Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 11.6s

Alternatives: 18

Speedup: 1.0×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{\sinh y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	return cos(x) * (sinh(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cos(x) * (sinh(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cos(x) * (Math.sinh(y) / y);
}

Python

def code(x, y):
	return math.cos(x) * (math.sinh(y) / y)

Julia

function code(x, y)
	return Float64(cos(x) * Float64(sinh(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cos(x) * (sinh(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cos[x], $MachinePrecision] * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 73.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.26:\\ \;\;\;\;\frac{\cos x}{1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.019444444444444445 + \left(y \cdot y\right) \cdot -0.00205026455026455\right)\right)\right)}\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.26)
   (/
    (cos x)
    (+
     1.0
     (*
      y
      (*
       y
       (+
        -0.16666666666666666
        (*
         (* y y)
         (+ 0.019444444444444445 (* (* y y) -0.00205026455026455))))))))
   (if (<= y 3.8e+77)
     (/ (sinh y) y)
     (*
      (cos x)
      (+
       1.0
       (*
        (* y y)
        (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.26) {
		tmp = cos(x) / (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.019444444444444445 + ((y * y) * -0.00205026455026455)))))));
	} else if (y <= 3.8e+77) {
		tmp = sinh(y) / y;
	} else {
		tmp = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.26d0) then
        tmp = cos(x) / (1.0d0 + (y * (y * ((-0.16666666666666666d0) + ((y * y) * (0.019444444444444445d0 + ((y * y) * (-0.00205026455026455d0))))))))
    else if (y <= 3.8d+77) then
        tmp = sinh(y) / y
    else
        tmp = cos(x) * (1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.26) {
		tmp = Math.cos(x) / (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.019444444444444445 + ((y * y) * -0.00205026455026455)))))));
	} else if (y <= 3.8e+77) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = Math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.26:
		tmp = math.cos(x) / (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.019444444444444445 + ((y * y) * -0.00205026455026455)))))))
	elif y <= 3.8e+77:
		tmp = math.sinh(y) / y
	else:
		tmp = math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.26)
		tmp = Float64(cos(x) / Float64(1.0 + Float64(y * Float64(y * Float64(-0.16666666666666666 + Float64(Float64(y * y) * Float64(0.019444444444444445 + Float64(Float64(y * y) * -0.00205026455026455))))))));
	elseif (y <= 3.8e+77)
		tmp = Float64(sinh(y) / y);
	else
		tmp = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.26)
		tmp = cos(x) / (1.0 + (y * (y * (-0.16666666666666666 + ((y * y) * (0.019444444444444445 + ((y * y) * -0.00205026455026455)))))));
	elseif (y <= 3.8e+77)
		tmp = sinh(y) / y;
	else
		tmp = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.26], N[(N[Cos[x], $MachinePrecision] / N[(1.0 + N[(y * N[(y * N[(-0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.019444444444444445 + N[(N[(y * y), $MachinePrecision] * -0.00205026455026455), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.8e+77], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.26000000000000001

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \color{blue}{\left(1 + {y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{{y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right)} - \frac{1}{6}\right)\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) - \frac{1}{6}\right)\right)}\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \left(y \cdot \left(\left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot \color{blue}{y}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(\left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) - \frac{1}{6}\right) \cdot y\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \left(y \cdot \color{blue}{\left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) - \frac{1}{6}\right)}\right)\right)\right)\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{6}\right)\right)}\right)\right)\right)\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right) + \frac{-1}{6}\right)\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left(\frac{-1}{6} + \color{blue}{{y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{7}{360} + \frac{-31}{15120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{7}{360}} + \frac{-31}{15120} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{7}{360}} + \frac{-31}{15120} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      15. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{7}{360}, \color{blue}{\left(\frac{-31}{15120} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      16. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{7}{360}, \left({y}^{2} \cdot \color{blue}{\frac{-31}{15120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      17. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{7}{360}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{-31}{15120}}\right)\right)\right)\right)\right)\right)\right)\right) \]

      18. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{7}{360}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{-31}{15120}\right)\right)\right)\right)\right)\right)\right)\right) \]

      19. *-lowering-*.f64 70.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{-1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{7}{360}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{-31}{15120}\right)\right)\right)\right)\right)\right)\right)\right) \]

   7. Simplified 70.1%

      \[\leadsto \frac{\cos x}{\color{blue}{1 + y \cdot \left(y \cdot \left(-0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.019444444444444445 + \left(y \cdot y\right) \cdot -0.00205026455026455\right)\right)\right)}} \]

   if 0.26000000000000001 < y < 3.8000000000000001e77

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 81.8%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sinh y}{\color{blue}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{y}\right) \]

