2sin (example 3.3)

Percentage Accurate: 42.0% → 99.4%

Time: 13.1s

Alternatives: 10

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 42.0% 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: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \sin \varepsilon, {\sin \varepsilon}^{2} \cdot \frac{1}{\frac{-1 - \cos \varepsilon}{\sin x}}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (cos x)
  (sin eps)
  (* (pow (sin eps) 2.0) (/ 1.0 (/ (- -1.0 (cos eps)) (sin x))))))

C

double code(double x, double eps) {
	return fma(cos(x), sin(eps), (pow(sin(eps), 2.0) * (1.0 / ((-1.0 - cos(eps)) / sin(x)))));
}

Julia

function code(x, eps)
	return fma(cos(x), sin(eps), Float64((sin(eps) ^ 2.0) * Float64(1.0 / Float64(Float64(-1.0 - cos(eps)) / sin(x)))))
end

Wolfram

code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * N[(1.0 / N[(N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.4%

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

   1. sin-sum 66.8%

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

4. Applied egg-rr 66.8%

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

5. Taylor expanded in x around inf 66.8%

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

   1. +-commutative 66.8%

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

   2. *-commutative 66.8%

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

   3. *-commutative 66.8%

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

   4. associate-+r- 99.2%

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

   5. *-commutative 99.2%

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

   6. fma-def 99.2%

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

   7. sub-neg 99.2%

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

   8. *-commutative 99.2%

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

   9. neg-mul-1 99.2%

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

   10. distribute-rgt-out 99.2%

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

   11. +-commutative 99.2%

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

7. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. flip-+ 99.0%

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

   3. metadata-eval 99.0%

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

   4. 1-sub-cos 99.3%

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

   5. associate-*l/ 99.3%

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

   6. pow2 99.3%

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

9. Applied egg-rr 99.3%

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

    1. associate-/l* 99.3%

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

    2. div-inv 99.3%

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

11. Applied egg-rr 99.3%

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

12. Final simplification 99.3%

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, {\sin \varepsilon}^{2} \cdot \frac{1}{\frac{-1 - \cos \varepsilon}{\sin x}}\right) \]
13. Add Preprocessing

Alternative 2: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{{\sin \varepsilon}^{2} \cdot \sin x}{-1 - \cos \varepsilon}\right) \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (fma
  (cos x)
  (sin eps)
  (/ (* (pow (sin eps) 2.0) (sin x)) (- -1.0 (cos eps)))))

C

double code(double x, double eps) {
	return fma(cos(x), sin(eps), ((pow(sin(eps), 2.0) * sin(x)) / (-1.0 - cos(eps))));
}

Julia

function code(x, eps)
	return fma(cos(x), sin(eps), Float64(Float64((sin(eps) ^ 2.0) * sin(x)) / Float64(-1.0 - cos(eps))))
end

Wolfram

code[x_, eps_] := N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision] + N[(N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(-1.0 - N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.4%

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

   1. sin-sum 66.8%

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

4. Applied egg-rr 66.8%

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

5. Taylor expanded in x around inf 66.8%

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

   1. +-commutative 66.8%

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

   2. *-commutative 66.8%

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

   3. *-commutative 66.8%

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

   4. associate-+r- 99.2%

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

   5. *-commutative 99.2%

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

   6. fma-def 99.2%

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

   7. sub-neg 99.2%

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

   8. *-commutative 99.2%

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

   9. neg-mul-1 99.2%

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

   10. distribute-rgt-out 99.2%

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

   11. +-commutative 99.2%

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

7. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. flip-+ 99.0%

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

   3. metadata-eval 99.0%

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

   4. 1-sub-cos 99.3%

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

   5. associate-*l/ 99.3%

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

   6. pow2 99.3%

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

9. Applied egg-rr 99.3%

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

10. Final simplification 99.3%

    \[\leadsto \mathsf{fma}\left(\cos x, \sin \varepsilon, \frac{{\sin \varepsilon}^{2} \cdot \sin x}{-1 - \cos \varepsilon}\right) \]
11. Add Preprocessing

