11. Simplified 97.3%

    \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(x \cdot -0.5, \varepsilon + x, 1\right)} \]
12. Add Preprocessing

Alternative 12: 97.8% accurate, 12.2× speedup

Math

\[\begin{array}{l} \\ \varepsilon \cdot \mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (* eps (fma -0.16666666666666666 (* eps eps) 1.0)))

C

double code(double x, double eps) {
	return eps * fma(-0.16666666666666666, (eps * eps), 1.0);
}

Julia

function code(x, eps)
	return Float64(eps * fma(-0.16666666666666666, Float64(eps * eps), 1.0))
end

Wolfram

code[x_, eps_] := N[(eps * N[(-0.16666666666666666 * N[(eps * eps), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.8%

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

3. Taylor expanded in eps around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-in N/A

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

   4. associate-+l+ N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \color{blue}{\frac{-1}{2} \cdot \sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   7. lower-sin.f64 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \color{blue}{\sin x}, \varepsilon \cdot \left(\frac{-1}{6} \cdot \left(\varepsilon \cdot \cos x\right)\right) + \cos x\right) \]

   8. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \varepsilon \cdot \color{blue}{\left(\left(\frac{-1}{6} \cdot \varepsilon\right) \cdot \cos x\right)} + \cos x\right) \]

   9. associate-*r* N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right)\right) \cdot \cos x} + \cos x\right) \]

   10. distribute-lft1-in N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, \frac{-1}{2} \cdot \sin x, \color{blue}{\left(\varepsilon \cdot \left(\frac{-1}{6} \cdot \varepsilon\right) + 1\right) \cdot \cos x}\right) \]

   12. lower-fma.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-cos.f64 99.6

       \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \color{blue}{\cos x}\right) \]

5. Simplified 99.6%

   \[\leadsto \color{blue}{\varepsilon \cdot \mathsf{fma}\left(\varepsilon, -0.5 \cdot \sin x, \mathsf{fma}\left(\varepsilon, \varepsilon \cdot -0.16666666666666666, 1\right) \cdot \cos x\right)} \]

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\left(\frac{-1}{6} \cdot {\varepsilon}^{2} + 1\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(\frac{-1}{6}, {\varepsilon}^{2}, 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(\frac{-1}{6}, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \]

   4. lower-*.f64 97.2

      \[\leadsto \varepsilon \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{\varepsilon \cdot \varepsilon}, 1\right) \]

8. Simplified 97.2%

   \[\leadsto \varepsilon \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \varepsilon \cdot \varepsilon, 1\right)} \]
9. Add Preprocessing

Alternative 13: 97.8% accurate, 207.0× 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 63.8%

   \[\sin \left(x + \varepsilon\right) - \sin x \]
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 98.4

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

5. Simplified 98.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \varepsilon \cdot \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 97.2%

   \[\leadsto \varepsilon \cdot \color{blue}{1} \]

9. Final simplification 97.2%

   \[\leadsto \varepsilon \]
10. 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 2024215 
(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)))