2sin (example 3.3)

Percentage Accurate: 41.7% → 99.7%

Time: 12.4s

Alternatives: 11

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: 41.7% 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, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. sin-sum 68.8%

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

   2. associate--l+ 68.8%

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

3. Applied egg-rr 68.8%

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

   1. +-commutative 68.8%

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

   2. sub-neg 68.8%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. flip-+ 99.3%

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

   2. frac-2neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. sub-1-cos 99.5%

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

   5. pow2 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. remove-double-neg 99.5%

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

   2. unpow2 99.5%

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

   3. neg-mul-1 99.5%

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

   4. times-frac 99.5%

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

   5. +-commutative 99.5%

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

   6. hang-0p-tan 99.7%

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

9. Simplified 99.7%

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

    1. +-commutative 99.7%

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

    2. associate-*r* 99.7%

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

    3. fma-def 99.7%

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

    4. frac-2neg 99.7%

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

    5. metadata-eval 99.7%

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

    6. /-rgt-identity 99.7%

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

    7. div-inv 99.7%

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

    8. metadata-eval 99.7%

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

11. Applied egg-rr 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. sin-sum 68.8%

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

   2. associate--l+ 68.8%

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

3. Applied egg-rr 68.8%

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

   1. +-commutative 68.8%

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

   2. sub-neg 68.8%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. flip-+ 99.3%

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

   2. frac-2neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. sub-1-cos 99.5%

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

   5. pow2 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. remove-double-neg 99.5%

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

   2. unpow2 99.5%

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

   3. neg-mul-1 99.5%

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

   4. times-frac 99.5%

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

   5. +-commutative 99.5%

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

   6. hang-0p-tan 99.7%

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

9. Simplified 99.7%

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

    1. expm1-log1p-u 97.8%

       \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right)\right)} \]

    2. expm1-udef 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right)} - 1\right)} \]

    3. associate-*r* 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sin x \cdot \frac{\sin \varepsilon}{-1}\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)}\right)} - 1\right) \]

    4. *-commutative 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\color{blue}{\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{-1}\right)}\right)} - 1\right) \]

    5. div-inv 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{-1}\right)\right)} - 1\right) \]

    6. metadata-eval 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot \color{blue}{0.5}\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{-1}\right)\right)} - 1\right) \]

    7. frac-2neg 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \color{blue}{\frac{-\sin \varepsilon}{--1}}\right)\right)} - 1\right) \]

    8. metadata-eval 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\color{blue}{1}}\right)\right)} - 1\right) \]

    9. /-rgt-identity 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \color{blue}{\left(-\sin \varepsilon\right)}\right)\right)} - 1\right) \]

11. Applied egg-rr 97.3%

    \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \left(-\sin \varepsilon\right)\right)\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 97.7%

       \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \left(-\sin \varepsilon\right)\right)\right)\right)} \]

    2. expm1-log1p 99.7%

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

    3. associate-*r* 99.7%

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

    4. *-commutative 99.7%

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

    5. *-commutative 99.7%

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

    6. *-commutative 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 3: 99.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

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) * sin(eps)))
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) * Math.sin(eps)));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. sin-sum 68.8%

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

   2. associate--l+ 68.8%

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

3. Applied egg-rr 68.8%

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

   1. +-commutative 68.8%

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

   2. sub-neg 68.8%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. flip-+ 99.3%

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

   2. frac-2neg 99.3%

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

   3. metadata-eval 99.3%

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

   4. sub-1-cos 99.5%

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

   5. pow2 99.5%

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

   6. sub-neg 99.5%

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

   7. metadata-eval 99.5%

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

7. Applied egg-rr 99.5%

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

   1. remove-double-neg 99.5%

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

   2. unpow2 99.5%

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

   3. neg-mul-1 99.5%

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

   4. times-frac 99.5%

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

   5. +-commutative 99.5%

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

   6. hang-0p-tan 99.7%

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

9. Simplified 99.7%

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

    1. expm1-log1p-u 97.8%

       \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right)\right)} \]

    2. expm1-udef 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(e^{\mathsf{log1p}\left(\sin x \cdot \left(\frac{\sin \varepsilon}{-1} \cdot \tan \left(\frac{\varepsilon}{2}\right)\right)\right)} - 1\right)} \]

    3. associate-*r* 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\color{blue}{\left(\sin x \cdot \frac{\sin \varepsilon}{-1}\right) \cdot \tan \left(\frac{\varepsilon}{2}\right)}\right)} - 1\right) \]

    4. *-commutative 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\color{blue}{\tan \left(\frac{\varepsilon}{2}\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{-1}\right)}\right)} - 1\right) \]

    5. div-inv 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \color{blue}{\left(\varepsilon \cdot \frac{1}{2}\right)} \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{-1}\right)\right)} - 1\right) \]

    6. metadata-eval 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot \color{blue}{0.5}\right) \cdot \left(\sin x \cdot \frac{\sin \varepsilon}{-1}\right)\right)} - 1\right) \]