      3. sinh-lowering-sinh.f64 81.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), y\right) \]

   9. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]

   if 3.8000000000000001e77 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)\\ \mathbf{if}\;y \leq 0.39:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (cos x)
          (+
           1.0
           (*
            (* y y)
            (+ 0.16666666666666666 (* (* y y) 0.008333333333333333)))))))
   (if (<= y 0.39) t_0 (if (<= y 3.8e+77) (/ (sinh y) y) t_0))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	double tmp;
	if (y <= 0.39) {
		tmp = t_0;
	} else if (y <= 3.8e+77) {
		tmp = sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(x) * (1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))
    if (y <= 0.39d0) then
        tmp = t_0
    else if (y <= 3.8d+77) then
        tmp = sinh(y) / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	double tmp;
	if (y <= 0.39) {
		tmp = t_0;
	} else if (y <= 3.8e+77) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))
	tmp = 0
	if y <= 0.39:
		tmp = t_0
	elif y <= 3.8e+77:
		tmp = math.sinh(y) / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))
	tmp = 0.0
	if (y <= 0.39)
		tmp = t_0;
	elseif (y <= 3.8e+77)
		tmp = Float64(sinh(y) / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
	tmp = 0.0;
	if (y <= 0.39)
		tmp = t_0;
	elseif (y <= 3.8e+77)
		tmp = sinh(y) / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.39], t$95$0, If[LessEqual[y, 3.8e+77], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.39000000000000001 or 3.8000000000000001e77 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

      2. *-rgt-identity N/A

         \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

      4. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

      5. associate-*r* N/A

         \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

      6. distribute-rgt-out N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

      7. +-commutative N/A

         \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

      8. associate-*l* N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

      9. *-commutative N/A

         \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

      10. distribute-lft-in N/A

          \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. Simplified 96.1%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

   if 0.39000000000000001 < y < 3.8000000000000001e77

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 81.8%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sinh y}{\color{blue}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{y}\right) \]

      3. sinh-lowering-sinh.f64 81.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), y\right) \]

   9. Applied egg-rr 81.8%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 84.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{if}\;y \leq 0.054:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (* (cos x) (+ 1.0 (* (* y y) 0.16666666666666666)))))
   (if (<= y 0.054) t_0 (if (<= y 1.32e+154) (/ (sinh y) y) t_0))))

C

double code(double x, double y) {
	double t_0 = cos(x) * (1.0 + ((y * y) * 0.16666666666666666));
	double tmp;
	if (y <= 0.054) {
		tmp = t_0;
	} else if (y <= 1.32e+154) {
		tmp = sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(x) * (1.0d0 + ((y * y) * 0.16666666666666666d0))
    if (y <= 0.054d0) then
        tmp = t_0
    else if (y <= 1.32d+154) then
        tmp = sinh(y) / y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.cos(x) * (1.0 + ((y * y) * 0.16666666666666666));
	double tmp;
	if (y <= 0.054) {
		tmp = t_0;
	} else if (y <= 1.32e+154) {
		tmp = Math.sinh(y) / y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.cos(x) * (1.0 + ((y * y) * 0.16666666666666666))
	tmp = 0
	if y <= 0.054:
		tmp = t_0
	elif y <= 1.32e+154:
		tmp = math.sinh(y) / y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(cos(x) * Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666)))
	tmp = 0.0
	if (y <= 0.054)
		tmp = t_0;
	elseif (y <= 1.32e+154)
		tmp = Float64(sinh(y) / y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = cos(x) * (1.0 + ((y * y) * 0.16666666666666666));
	tmp = 0.0;
	if (y <= 0.054)
		tmp = t_0;
	elseif (y <= 1.32e+154)
		tmp = sinh(y) / y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Cos[x], $MachinePrecision] * N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.054], t$95$0, If[LessEqual[y, 1.32e+154], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0539999999999999994 or 1.31999999999999998e154 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 90.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 90.8%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   if 0.0539999999999999994 < y < 1.31999999999999998e154

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 84.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sinh y}{\color{blue}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{y}\right) \]

      3. sinh-lowering-sinh.f64 84.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), y\right) \]

