2sin (example 3.3)

Percentage Accurate: 41.9% → 99.6%

Time: 15.7s

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: 41.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. sin-sum 71.0%

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

   2. associate--l+ 71.0%

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

3. Applied egg-rr 71.0%

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

   1. associate-+r- 71.0%

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

   2. +-commutative 71.0%

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

   3. associate-+r- 99.3%

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

   4. *-commutative 99.3%

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

   5. sub-neg 99.3%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around inf 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. sub-neg 99.4%

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

   4. metadata-eval 99.4%

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

   5. +-commutative 99.4%

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

   6. fma-def 99.4%

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

   7. *-commutative 99.4%

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

8. Simplified 99.4%

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

   1. flip-+ 99.3%

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

   2. clear-num 99.3%

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

   3. sub-neg 99.3%

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

   4. metadata-eval 99.3%

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

   5. metadata-eval 99.3%

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

   6. sub-1-cos 99.6%

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

   7. pow2 99.6%

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

10. Applied egg-rr 99.6%

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

11. Taylor expanded in eps around inf 99.6%

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

    1. mul-1-neg 99.6%

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

    2. unpow2 99.6%

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

    3. associate-*r/ 99.6%

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

    4. hang-0p-tan 99.7%

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

13. Simplified 99.7%

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

14. Final simplification 99.7%

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

Alternative 2: 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 44.8%

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

   1. sin-sum 71.0%

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

   2. associate--l+ 71.0%

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

3. Applied egg-rr 71.0%

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

   1. associate-+r- 71.0%

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

   2. +-commutative 71.0%

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

   3. associate-+r- 99.3%

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

   4. *-commutative 99.3%

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

   5. sub-neg 99.3%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.4%

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

5. Simplified 99.4%

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

6. Final simplification 99.4%

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

Alternative 3: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. sin-sum 71.0%

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

   2. associate--l+ 71.0%

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

3. Applied egg-rr 71.0%

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

   1. associate-+r- 71.0%

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

   2. +-commutative 71.0%

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

   3. associate-+r- 99.3%

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

   4. *-commutative 99.3%

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

   5. sub-neg 99.3%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.4%

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

5. Simplified 99.4%

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

6. Taylor expanded in eps around inf 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. sub-neg 99.4%

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

   4. metadata-eval 99.4%

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

   5. +-commutative 99.4%

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

   6. fma-def 99.4%

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

   7. *-commutative 99.4%

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

8. Simplified 99.4%

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

9. Final simplification 99.4%

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

Alternative 4: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. sin-sum 71.0%

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

   2. associate--l+ 71.0%

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

3. Applied egg-rr 71.0%

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

   1. associate-+r- 71.0%

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

   2. +-commutative 71.0%

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

   3. associate-+r- 99.3%

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

   4. *-commutative 99.3%

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

   5. sub-neg 99.3%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.4%

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

5. Simplified 99.4%

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

   1. fma-udef 99.4%

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

   2. +-commutative 99.4%

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

   3. *-commutative 99.4%

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

7. Applied egg-rr 99.4%

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

8. Final simplification 99.4%

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

Alternative 5: 77.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. sin-sum 71.0%

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

   2. associate--l+ 71.0%

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

3. Applied egg-rr 71.0%

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

   1. associate-+r- 71.0%

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

   2. +-commutative 71.0%

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

   3. associate-+r- 99.3%

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

   4. *-commutative 99.3%

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

   5. sub-neg 99.3%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.4%

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

5. Simplified 99.4%

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

   1. fma-udef 99.4%

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

   2. distribute-lft-in 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-+r+ 71.0%

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

   5. *-commutative 71.0%

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

   6. sin-sum 44.8%

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

   7. +-commutative 44.8%

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

   8. expm1-log1p 44.8%

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

   9. metadata-eval 44.8%

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

   10. cancel-sign-sub-inv 44.8%

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

   11. *-un-lft-identity 44.8%

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

   12. expm1-log1p 44.8%

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

   13. +-commutative 44.8%

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

   14. sin-sum 71.0%

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

   15. *-commutative 71.0%

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

   16. associate--l+ 99.3%

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

7. Applied egg-rr 99.3%

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

8. Taylor expanded in eps around 0 75.5%

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

9. Final simplification 75.5%

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

Alternative 6: 76.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. sin-sum 71.0%