Alternative 3: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sin \varepsilon - \frac{{\sin \varepsilon}^{2} \cdot \sin x}{1 + \cos \varepsilon} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (-
  (* (cos x) (sin eps))
  (/ (* (pow (sin eps) 2.0) (sin x)) (+ 1.0 (cos eps)))))

C

double code(double x, double eps) {
	return (cos(x) * sin(eps)) - ((pow(sin(eps), 2.0) * sin(x)) / (1.0 + cos(eps)));
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    code = (cos(x) * sin(eps)) - (((sin(eps) ** 2.0d0) * sin(x)) / (1.0d0 + cos(eps)))
end function

Java

public static double code(double x, double eps) {
	return (Math.cos(x) * Math.sin(eps)) - ((Math.pow(Math.sin(eps), 2.0) * Math.sin(x)) / (1.0 + Math.cos(eps)));
}

Python

def code(x, eps):
	return (math.cos(x) * math.sin(eps)) - ((math.pow(math.sin(eps), 2.0) * math.sin(x)) / (1.0 + math.cos(eps)))

Julia

function code(x, eps)
	return Float64(Float64(cos(x) * sin(eps)) - Float64(Float64((sin(eps) ^ 2.0) * sin(x)) / Float64(1.0 + cos(eps))))
end

MATLAB

function tmp = code(x, eps)
	tmp = (cos(x) * sin(eps)) - (((sin(eps) ^ 2.0) * sin(x)) / (1.0 + cos(eps)));
end

Wolfram

code[x_, eps_] := N[(N[(N[Cos[x], $MachinePrecision] * N[Sin[eps], $MachinePrecision]), $MachinePrecision] - N[(N[(N[Power[N[Sin[eps], $MachinePrecision], 2.0], $MachinePrecision] * N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.4%

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

   1. sin-sum 66.8%

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

4. Applied egg-rr 66.8%

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

5. Taylor expanded in x around inf 66.8%

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

   1. +-commutative 66.8%

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

   2. *-commutative 66.8%

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

   3. *-commutative 66.8%

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

   4. associate-+r- 99.2%

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

   5. *-commutative 99.2%

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

   6. fma-def 99.2%

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

   7. sub-neg 99.2%

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

   8. *-commutative 99.2%

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

   9. neg-mul-1 99.2%

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

   10. distribute-rgt-out 99.2%

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

   11. +-commutative 99.2%

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

7. Simplified 99.2%

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

   1. *-commutative 99.2%

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

   2. flip-+ 99.0%

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

   3. metadata-eval 99.0%

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

   4. 1-sub-cos 99.3%

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

   5. associate-*l/ 99.3%

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

   6. pow2 99.3%

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

9. Applied egg-rr 99.3%

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

10. Taylor expanded in x around inf 99.3%

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

11. Final simplification 99.3%

    \[\leadsto \cos x \cdot \sin \varepsilon - \frac{{\sin \varepsilon}^{2} \cdot \sin x}{1 + \cos \varepsilon} \]
12. Add Preprocessing