   9. Applied egg-rr 84.0%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.054:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.00096:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 0.00096) (cos x) (/ (sinh y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.00096) {
		tmp = cos(x);
	} else {
		tmp = sinh(y) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.00096d0) then
        tmp = cos(x)
    else
        tmp = sinh(y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.00096) {
		tmp = Math.cos(x);
	} else {
		tmp = Math.sinh(y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.00096:
		tmp = math.cos(x)
	else:
		tmp = math.sinh(y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.00096)
		tmp = cos(x);
	else
		tmp = Float64(sinh(y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.00096)
		tmp = cos(x);
	else
		tmp = sinh(y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.00096], N[Cos[x], $MachinePrecision], N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 9.60000000000000024e-4

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 69.7%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 69.7%

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

   if 9.60000000000000024e-4 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 80.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]
   8. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \frac{\sinh y}{\color{blue}{y}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sinh y, \color{blue}{y}\right) \]

      3. sinh-lowering-sinh.f64 80.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sinh.f64}\left(y\right), y\right) \]

   9. Applied egg-rr 80.0%

      \[\leadsto \color{blue}{\frac{\sinh y}{y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 66.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\\ \mathbf{if}\;y \leq 0.00096:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{\left(y \cdot y\right) \cdot \left(0.004629629629629629 + t\_0 \cdot \left(t\_0 \cdot t\_0\right)\right)}{0.027777777777777776 + t\_0 \cdot \left(t\_0 - 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + t\_0\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (* y y)
          (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))
   (if (<= y 0.00096)
     (cos x)
     (if (<= y 2.8e+39)
       (+
        1.0
        (/
         (* (* y y) (+ 0.004629629629629629 (* t_0 (* t_0 t_0))))
         (+ 0.027777777777777776 (* t_0 (- t_0 0.16666666666666666)))))
       (/ 1.0 (/ y (* y (+ 1.0 (* (* y y) (+ 0.16666666666666666 t_0))))))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984));
	double tmp;
	if (y <= 0.00096) {
		tmp = cos(x);
	} else if (y <= 2.8e+39) {
		tmp = 1.0 + (((y * y) * (0.004629629629629629 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776 + (t_0 * (t_0 - 0.16666666666666666))));
	} else {
		tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * y) * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))
    if (y <= 0.00096d0) then
        tmp = cos(x)
    else if (y <= 2.8d+39) then
        tmp = 1.0d0 + (((y * y) * (0.004629629629629629d0 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776d0 + (t_0 * (t_0 - 0.16666666666666666d0))))
    else
        tmp = 1.0d0 / (y / (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984));
	double tmp;
	if (y <= 0.00096) {
		tmp = Math.cos(x);
	} else if (y <= 2.8e+39) {
		tmp = 1.0 + (((y * y) * (0.004629629629629629 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776 + (t_0 * (t_0 - 0.16666666666666666))));
	} else {
		tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))
	tmp = 0
	if y <= 0.00096:
		tmp = math.cos(x)
	elif y <= 2.8e+39:
		tmp = 1.0 + (((y * y) * (0.004629629629629629 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776 + (t_0 * (t_0 - 0.16666666666666666))))
	else:
		tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))
	tmp = 0.0
	if (y <= 0.00096)
		tmp = cos(x);
	elseif (y <= 2.8e+39)
		tmp = Float64(1.0 + Float64(Float64(Float64(y * y) * Float64(0.004629629629629629 + Float64(t_0 * Float64(t_0 * t_0)))) / Float64(0.027777777777777776 + Float64(t_0 * Float64(t_0 - 0.16666666666666666)))));
	else
		tmp = Float64(1.0 / Float64(y / Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984));
	tmp = 0.0;
	if (y <= 0.00096)
		tmp = cos(x);
	elseif (y <= 2.8e+39)
		tmp = 1.0 + (((y * y) * (0.004629629629629629 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776 + (t_0 * (t_0 - 0.16666666666666666))));
	else
		tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 0.00096], N[Cos[x], $MachinePrecision], If[LessEqual[y, 2.8e+39], N[(1.0 + N[(N[(N[(y * y), $MachinePrecision] * N[(0.004629629629629629 + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(t$95$0 * N[(t$95$0 - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < 9.60000000000000024e-4

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 69.7%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 69.7%

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

   if 9.60000000000000024e-4 < y < 2.80000000000000001e39

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 100.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 4.4%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 4.4%

       \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{{\frac{1}{6}}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}^{3}}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)} \cdot \left(\color{blue}{y} \cdot y\right)\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left({\frac{1}{6}}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}^{3}\right) \cdot \left(y \cdot y\right)}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left({\frac{1}{6}}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}^{3}\right) \cdot \left(y \cdot y\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right) \]

