2sin (example 3.3)

Percentage Accurate: 42.7% → 99.7%

Time: 11.9s

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: 42.7% 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, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \varepsilon \cdot \sin x\\ t_1 := \tan \left(\varepsilon \cdot 0.5\right)\\ \mathsf{fma}\left(1, \sin \varepsilon \cdot \cos x, t_1 \cdot \left(\sin x \cdot \left(-\sin \varepsilon\right)\right)\right) + \mathsf{fma}\left(-t_1, t_0, t_1 \cdot t_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (let* ((t_0 (* (sin eps) (sin x))) (t_1 (tan (* eps 0.5))))
   (+
    (fma 1.0 (* (sin eps) (cos x)) (* t_1 (* (sin x) (- (sin eps)))))
    (fma (- t_1) t_0 (* t_1 t_0)))))

C

double code(double x, double eps) {
	double t_0 = sin(eps) * sin(x);
	double t_1 = tan((eps * 0.5));
	return fma(1.0, (sin(eps) * cos(x)), (t_1 * (sin(x) * -sin(eps)))) + fma(-t_1, t_0, (t_1 * t_0));
}

Julia

function code(x, eps)
	t_0 = Float64(sin(eps) * sin(x))
	t_1 = tan(Float64(eps * 0.5))
	return Float64(fma(1.0, Float64(sin(eps) * cos(x)), Float64(t_1 * Float64(sin(x) * Float64(-sin(eps))))) + fma(Float64(-t_1), t_0, Float64(t_1 * t_0)))
end

Wolfram

code[x_, eps_] := Block[{t$95$0 = N[(N[Sin[eps], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Tan[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(N[(1.0 * N[(N[Sin[eps], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[Sin[x], $MachinePrecision] * (-N[Sin[eps], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-t$95$1) * t$95$0 + N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 37.1%