Alternative 4: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.4%

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

   1. sin-sum 66.8%

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

4. Applied egg-rr 66.8%

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

5. Taylor expanded in x around inf 66.8%

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

   1. +-commutative 66.8%

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

   2. *-commutative 66.8%

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

   3. *-commutative 66.8%

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

   4. associate-+r- 99.2%

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

   5. *-commutative 99.2%

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

   6. fma-def 99.2%

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

   7. sub-neg 99.2%

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

   8. *-commutative 99.2%

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

   9. neg-mul-1 99.2%

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

   10. distribute-rgt-out 99.2%

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

   11. +-commutative 99.2%

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

7. Simplified 99.2%

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

8. Final simplification 99.2%

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

Alternative 5: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.4%

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

   1. sin-sum 66.8%

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

4. Applied egg-rr 66.8%

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

5. Taylor expanded in x around inf 66.8%

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

   1. +-commutative 66.8%

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

   2. *-commutative 66.8%

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

   3. *-commutative 66.8%

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

   4. associate-+r- 99.2%

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

   5. *-commutative 99.2%

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

   6. fma-def 99.2%

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

   7. sub-neg 99.2%

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

   8. *-commutative 99.2%

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

   9. neg-mul-1 99.2%

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

   10. distribute-rgt-out 99.2%

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

   11. +-commutative 99.2%

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

7. Simplified 99.2%

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

   1. fma-udef 99.2%

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

   2. +-commutative 99.2%

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

   3. *-commutative 99.2%

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

9. Applied egg-rr 99.2%

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

10. Final simplification 99.2%

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

Alternative 6: 75.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.4%

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

   1. diff-sin 38.8%

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

   2. div-inv 38.8%

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

   3. associate--l+ 38.8%

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

   4. metadata-eval 38.8%

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

   5. div-inv 38.8%

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

   6. +-commutative 38.8%

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

   7. associate-+l+ 38.9%

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

   8. metadata-eval 38.9%

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

4. Applied egg-rr 38.9%

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

   1. associate-*r* 38.9%

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

   2. *-commutative 38.9%

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

   3. *-commutative 38.9%

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

   4. +-commutative 38.9%

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

   5. count-2 38.9%

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

   6. fma-def 38.9%

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

   7. associate-+r- 38.9%

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

   8. +-commutative 38.9%

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

   9. associate--l+ 72.7%

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

   10. +-inverses 72.7%

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

6. Simplified 72.7%

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

7. Final simplification 72.7%

   \[\leadsto \cos \left(0.5 \cdot \mathsf{fma}\left(2, x, \varepsilon\right)\right) \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right) \]
8. Add Preprocessing

Alternative 7: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0032 \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(-1 + \cos \varepsilon\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= x -0.0032) (not (<= x 0.37)))
   (* (cos x) (* 2.0 (sin (* eps 0.5))))
   (+ (sin eps) (* x (+ -1.0 (cos eps))))))

C

double code(double x, double eps) {
	double tmp;
	if ((x <= -0.0032) || !(x <= 0.37)) {
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	} else {
		tmp = sin(eps) + (x * (-1.0 + cos(eps)));
	}
	return tmp;
}

Fortran

real(8) function code(x, eps)
    real(8), intent (in) :: x
    real(8), intent (in) :: eps
    real(8) :: tmp
    if ((x <= (-0.0032d0)) .or. (.not. (x <= 0.37d0))) then
        tmp = cos(x) * (2.0d0 * sin((eps * 0.5d0)))
    else
        tmp = sin(eps) + (x * ((-1.0d0) + cos(eps)))
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((x <= -0.0032) || !(x <= 0.37)) {
		tmp = Math.cos(x) * (2.0 * Math.sin((eps * 0.5)));
	} else {
		tmp = Math.sin(eps) + (x * (-1.0 + Math.cos(eps)));
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (x <= -0.0032) or not (x <= 0.37):
		tmp = math.cos(x) * (2.0 * math.sin((eps * 0.5)))
	else:
		tmp = math.sin(eps) + (x * (-1.0 + math.cos(eps)))
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((x <= -0.0032) || !(x <= 0.37))
		tmp = Float64(cos(x) * Float64(2.0 * sin(Float64(eps * 0.5))));
	else
		tmp = Float64(sin(eps) + Float64(x * Float64(-1.0 + cos(eps))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((x <= -0.0032) || ~((x <= 0.37)))
		tmp = cos(x) * (2.0 * sin((eps * 0.5)));
	else
		tmp = sin(eps) + (x * (-1.0 + cos(eps)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[x, -0.0032], N[Not[LessEqual[x, 0.37]], $MachinePrecision]], N[(N[Cos[x], $MachinePrecision] * N[(2.0 * N[Sin[N[(eps * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[eps], $MachinePrecision] + N[(x * N[(-1.0 + N[Cos[eps], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00320000000000000015 or 0.37 < x

