2sin (example 3.3)

Percentage Accurate: 41.9% → 99.7%

Time: 16.9s

Alternatives: 6

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.9% 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, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.3%

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

   1. sin-sum 67.9%

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

   2. associate--l+ 67.9%

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

4. Applied egg-rr 67.9%

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

   1. +-commutative 67.9%

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

   2. associate-+l- 99.3%

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

   3. *-commutative 99.3%

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

   4. *-rgt-identity 99.3%

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

   5. distribute-lft-out-- 99.3%

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

6. Simplified 99.3%

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

   1. flip-- 99.2%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{1 \cdot 1 - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon}} \]

   2. associate-*r/ 99.2%

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

   3. metadata-eval 99.2%

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

   4. 1-sub-cos 99.5%

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

   5. pow2 99.5%

      \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\sin x \cdot \color{blue}{{\sin \varepsilon}^{2}}}{1 + \cos \varepsilon} \]

8. Applied egg-rr 99.5%

   \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
9. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\color{blue}{{\sin \varepsilon}^{2} \cdot \sin x}}{1 + \cos \varepsilon} \]

   2. associate-/l* 99.5%

      \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

10. Simplified 99.5%

    \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

11. Taylor expanded in eps around inf 99.5%

    \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2} \cdot \sin x}{1 + \cos \varepsilon}} \]
12. Step-by-step derivation

    1. associate-/l* 99.5%

       \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

    2. unpow2 99.5%

       \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{\frac{1 + \cos \varepsilon}{\sin x}} \]

    3. associate-*r/ 99.5%

       \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\sin \varepsilon \cdot \frac{\sin \varepsilon}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

    4. associate-/r/ 99.5%

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

    5. hang-0p-tan 99.6%

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

13. Simplified 99.6%

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

14. Final simplification 99.6%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.3%

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

   1. sin-sum 67.9%

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

   2. associate--l+ 67.9%

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

4. Applied egg-rr 67.9%

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

   1. +-commutative 67.9%

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

   2. associate-+l- 99.3%

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

   3. *-commutative 99.3%

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

   4. *-rgt-identity 99.3%

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

   5. distribute-lft-out-- 99.3%

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

6. Simplified 99.3%

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

   1. flip-- 99.2%

      \[\leadsto \sin \varepsilon \cdot \cos x - \sin x \cdot \color{blue}{\frac{1 \cdot 1 - \cos \varepsilon \cdot \cos \varepsilon}{1 + \cos \varepsilon}} \]

   2. associate-*r/ 99.2%

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

   3. metadata-eval 99.2%

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

   4. 1-sub-cos 99.5%

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

   5. pow2 99.5%

      \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\sin x \cdot \color{blue}{{\sin \varepsilon}^{2}}}{1 + \cos \varepsilon} \]

8. Applied egg-rr 99.5%

   \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{\sin x \cdot {\sin \varepsilon}^{2}}{1 + \cos \varepsilon}} \]
9. Step-by-step derivation

   1. *-commutative 99.5%

      \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\color{blue}{{\sin \varepsilon}^{2} \cdot \sin x}}{1 + \cos \varepsilon} \]

   2. associate-/l* 99.5%

      \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

10. Simplified 99.5%

    \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

11. Taylor expanded in eps around inf 99.5%

    \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2} \cdot \sin x}{1 + \cos \varepsilon}} \]
12. Step-by-step derivation

    1. associate-/l* 99.5%

       \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\frac{{\sin \varepsilon}^{2}}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

    2. unpow2 99.5%

       \[\leadsto \sin \varepsilon \cdot \cos x - \frac{\color{blue}{\sin \varepsilon \cdot \sin \varepsilon}}{\frac{1 + \cos \varepsilon}{\sin x}} \]

    3. associate-*r/ 99.5%

       \[\leadsto \sin \varepsilon \cdot \cos x - \color{blue}{\sin \varepsilon \cdot \frac{\sin \varepsilon}{\frac{1 + \cos \varepsilon}{\sin x}}} \]

    4. associate-/r/ 99.5%

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

    5. hang-0p-tan 99.6%

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

13. Simplified 99.6%

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

    1. distribute-lft-out-- 99.6%

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

    2. *-commutative 99.6%

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

    3. div-inv 99.6%

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

    4. metadata-eval 99.6%

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

15. Applied egg-rr 99.6%

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

16. Final simplification 99.6%

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

Alternative 3: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.3%

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

   1. add-sqr-sqrt 22.6%

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

   2. sqrt-unprod 22.4%

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

   3. pow2 22.4%

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

4. Applied egg-rr 22.4%

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

   1. sqrt-pow1 46.3%

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

   2. diff-sin 45.9%

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

   3. metadata-eval 45.9%

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

   4. pow1 45.9%

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

   5. div-inv 45.9%

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

   6. +-commutative 45.9%

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

   7. associate--l+ 78.7%

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

   8. metadata-eval 78.7%

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

   9. div-inv 78.7%

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

   10. +-commutative 78.7%

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

   11. associate-+l+ 78.7%

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

   12. metadata-eval 78.7%

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

6. Applied egg-rr 78.7%

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

   1. *-commutative 78.7%

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

   2. +-inverses 78.7%

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

   3. *-commutative 78.7%

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

   4. count-2 78.7%

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

8. Simplified 78.7%

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

9. Final simplification 78.7%

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

Alternative 4: 77.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -8e-6) (not (<= eps 1.9e-5))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -8e-6) || !(eps <= 1.9e-5)) {
		tmp = sin(eps);
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-8d-6)) .or. (.not. (eps <= 1.9d-5))) then
        tmp = sin(eps)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -8e-6) || !(eps <= 1.9e-5)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -8e-6) or not (eps <= 1.9e-5):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -8e-6) || !(eps <= 1.9e-5))
		tmp = sin(eps);
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -8e-6) || ~((eps <= 1.9e-5)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -8e-6], N[Not[LessEqual[eps, 1.9e-5]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -7.99999999999999964e-6 or 1.9000000000000001e-5 < eps

   1. Initial program 60.7%

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

   3. Taylor expanded in x around 0 61.7%

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

   if -7.99999999999999964e-6 < eps < 1.9000000000000001e-5

   1. Initial program 29.8%

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

   3. Taylor expanded in eps around 0 98.8%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -8 \cdot 10^{-6} \lor \neg \left(\varepsilon \leq 1.9 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in x around 0 56.5%

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

4. Final simplification 56.5%

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

Alternative 6: 29.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x eps) :precision binary64 eps)

C

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

Fortran

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

Java

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

Python

def code(x, eps):
	return eps

Julia

function code(x, eps)
	return eps
end

MATLAB

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

Wolfram

code[x_, eps_] := eps

Derivation

1. Initial program 46.3%

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

3. Taylor expanded in eps around 0 16.8%

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

4. Taylor expanded in x around 0 25.8%

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

5. Final simplification 25.8%

   \[\leadsto \varepsilon \]
6. Add Preprocessing

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024021 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (fma (sin x) (- (cos eps) 1.0) (* (sin eps) (cos x)))

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