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

   1. sin-sum 60.5%

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

   2. associate--l+ 60.6%

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

3. Applied egg-rr 60.6%

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

   1. +-commutative 60.6%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.2%

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

5. Simplified 99.2%

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

   1. flip-- 99.0%

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

   2. div-inv 99.0%

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

   3. metadata-eval 99.0%

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

   4. 1-sub-cos 99.4%

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

   5. pow2 99.4%

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

7. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

    1. *-commutative 99.4%

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

    2. *-lft-identity 99.4%

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

    3. times-frac 99.4%

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

    4. /-rgt-identity 99.4%

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

    5. unpow2 99.4%

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

    6. associate-*r/ 99.4%

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

    7. hang-0p-tan 99.7%

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

12. Simplified 99.7%

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

    1. *-un-lft-identity 99.7%

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

    2. associate-*r* 99.7%

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

    3. prod-diff 99.7%

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

    4. div-inv 99.7%

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

    5. metadata-eval 99.7%

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

    6. div-inv 99.7%

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

    7. metadata-eval 99.7%

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

14. Applied egg-rr 99.7%

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

15. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. sin-sum 60.5%

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

   2. associate--l+ 60.6%

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

3. Applied egg-rr 60.6%

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

   1. +-commutative 60.6%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.2%

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

5. Simplified 99.2%

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

   1. flip-- 99.0%

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

   2. div-inv 99.0%

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

   3. metadata-eval 99.0%

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

   4. 1-sub-cos 99.4%

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

   5. pow2 99.4%

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

7. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.4%

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

   2. *-rgt-identity 99.4%

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

9. Simplified 99.4%

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

10. Taylor expanded in x around inf 99.4%

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

    1. *-commutative 99.4%

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

    2. *-lft-identity 99.4%

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

    3. times-frac 99.4%

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

    4. /-rgt-identity 99.4%

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

    5. unpow2 99.4%

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

    6. associate-*r/ 99.4%

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

    7. hang-0p-tan 99.7%

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

12. Simplified 99.7%

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

    1. sub-neg 99.7%

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

    2. div-inv 99.7%

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

    3. metadata-eval 99.7%

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

14. Applied egg-rr 99.7%

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

    1. sub-neg 99.7%

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

    2. *-commutative 99.7%

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

    3. associate-*r* 99.7%

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

    4. *-commutative 99.7%

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

    5. associate-*r* 99.7%

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

    6. distribute-rgt-out-- 99.7%

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

16. Simplified 99.7%

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

17. Final simplification 99.7%

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

Alternative 3: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 37.1%

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

   1. add-cbrt-cube 30.8%

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

   2. pow3 30.8%

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

3. Applied egg-rr 30.8%

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

   1. rem-cbrt-cube 37.1%

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

   2. diff-sin 36.6%

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

   3. div-inv 36.6%

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

   4. +-commutative 36.6%

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

   5. associate--l+ 77.2%

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

   6. metadata-eval 77.2%

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

   7. div-inv 77.2%

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

   8. +-commutative 77.2%

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

   9. associate-+l+ 77.0%

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

   10. metadata-eval 77.0%

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

5. Applied egg-rr 77.0%

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

   1. associate-*r* 77.0%

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

   2. *-commutative 77.0%

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

   3. *-commutative 77.0%

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

   4. count-2 77.0%

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

   5. *-commutative 77.0%

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

   6. +-inverses 77.0%

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

   7. +-rgt-identity 77.0%

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

7. Simplified 77.0%

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

8. Final simplification 77.0%

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

Alternative 4: 76.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.106:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0033:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.106)
   (sin eps)
   (if (<= eps 0.0033) (* eps (cos x)) (sin (+ eps x)))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.106) {
		tmp = sin(eps);
	} else if (eps <= 0.0033) {
		tmp = eps * cos(x);
	} else {
		tmp = sin((eps + 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 <= (-0.106d0)) then
        tmp = sin(eps)
    else if (eps <= 0.0033d0) then
        tmp = eps * cos(x)
    else
        tmp = sin((eps + x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.106) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.0033) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin((eps + x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.106:
		tmp = math.sin(eps)
	elif eps <= 0.0033:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin((eps + x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.106)
		tmp = sin(eps);
	elseif (eps <= 0.0033)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(Float64(eps + x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.106)
		tmp = sin(eps);
	elseif (eps <= 0.0033)
		tmp = eps * cos(x);
	else
		tmp = sin((eps + x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.106], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.0033], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[N[(eps + x), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if eps < -0.105999999999999997

   1. Initial program 55.0%

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

   2. Taylor expanded in x around 0 57.0%

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

   if -0.105999999999999997 < eps < 0.0033

   1. Initial program 24.3%

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

   2. Taylor expanded in eps around 0 97.6%

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

   if 0.0033 < eps

   1. Initial program 48.6%

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

      1. log1p-expm1-u 48.6%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(x + \varepsilon\right)\right)\right)} - \sin x \]

   3. Applied egg-rr 48.6%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(x + \varepsilon\right)\right)\right)} - \sin x \]
   4. Step-by-step derivation

      1. log1p-expm1-u 48.6%

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

      2. add-sqr-sqrt 21.5%

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

      3. sqrt-unprod 27.1%

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

      4. pow2 27.1%

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

      5. +-commutative 27.1%

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

   5. Applied egg-rr 27.1%

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

      1. unpow2 27.1%

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

      2. rem-sqrt-square 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in x around 0 26.0%

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

      1. +-commutative 26.0%

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

      2. rem-square-sqrt 21.3%

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

      3. fabs-sqr 21.3%

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

      4. rem-square-sqrt 49.4%

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

   10. Simplified 49.4%

       \[\leadsto \color{blue}{\sin \left(x + \varepsilon\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.106:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0033:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\varepsilon + x\right)\\ \end{array} \]

Alternative 5: 56.1% 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 37.1%

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

2. Taylor expanded in x around 0 53.5%

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

3. Final simplification 53.5%

   \[\leadsto \sin \varepsilon \]

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 37.1%

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

   1. sin-sum 60.5%

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

   2. associate--l+ 60.6%

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

3. Applied egg-rr 60.6%

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

   1. +-commutative 60.6%

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

   2. associate-+l- 99.2%

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

   3. *-commutative 99.2%

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

   4. *-rgt-identity 99.2%

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

   5. distribute-lft-out-- 99.2%

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

5. Simplified 99.2%

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

6. Taylor expanded in x around 0 54.6%

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

7. Taylor expanded in eps around 0 30.7%

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

8. Final simplification 30.7%

   \[\leadsto \varepsilon \]

Developer target: 99.5% 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 2023318 
(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)))