   1. Initial program 6.9%

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

      1. diff-sin 6.1%

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

      2. div-inv 6.1%

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

      3. associate--l+ 6.1%

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

      4. metadata-eval 6.1%

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

      5. div-inv 6.1%

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

      6. +-commutative 6.1%

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

      7. associate-+l+ 6.3%

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

      8. metadata-eval 6.3%

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

   4. Applied egg-rr 6.3%

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

      1. associate-*r* 6.3%

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

      2. *-commutative 6.3%

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

      3. *-commutative 6.3%

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

      4. +-commutative 6.3%

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

      5. count-2 6.3%

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

      6. fma-def 6.3%

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

      7. associate-+r- 6.3%

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

      8. +-commutative 6.3%

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

      9. associate--l+ 52.9%

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

      10. +-inverses 52.9%

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

   6. Simplified 52.9%

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

   7. Taylor expanded in eps around 0 53.7%

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

   if -0.00320000000000000015 < x < 0.37

   1. Initial program 80.5%

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

   3. Taylor expanded in x around 0 98.8%

      \[\leadsto \color{blue}{\sin \varepsilon + x \cdot \left(\cos \varepsilon - 1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0032 \lor \neg \left(x \leq 0.37\right):\\ \;\;\;\;\cos x \cdot \left(2 \cdot \sin \left(\varepsilon \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \varepsilon + x \cdot \left(-1 + \cos \varepsilon\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 0.00385\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \end{array} \]

FPCore

(FPCore (x eps)
 :precision binary64
 (if (or (<= eps -0.0082) (not (<= eps 0.00385))) (sin eps) (* (cos x) eps)))

C

double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 0.00385)) {
		tmp = sin(eps);
	} else {
		tmp = cos(x) * 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 <= (-0.0082d0)) .or. (.not. (eps <= 0.00385d0))) then
        tmp = sin(eps)
    else
        tmp = cos(x) * eps
    end if
    code = tmp
end function

Java

public static double code(double x, double eps) {
	double tmp;
	if ((eps <= -0.0082) || !(eps <= 0.00385)) {
		tmp = Math.sin(eps);
	} else {
		tmp = Math.cos(x) * eps;
	}
	return tmp;
}

Python

def code(x, eps):
	tmp = 0
	if (eps <= -0.0082) or not (eps <= 0.00385):
		tmp = math.sin(eps)
	else:
		tmp = math.cos(x) * eps
	return tmp

Julia

function code(x, eps)
	tmp = 0.0
	if ((eps <= -0.0082) || !(eps <= 0.00385))
		tmp = sin(eps);
	else
		tmp = Float64(cos(x) * eps);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, eps)
	tmp = 0.0;
	if ((eps <= -0.0082) || ~((eps <= 0.00385)))
		tmp = sin(eps);
	else
		tmp = cos(x) * eps;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, eps_] := If[Or[LessEqual[eps, -0.0082], N[Not[LessEqual[eps, 0.00385]], $MachinePrecision]], N[Sin[eps], $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * eps), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eps < -0.00820000000000000069 or 0.0038500000000000001 < eps

   1. Initial program 49.3%

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

   3. Taylor expanded in x around 0 50.3%

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

   if -0.00820000000000000069 < eps < 0.0038500000000000001

   1. Initial program 28.8%

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

   3. Taylor expanded in eps around 0 97.4%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \leq -0.0082 \lor \neg \left(\varepsilon \leq 0.00385\right):\\ \;\;\;\;\sin \varepsilon\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \varepsilon\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 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 39.4%

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

3. Taylor expanded in x around 0 49.9%

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

4. Final simplification 49.9%

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

Alternative 10: 29.3% 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 39.4%

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

3. Taylor expanded in x around 0 49.9%

   \[\leadsto \color{blue}{\sin \varepsilon} \]
4. Step-by-step derivation

   1. add-cbrt-cube 37.1%

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

   2. pow3 37.2%

      \[\leadsto \sqrt[3]{\color{blue}{{\sin \varepsilon}^{3}}} \]

5. Applied egg-rr 37.2%

   \[\leadsto \color{blue}{\sqrt[3]{{\sin \varepsilon}^{3}}} \]

6. Taylor expanded in eps around 0 25.4%

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

7. Final simplification 25.4%

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

Developer target: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Reproduce

herbie shell --seed 2024010 
(FPCore (x eps)
  :name "2sin (example 3.3)"
  :precision binary64

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

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