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

   2. associate--l+ 71.0%

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

3. Applied egg-rr 71.0%

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

   1. associate-+r- 71.0%

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

   2. +-commutative 71.0%

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

   3. associate-+r- 99.3%

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

   4. *-commutative 99.3%

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

   5. sub-neg 99.3%

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

   6. fma-def 99.3%

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

   7. *-commutative 99.3%

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

   8. neg-mul-1 99.3%

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

   9. distribute-rgt-out 99.4%

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

5. Simplified 99.4%

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

   1. fma-udef 99.4%

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

   2. distribute-lft-in 99.3%

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

   3. *-commutative 99.3%

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

   4. associate-+r+ 71.0%

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

   5. *-commutative 71.0%

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

   6. sin-sum 44.8%

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

   7. +-commutative 44.8%

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

   8. expm1-log1p 44.8%

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

   9. metadata-eval 44.8%

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

   10. cancel-sign-sub-inv 44.8%

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

   11. *-un-lft-identity 44.8%

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

   12. expm1-log1p 44.8%

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

   13. diff-sin 44.2%

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

   14. +-commutative 44.2%

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

7. Applied egg-rr 44.2%

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

   1. associate--l+ 74.2%

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

9. Simplified 74.2%

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

10. Final simplification 74.2%

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

Alternative 7: 76.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 44.8%

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

   1. diff-sin 44.2%

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

   2. div-inv 44.2%

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

   3. +-commutative 44.2%

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

   4. associate--l+ 74.2%

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

   5. *-un-lft-identity 74.2%

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

   6. *-un-lft-identity 74.2%

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

   7. distribute-rgt-out-- 74.2%

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

   8. metadata-eval 74.2%

      \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot \color{blue}{0}\right) \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\left(x + \varepsilon\right) + x}{2}\right)\right) \]

   9. metadata-eval 74.2%

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

   10. div-inv 74.2%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \color{blue}{\left(\left(\left(x + \varepsilon\right) + x\right) \cdot \frac{1}{2}\right)}\right) \]

   11. +-commutative 74.2%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\left(\color{blue}{\left(\varepsilon + x\right)} + x\right) \cdot \frac{1}{2}\right)\right) \]

   12. associate-+l+ 74.1%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\color{blue}{\left(\varepsilon + \left(x + x\right)\right)} \cdot \frac{1}{2}\right)\right) \]

   13. count-2 74.1%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\left(\varepsilon + \color{blue}{2 \cdot x}\right) \cdot \frac{1}{2}\right)\right) \]

   14. *-commutative 74.1%

       \[\leadsto 2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\left(\varepsilon + \color{blue}{x \cdot 2}\right) \cdot \frac{1}{2}\right)\right) \]

   15. metadata-eval 74.1%

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

3. Applied egg-rr 74.1%

   \[\leadsto \color{blue}{2 \cdot \left(\sin \left(\left(\varepsilon + x \cdot 0\right) \cdot 0.5\right) \cdot \cos \left(\left(\varepsilon + x \cdot 2\right) \cdot 0.5\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative 74.1%

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

   2. associate-*l* 74.1%

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

   3. mul0-rgt 74.1%

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

   4. *-commutative 74.1%

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

5. Simplified 74.1%

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

6. Final simplification 74.1%

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

Alternative 8: 75.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.76 \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.76) (not (<= eps 4.5e-6))) (sin eps) (* eps (cos x))))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.76) || !(eps <= 4.5e-6)) {
		tmp = sin(eps);
	} else {
		tmp = eps * cos(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((eps <= (-0.76d0)) .or. (.not. (eps <= 4.5d-6))) then
        tmp = sin(eps)
    else
        tmp = eps * cos(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.76) || !(eps <= 4.5e-6)) {
		tmp = Math.sin(eps);
	} else {
		tmp = eps * Math.cos(x);
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.76) or not (eps <= 4.5e-6):
		tmp = math.sin(eps)
	else:
		tmp = eps * math.cos(x)
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.76) || !(eps <= 4.5e-6))
		tmp = sin(eps);
	else
		tmp = Float64(eps * cos(x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.76) || ~((eps <= 4.5e-6)))
		tmp = sin(eps);
	else
		tmp = eps * cos(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.76], N[Not[LessEqual[eps, 4.5e-6]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(eps * N[Cos[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.76000000000000001 or 4.50000000000000011e-6 < eps

   1. Initial program 52.3%

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

   2. Taylor expanded in x around 0 53.2%

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

   if -0.76000000000000001 < eps < 4.50000000000000011e-6

   1. Initial program 36.0%

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

   2. Taylor expanded in eps around 0 98.5%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.76 \lor \neg \left(\varepsilon \leq 4.5 \cdot 10^{-6}\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\varepsilon \cdot \cos x\\ \end{array} \]

Alternative 9: 29.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\varepsilon\right) \end{array} \]

FPCore

(FPCore (x eps) :precision binary64 (log1p eps))

C

double code(double x, double eps) {
	return log1p(eps);
}

Java

public static double code(double x, double eps) {
	return Math.log1p(eps);
}

Python

def code(x, eps):
	return math.log1p(eps)

Julia

function code(x, eps)
	return log1p(eps)
end

Wolfram

code[x_, eps_] := N[Log[1 + eps], $MachinePrecision]

Derivation

1. Initial program 44.8%

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

   1. log1p-expm1-u 44.7%

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

3. Applied egg-rr 44.7%

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

4. Taylor expanded in x around 0 31.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{e^{\sin \varepsilon} - 1}\right) \]

5. Taylor expanded in eps around 0 29.2%

   \[\leadsto \mathsf{log1p}\left(\color{blue}{\varepsilon}\right) \]

6. Final simplification 29.2%

   \[\leadsto \mathsf{log1p}\left(\varepsilon\right) \]

Alternative 10: 54.8% 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 44.8%

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

2. Taylor expanded in x around 0 55.6%

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

3. Final simplification 55.6%

   \[\leadsto \sin \varepsilon \]

Alternative 11: 29.0% 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 44.8%

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

2. Taylor expanded in eps around 0 47.5%

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

3. Taylor expanded in x around 0 29.0%

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

4. Final simplification 29.0%

   \[\leadsto \varepsilon \]

Developer target: 76.0% 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 2023301 
(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)))