    7. frac-2neg 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \color{blue}{\frac{-\sin \varepsilon}{--1}}\right)\right)} - 1\right) \]

    8. metadata-eval 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \frac{-\sin \varepsilon}{\color{blue}{1}}\right)\right)} - 1\right) \]

    9. /-rgt-identity 97.3%

       \[\leadsto \sin \varepsilon \cdot \cos x + \left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \color{blue}{\left(-\sin \varepsilon\right)}\right)\right)} - 1\right) \]

11. Applied egg-rr 97.3%

    \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\left(e^{\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \left(-\sin \varepsilon\right)\right)\right)} - 1\right)} \]
12. Step-by-step derivation

    1. expm1-def 97.7%

       \[\leadsto \sin \varepsilon \cdot \cos x + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(\varepsilon \cdot 0.5\right) \cdot \left(\sin x \cdot \left(-\sin \varepsilon\right)\right)\right)\right)} \]

    2. expm1-log1p 99.7%

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

    3. *-commutative 99.7%

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

    4. distribute-rgt-neg-out 99.7%

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

    5. *-commutative 99.7%

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

    6. distribute-rgt-neg-out 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. sin-sum 68.8%

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

   2. associate--l+ 68.8%

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

3. Applied egg-rr 68.8%

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

   1. +-commutative 68.8%

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

   2. sub-neg 68.8%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

   1. fma-def 99.5%

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

   2. *-commutative 99.5%

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

7. Applied egg-rr 99.5%

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

8. Final simplification 99.5%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.5%

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

   1. sin-sum 68.8%

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

   2. associate--l+ 68.8%

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

3. Applied egg-rr 68.8%

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

   1. +-commutative 68.8%

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

   2. sub-neg 68.8%

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

   3. associate-+l+ 99.4%

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

   4. *-commutative 99.4%

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

   5. neg-mul-1 99.4%

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

   6. *-commutative 99.4%

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

   7. distribute-rgt-out 99.4%

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

   8. +-commutative 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 6: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0022 \lor \neg \left(\varepsilon \leq 0.00105\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0022) (not (<= eps 0.00105)))
   (- (sin eps) (sin x))
   (+ (* -0.5 (* (sin x) (* eps eps))) (* eps (cos x)))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0022) || !(eps <= 0.00105)) {
		tmp = sin(eps) - sin(x);
	} else {
		tmp = (-0.5 * (sin(x) * (eps * eps))) + (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 <= (-0.0022d0)) .or. (.not. (eps <= 0.00105d0))) then
        tmp = sin(eps) - sin(x)
    else
        tmp = ((-0.5d0) * (sin(x) * (eps * eps))) + (eps * cos(x))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0022) || !(eps <= 0.00105)) {
		tmp = Math.sin(eps) - Math.sin(x);
	} else {
		tmp = (-0.5 * (Math.sin(x) * (eps * eps))) + (eps * Math.cos(x));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0022) or not (eps <= 0.00105):
		tmp = math.sin(eps) - math.sin(x)
	else:
		tmp = (-0.5 * (math.sin(x) * (eps * eps))) + (eps * math.cos(x))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0022) || !(eps <= 0.00105))
		tmp = Float64(sin(eps) - sin(x));
	else
		tmp = Float64(Float64(-0.5 * Float64(sin(x) * Float64(eps * eps))) + Float64(eps * cos(x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0022) || ~((eps <= 0.00105)))
		tmp = sin(eps) - sin(x);
	else
		tmp = (-0.5 * (sin(x) * (eps * eps))) + (eps * cos(x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0022], N[Not[LessEqual[eps, 0.00105]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(N[Sin[x], $MachinePrecision] * N[(eps * eps), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00220000000000000013 or 0.00104999999999999994 < eps

   1. Initial program 59.4%

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

      1. log1p-expm1-u 59.3%

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

   3. Applied egg-rr 59.3%

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

   4. Taylor expanded in x around 0 60.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin \varepsilon} - 1}\right) - \sin x \]
   5. Step-by-step derivation

      1. expm1-def 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in eps around inf 61.1%

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

   if -0.00220000000000000013 < eps < 0.00104999999999999994

   1. Initial program 31.5%

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

   2. Taylor expanded in eps around 0 99.2%

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

      1. +-commutative 99.2%

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

      2. fma-def 99.2%

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

      3. unpow2 99.2%

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

      4. associate-*l* 99.2%

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

      5. *-commutative 99.2%

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

   4. Simplified 99.2%

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

      1. fma-udef 99.2%

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

      2. associate-*r* 99.2%

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

   6. Applied egg-rr 99.2%

      \[\leadsto \color{blue}{-0.5 \cdot \left(\left(\varepsilon \cdot \varepsilon\right) \cdot \sin x\right) + \varepsilon \cdot \cos x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0022 \lor \neg \left(\varepsilon \leq 0.00105\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\sin x \cdot \left(\varepsilon \cdot \varepsilon\right)\right) + \varepsilon \cdot \cos x\\ \end{array} \]

Alternative 7: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 + x)))))
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 + x)))));
}

