2cos (problem 3.3.5)

Percentage Accurate: 39.2% → 90.6%

Time: 14.3s

Alternatives: 9

Speedup: 2.0×

Specification

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: 39.2% 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: 90.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. cos-sum 65.7%

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

   2. cancel-sign-sub-inv 65.7%

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

   3. fma-def 65.8%

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

4. Applied egg-rr 65.8%

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

5. Taylor expanded in x around inf 65.7%

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

   1. associate--l+ 91.5%

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

   2. mul-1-neg 91.5%

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

   3. *-commutative 91.5%

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

   4. distribute-rgt-neg-in 91.5%

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

   5. fma-def 91.5%

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

   6. *-lft-identity 91.5%

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

   7. distribute-rgt-out-- 91.6%

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

   8. sub-neg 91.6%

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

   9. metadata-eval 91.6%

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

   10. +-commutative 91.6%

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

7. Simplified 91.6%

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

8. Final simplification 91.6%

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

Alternative 2: 90.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. cos-sum 65.7%

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

   2. cancel-sign-sub-inv 65.7%

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

   3. fma-def 65.8%

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

4. Applied egg-rr 65.8%

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

5. Taylor expanded in x around inf 65.7%

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

   1. associate--l+ 91.5%

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

   2. mul-1-neg 91.5%

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

   3. *-commutative 91.5%

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

   4. distribute-rgt-neg-in 91.5%

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

   5. fma-def 91.5%

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

   6. *-lft-identity 91.5%

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

   7. distribute-rgt-out-- 91.6%

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

   8. sub-neg 91.6%

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

   9. metadata-eval 91.6%

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

   10. +-commutative 91.6%

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

7. Simplified 91.6%

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

8. Taylor expanded in x around inf 91.5%

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

   1. +-commutative 91.5%

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

   2. *-commutative 91.5%

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

   3. sub-neg 91.5%

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

   4. metadata-eval 91.5%

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

   5. +-commutative 91.5%

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

   6. mul-1-neg 91.5%

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

   7. *-commutative 91.5%

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

   8. unsub-neg 91.5%

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

   9. +-commutative 91.5%

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

   10. *-commutative 91.5%

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

10. Simplified 91.5%

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

11. Final simplification 91.5%

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

Alternative 3: 77.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. add-cube-cbrt 38.8%

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}} \]

   2. pow3 38.8%

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\cos x}\right)}^{3}} \]

4. Applied egg-rr 38.8%

   \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\cos x}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 38.8%

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

   2. diff-cos 45.8%

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

6. Applied egg-rr 45.8%

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

   1. *-commutative 45.8%

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

   2. *-commutative 45.8%

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

   3. associate-*l* 45.8%

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

   4. +-commutative 45.8%

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

   5. associate-+l+ 45.8%

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

   6. +-commutative 45.8%

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

   7. associate--l+ 72.6%

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

8. Simplified 72.6%

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

9. Final simplification 72.6%

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

Alternative 4: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. add-cube-cbrt 38.5%

      \[\leadsto \color{blue}{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x} \cdot \sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right) \cdot \sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}} \]

   2. pow3 38.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^{3}} \]

4. Applied egg-rr 38.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\cos \left(x + \varepsilon\right) - \cos x}\right)}^{3}} \]
5. Step-by-step derivation

   1. rem-cube-cbrt 38.8%

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

   2. diff-cos 45.8%

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

   3. div-inv 45.8%

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

   4. +-commutative 45.8%

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

   5. associate--l+ 72.7%

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

   6. metadata-eval 72.7%

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

   7. div-inv 72.7%

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

   8. +-commutative 72.7%

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

   9. associate-+l+ 72.6%

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

   10. metadata-eval 72.6%

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

6. Applied egg-rr 72.6%

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

   1. *-commutative 72.6%

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

   2. +-inverses 72.6%

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

   3. *-commutative 72.6%

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

   4. count-2 72.6%

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

8. Simplified 72.6%

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

9. Final simplification 72.6%

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

Alternative 5: 47.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

   1. diff-cos 45.8%

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

   2. div-inv 45.8%

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

   3. associate--l+ 45.8%

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

   4. metadata-eval 45.8%

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

   5. div-inv 45.8%

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

   6. +-commutative 45.8%

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

   7. associate-+l+ 45.8%

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

   8. metadata-eval 45.8%

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

4. Applied egg-rr 45.8%

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

   1. associate-*r* 45.8%

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

   2. *-commutative 45.8%

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

   3. *-commutative 45.8%

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

   4. +-commutative 45.8%

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

   5. count-2 45.8%

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

   6. fma-def 45.8%

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

   7. *-commutative 45.8%

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

   8. associate-+r- 45.8%

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

   9. +-commutative 45.8%

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

   10. associate--l+ 72.6%

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

   11. +-inverses 72.6%

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

6. Simplified 72.6%

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

7. Taylor expanded in x around 0 46.7%

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

8. Final simplification 46.7%

   \[\leadsto -2 \cdot {\sin \left(\varepsilon \cdot 0.5\right)}^{2} \]
9. Add Preprocessing

Alternative 6: 41.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in eps around 0 39.7%

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

   1. mul-1-neg 39.7%

      \[\leadsto \color{blue}{-\varepsilon \cdot \sin x} \]

   2. *-commutative 39.7%

      \[\leadsto -\color{blue}{\sin x \cdot \varepsilon} \]

   3. distribute-rgt-neg-in 39.7%

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

5. Simplified 39.7%

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

6. Final simplification 39.7%

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

Alternative 7: 39.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -1 + \cos \varepsilon \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in x around 0 38.9%

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

4. Final simplification 38.9%

   \[\leadsto -1 + \cos \varepsilon \]
5. Add Preprocessing

Alternative 8: 18.3% accurate, 51.3× speedup

Math

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

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(eps * Float64(-x))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.8%

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

3. Taylor expanded in eps around 0 13.9%

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

   1. mul-1-neg 13.9%

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

   2. unsub-neg 13.9%

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

5. Simplified 13.9%

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

6. Taylor expanded in x around 0 18.1%

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

   1. associate-*r* 18.1%

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

   2. neg-mul-1 18.1%

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

8. Simplified 18.1%

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

9. Final simplification 18.1%

   \[\leadsto \varepsilon \cdot \left(-x\right) \]
10. Add Preprocessing

Alternative 9: 13.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 0.0)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return 0.0

Julia

function code(x, eps)
	return 0.0
end

MATLAB

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

Wolfram

code[x_, eps_] := 0.0

Derivation

1. Initial program 38.8%

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

   1. add-cube-cbrt 38.8%

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}} \]

   2. pow3 38.8%

      \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\cos x}\right)}^{3}} \]

4. Applied egg-rr 38.8%

   \[\leadsto \cos \left(x + \varepsilon\right) - \color{blue}{{\left(\sqrt[3]{\cos x}\right)}^{3}} \]

5. Taylor expanded in eps around 0 13.5%

   \[\leadsto \color{blue}{\cos x - {1}^{0.3333333333333333} \cdot \cos x} \]
6. Step-by-step derivation

   1. pow-base-1 13.5%

      \[\leadsto \cos x - \color{blue}{1} \cdot \cos x \]

   2. *-lft-identity 13.5%

      \[\leadsto \cos x - \color{blue}{\cos x} \]

   3. +-inverses 13.5%

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

7. Simplified 13.5%

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

8. Final simplification 13.5%

   \[\leadsto 0 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(FPCore (x eps)
  :name "2cos (problem 3.3.5)"
  :precision binary64
  (- (cos (+ x eps)) (cos x)))