   12. Applied egg-rr 61.4%

       \[\leadsto 1 + \color{blue}{\frac{\left(0.004629629629629629 + \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \left(y \cdot y\right)}{0.027777777777777776 + \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right) - 0.16666666666666666\right)}} \]

   if 2.80000000000000001e39 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 78.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 78.0%

       \[\leadsto \frac{1}{\frac{y}{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.00096:\\ \;\;\;\;\cos x\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{\left(y \cdot y\right) \cdot \left(0.004629629629629629 + \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{0.027777777777777776 + \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right) - 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\\ \mathbf{if}\;y \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{\left(y \cdot y\right) \cdot \left(0.004629629629629629 + t\_0 \cdot \left(t\_0 \cdot t\_0\right)\right)}{0.027777777777777776 + t\_0 \cdot \left(t\_0 - 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + t\_0\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0
         (*
          (* y y)
          (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))
   (if (<= y 2.8e+39)
     (+
      1.0
      (/
       (* (* y y) (+ 0.004629629629629629 (* t_0 (* t_0 t_0))))
       (+ 0.027777777777777776 (* t_0 (- t_0 0.16666666666666666)))))
     (/ 1.0 (/ y (* y (+ 1.0 (* (* y y) (+ 0.16666666666666666 t_0)))))))))

C

double code(double x, double y) {
	double t_0 = (y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984));
	double tmp;
	if (y <= 2.8e+39) {
		tmp = 1.0 + (((y * y) * (0.004629629629629629 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776 + (t_0 * (t_0 - 0.16666666666666666))));
	} else {
		tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y * y) * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))
    if (y <= 2.8d+39) then
        tmp = 1.0d0 + (((y * y) * (0.004629629629629629d0 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776d0 + (t_0 * (t_0 - 0.16666666666666666d0))))
    else
        tmp = 1.0d0 / (y / (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984));
	double tmp;
	if (y <= 2.8e+39) {
		tmp = 1.0 + (((y * y) * (0.004629629629629629 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776 + (t_0 * (t_0 - 0.16666666666666666))));
	} else {
		tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + t_0)))));
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))
	tmp = 0
	if y <= 2.8e+39:
		tmp = 1.0 + (((y * y) * (0.004629629629629629 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776 + (t_0 * (t_0 - 0.16666666666666666))))
	else:
		tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + t_0)))))
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))
	tmp = 0.0
	if (y <= 2.8e+39)
		tmp = Float64(1.0 + Float64(Float64(Float64(y * y) * Float64(0.004629629629629629 + Float64(t_0 * Float64(t_0 * t_0)))) / Float64(0.027777777777777776 + Float64(t_0 * Float64(t_0 - 0.16666666666666666)))));
	else
		tmp = Float64(1.0 / Float64(y / Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984));
	tmp = 0.0;
	if (y <= 2.8e+39)
		tmp = 1.0 + (((y * y) * (0.004629629629629629 + (t_0 * (t_0 * t_0)))) / (0.027777777777777776 + (t_0 * (t_0 - 0.16666666666666666))));
	else
		tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, 2.8e+39], N[(1.0 + N[(N[(N[(y * y), $MachinePrecision] * N[(0.004629629629629629 + N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(0.027777777777777776 + N[(t$95$0 * N[(t$95$0 - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(y / N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < 2.80000000000000001e39

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 67.5%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 63.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

   10. Simplified 63.3%

       \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\left(\frac{1}{6} + \left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) \cdot \color{blue}{\left(y \cdot y\right)}\right)\right) \]

       2. flip3-+ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{{\frac{1}{6}}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}^{3}}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)} \cdot \left(\color{blue}{y} \cdot y\right)\right)\right) \]

       3. associate-*l/ N/A

          \[\leadsto \mathsf{+.f64}\left(1, \left(\frac{\left({\frac{1}{6}}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}^{3}\right) \cdot \left(y \cdot y\right)}{\color{blue}{\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)}}\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\left({\frac{1}{6}}^{3} + {\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)}^{3}\right) \cdot \left(y \cdot y\right)\right), \color{blue}{\left(\frac{1}{6} \cdot \frac{1}{6} + \left(\left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right) - \frac{1}{6} \cdot \left(\left(y \cdot y\right) \cdot \left(\frac{1}{120} + \left(y \cdot y\right) \cdot \frac{1}{5040}\right)\right)\right)\right)}\right)\right) \]