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 46.5%

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

   1. diff-sin 45.8%

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

   2. div-inv 45.8%

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

   3. metadata-eval 45.8%

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

   4. div-inv 45.8%

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

   5. +-commutative 45.8%

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

   6. metadata-eval 45.8%

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

3. Applied egg-rr 45.8%

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

   1. *-commutative 45.8%

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

   2. +-commutative 45.8%

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

   3. associate--l+ 77.6%

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

   4. +-inverses 77.6%

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

   5. distribute-lft-in 77.6%

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

   6. metadata-eval 77.6%

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

   7. *-commutative 77.6%

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

   8. associate-+r+ 77.7%

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

   9. +-commutative 77.7%

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

5. Simplified 77.7%

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

6. Final simplification 77.7%

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

Alternative 8: 77.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00024 \lor \neg \left(\varepsilon \leq 0.001\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.00024) (not (<= eps 0.001)))
   (- (sin eps) (sin x))
   (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00024) || !(eps <= 0.001)) {
		tmp = sin(eps) - sin(x);
	} 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 <= (-0.00024d0)) .or. (.not. (eps <= 0.001d0))) then
        tmp = sin(eps) - sin(x)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.00024) || !(eps <= 0.001)) {
		tmp = Math.sin(eps) - Math.sin(x);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.00024) or not (eps <= 0.001):
		tmp = math.sin(eps) - math.sin(x)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.00024) || !(eps <= 0.001))
		tmp = Float64(sin(eps) - sin(x));
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.00024) || ~((eps <= 0.001)))
		tmp = sin(eps) - sin(x);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.00024], N[Not[LessEqual[eps, 0.001]], $MachinePrecision]], N[(N[Sin[eps], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.40000000000000006e-4 or 1e-3 < eps

   1. Initial program 59.4%

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

      1. log1p-expm1-u 59.3%

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

   3. Applied egg-rr 59.3%

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

   4. Taylor expanded in x around 0 60.5%

      \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin \varepsilon} - 1}\right) - \sin x \]
   5. Step-by-step derivation

      1. expm1-def 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in eps around inf 61.1%

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

   if -2.40000000000000006e-4 < eps < 1e-3

   1. Initial program 31.5%

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

   2. Taylor expanded in eps around 0 98.5%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.00024 \lor \neg \left(\varepsilon \leq 0.001\right):\\ \;\;\;\;\sin \varepsilon - \sin x\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9: 76.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000235:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0013:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (<= eps -0.000235)
   (sin eps)
   (if (<= eps 0.0013) (* eps (cos x)) (sin eps))))

C

double code(double x, double eps) {
	double tmp;
	if (eps <= -0.000235) {
		tmp = sin(eps);
	} else if (eps <= 0.0013) {
		tmp = eps * cos(x);
	} else {
		tmp = sin(eps);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if (eps <= (-0.000235d0)) then
        tmp = sin(eps)
    else if (eps <= 0.0013d0) then
        tmp = eps * cos(x)
    else
        tmp = sin(eps)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if (eps <= -0.000235) {
		tmp = Math.sin(eps);
	} else if (eps <= 0.0013) {
		tmp = eps * Math.cos(x);
	} else {
		tmp = Math.sin(eps);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if eps <= -0.000235:
		tmp = math.sin(eps)
	elif eps <= 0.0013:
		tmp = eps * math.cos(x)
	else:
		tmp = math.sin(eps)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if (eps <= -0.000235)
		tmp = sin(eps);
	elseif (eps <= 0.0013)
		tmp = Float64(eps * cos(x));
	else
		tmp = sin(eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if (eps <= -0.000235)
		tmp = sin(eps);
	elseif (eps <= 0.0013)
		tmp = eps * cos(x);
	else
		tmp = sin(eps);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[LessEqual[eps, -0.000235], N[Sin[eps], $MachinePrecision], If[LessEqual[eps, 0.0013], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision], N[Sin[eps], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if eps < -2.34999999999999993e-4 or 0.0012999999999999999 < eps

   1. Initial program 59.4%

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

   2. Taylor expanded in x around 0 59.7%

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

   if -2.34999999999999993e-4 < eps < 0.0012999999999999999

   1. Initial program 31.5%

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

   2. Taylor expanded in eps around 0 98.5%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.000235:\\ \;\;\;\;\sin \varepsilon\\ \mathbf{elif}\;\varepsilon \leq 0.0013:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon\\ \end{array} \]

Alternative 10: 54.6% 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 46.5%

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

2. Taylor expanded in x around 0 55.9%

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

3. Final simplification 55.9%

   \[\leadsto \sin \varepsilon \]

Alternative 11: 28.9% 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 46.5%

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

2. Taylor expanded in eps around 0 47.7%

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

3. Taylor expanded in x around 0 25.8%

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

4. Final simplification 25.8%

   \[\leadsto \varepsilon \]

Developer target: 76.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023255 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

  :herbie-target
  (* 2.0 (* (cos (+ x (/ eps 2.0))) (sin (/ eps 2.0))))

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