2cos (problem 3.3.5)

Percentage Accurate: 52.2% → 99.8%

Time: 15.2s

Alternatives: 11

Speedup: 25.9×

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. lower-sin.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\left(\sin \left(\mathsf{fma}\left(0.5, \varepsilon, x\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)\right)} \cdot -2 \]
8. Step-by-step derivation

9. Applied rewrites 99.7%

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

Alternative 2: 99.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

   1. lift--.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. lift-cos.f64 N/A

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

   4. diff-cos N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

4. Applied rewrites 78.4%

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

5. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. distribute-rgt-in N/A

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

   5. *-commutative N/A

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

   6. associate-*l* N/A

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

   7. metadata-eval N/A

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

   8. *-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. lower-sin.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-sin.f64 N/A

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

   14. lower-*.f64 99.7

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

7. Applied rewrites 99.7%

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

Alternative 3: 99.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.1%

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

Alternative 4: 98.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in eps around 0

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(\left(\frac{1}{24} \cdot \left(\varepsilon \cdot \varepsilon\right) - \frac{1}{2}\right) \cdot \varepsilon\right) \cdot x, \frac{-1}{2}, \left(\varepsilon \cdot \varepsilon\right) \cdot \frac{1}{6}\right), x, \frac{-1}{2} \cdot \varepsilon\right) - \sin x\right) \cdot \varepsilon \]
10. Step-by-step derivation

11. Applied rewrites 98.4%

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

Alternative 5: 98.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.16666666666666666 \cdot \varepsilon, x, 0.25 \cdot \left(x \cdot x\right) - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (-
   (* (fma (* 0.16666666666666666 eps) x (- (* 0.25 (* x x)) 0.5)) eps)
   (sin x))
  eps))

C

double code(double x, double eps) {
	return ((fma((0.16666666666666666 * eps), x, ((0.25 * (x * x)) - 0.5)) * eps) - sin(x)) * eps;
}

Julia

function code(x, eps)
	return Float64(Float64(Float64(fma(Float64(0.16666666666666666 * eps), x, Float64(Float64(0.25 * Float64(x * x)) - 0.5)) * eps) - sin(x)) * eps)
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(N[(0.16666666666666666 * eps), $MachinePrecision] * x + N[(N[(0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] * eps), $MachinePrecision]

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.4%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.4%

    \[\leadsto \left(\mathsf{fma}\left(0.16666666666666666 \cdot \varepsilon, x, 0.25 \cdot \left(x \cdot x\right) - 0.5\right) \cdot \varepsilon - \sin x\right) \cdot \varepsilon \]
12. Add Preprocessing

Alternative 6: 98.9% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in eps around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.3%

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

9. Taylor expanded in eps around 0

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

11. Applied rewrites 98.3%

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

Alternative 7: 98.3% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \varepsilon, -0.16666666666666666 \cdot x\right), x, 1\right), x, 0.5 \cdot \varepsilon\right) \cdot \left(-\varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (*
  (fma (fma (fma -0.25 eps (* -0.16666666666666666 x)) x 1.0) x (* 0.5 eps))
  (- eps)))

C

double code(double x, double eps) {
	return fma(fma(fma(-0.25, eps, (-0.16666666666666666 * x)), x, 1.0), x, (0.5 * eps)) * -eps;
}

Julia

function code(x, eps)
	return Float64(fma(fma(fma(-0.25, eps, Float64(-0.16666666666666666 * x)), x, 1.0), x, Float64(0.5 * eps)) * Float64(-eps))
end

Wolfram

code[x_, eps_] := N[(N[(N[(N[(-0.25 * eps + N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] * x + 1.0), $MachinePrecision] * x + N[(0.5 * eps), $MachinePrecision]), $MachinePrecision] * (-eps)), $MachinePrecision]

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 97.2%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.25, \varepsilon, -0.16666666666666666 \cdot x\right), x, 1\right), x, 0.5 \cdot \varepsilon\right) \cdot \left(-\color{blue}{\varepsilon}\right) \]
9. Add Preprocessing

Alternative 8: 98.0% accurate, 10.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.4%

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

Alternative 9: 97.8% accurate, 14.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in eps around 0

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

   1. remove-double-neg N/A

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

   2. distribute-lft-neg-out N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-*.f64 N/A

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

5. Applied rewrites 99.1%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 96.2%

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

Alternative 10: 79.0% accurate, 25.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in eps around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. lower-*.f64 N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-sin.f64 78.8

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

5. Applied rewrites 78.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 77.1%

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

Alternative 11: 50.9% accurate, 51.8× speedup

Math

\[\begin{array}{l} \\ 1 - 1 \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (- 1.0 1.0))

C

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

Fortran

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

Java

public static double code(double x, double eps) {
	return 1.0 - 1.0;
}

Python

def code(x, eps):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

code[x_, eps_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 49.1%

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

3. Taylor expanded in x around 0

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

   1. lower--.f64 N/A

      \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

   2. lower-cos.f64 47.0

      \[\leadsto \color{blue}{\cos \varepsilon} - 1 \]

5. Applied rewrites 47.0%

   \[\leadsto \color{blue}{\cos \varepsilon - 1} \]

6. Taylor expanded in eps around 0

   \[\leadsto 1 - 1 \]
7. Step-by-step derivation

8. Applied rewrites 46.9%

   \[\leadsto 1 - 1 \]
9. Add Preprocessing

Developer Target 1: 98.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt[3]{\left(-2 \cdot \sin \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right)\right) \cdot \sin \left(0.5 \cdot \varepsilon\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (pow (cbrt (* (* -2.0 (sin (* 0.5 (fma 2.0 x eps)))) (sin (* 0.5 eps)))) 3.0))

C

double code(double x, double eps) {
	return pow(cbrt(((-2.0 * sin((0.5 * fma(2.0, x, eps)))) * sin((0.5 * eps)))), 3.0);
}

Julia

function code(x, eps)
	return cbrt(Float64(Float64(-2.0 * sin(Float64(0.5 * fma(2.0, x, eps)))) * sin(Float64(0.5 * eps)))) ^ 3.0
end

Wolfram

code[x_, eps_] := N[Power[N[Power[N[(N[(-2.0 * N[Sin[N[(0.5 * N[(2.0 * x + eps), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(0.5 * eps), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Reproduce

herbie shell --seed 2024337 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  :pre (and (and (and (<= -10000.0 x) (<= x 10000.0)) (< (* 1e-16 (fabs x)) eps)) (< eps (fabs x)))

  :alt
  (! :herbie-platform default (pow (cbrt (* -2 (sin (* 1/2 (fma 2 x eps))) (sin (* 1/2 eps)))) 3))

  (- (cos (+ x eps)) (cos x)))