   12. Applied egg-rr 43.2%

       \[\leadsto 1 + \color{blue}{\frac{\left(0.004629629629629629 + \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right) \cdot \left(y \cdot y\right)}{0.027777777777777776 + \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right) - 0.16666666666666666\right)}} \]

   if 2.80000000000000001e39 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

      4. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

      6. sinh-lowering-sinh.f64 100.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
   6. Step-by-step derivation

   7. Simplified 78.0%

      \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

      14. *-lowering-*.f64 78.0%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. Simplified 78.0%

       \[\leadsto \frac{1}{\frac{y}{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;1 + \frac{\left(y \cdot y\right) \cdot \left(0.004629629629629629 + \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)\right)}{0.027777777777777776 + \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \cdot \left(\left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right) - 0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.4% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{y}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (/
  1.0
  (/
   y
   (*
    y
    (+
     1.0
     (*
      (* y y)
      (+
       0.16666666666666666
       (*
        (* y y)
        (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984))))))))))

C

double code(double x, double y) {
	return 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 / (y / (y * (1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0))))))))
end function

Java

public static double code(double x, double y) {
	return 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
}

Python

def code(x, y):
	return 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))))

Julia

function code(x, y)
	return Float64(1.0 / Float64(y / Float64(y * Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984)))))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 / (y / (y * (1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984))))))));
end

Wolfram

code[x_, y_] := N[(1.0 / N[(y / N[(y * N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

   6. sinh-lowering-sinh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 69.5%

   \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]

8. Taylor expanded in y around 0

   \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \color{blue}{\left(y \cdot \left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)}\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right) \]

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right)\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right)\right)\right)\right) \]

   10. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right)\right)\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

   14. *-lowering-*.f64 66.5%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right)\right)\right)\right) \]

10. Simplified 66.5%

    \[\leadsto \frac{1}{\frac{y}{\color{blue}{y \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)}}} \]
11. Add Preprocessing

Alternative 9: 49.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+27}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.006944444444444444\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 8e+27)
   (+ 1.0 (* (* y y) 0.16666666666666666))
   (if (<= y 1.65e+154)
     (* (* (* y y) 0.006944444444444444) (* (* x x) (* x x)))
     (+ 1.0 (* y (* y 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 8e+27) {
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	} else if (y <= 1.65e+154) {
		tmp = ((y * y) * 0.006944444444444444) * ((x * x) * (x * x));
	} else {
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 8d+27) then
        tmp = 1.0d0 + ((y * y) * 0.16666666666666666d0)
    else if (y <= 1.65d+154) then
        tmp = ((y * y) * 0.006944444444444444d0) * ((x * x) * (x * x))
    else
        tmp = 1.0d0 + (y * (y * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 8e+27) {
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	} else if (y <= 1.65e+154) {
		tmp = ((y * y) * 0.006944444444444444) * ((x * x) * (x * x));
	} else {
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 8e+27:
		tmp = 1.0 + ((y * y) * 0.16666666666666666)
	elif y <= 1.65e+154:
		tmp = ((y * y) * 0.006944444444444444) * ((x * x) * (x * x))
	else:
		tmp = 1.0 + (y * (y * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 8e+27)
		tmp = Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666));
	elseif (y <= 1.65e+154)
		tmp = Float64(Float64(Float64(y * y) * 0.006944444444444444) * Float64(Float64(x * x) * Float64(x * x)));
	else
		tmp = Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 8e+27)
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	elseif (y <= 1.65e+154)
		tmp = ((y * y) * 0.006944444444444444) * ((x * x) * (x * x));
	else
		tmp = 1.0 + (y * (y * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 8e+27], N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e+154], N[(N[(N[(y * y), $MachinePrecision] * 0.006944444444444444), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 8.0000000000000001e27

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 87.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 87.7%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cos x}{1}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1}{\cos x}}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\cos x}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \cos x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 87.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   7. Applied egg-rr 87.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\cos x}}} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      4. *-lowering-*.f64 58.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   10. Simplified 58.0%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   if 8.0000000000000001e27 < y < 1.65e154

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 6.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 6.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left(\frac{1}{24} \cdot {x}^{2}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      12. *-lowering-*.f64 36.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   8. Simplified 36.1%

      \[\leadsto \color{blue}{\left(1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)\right)} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \left({x}^{4} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right)} \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{1}{24} \cdot \left(\left(1 + \frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{{x}^{4}}\right) \]

       2. associate-*r* N/A

          \[\leadsto \left(\frac{1}{24} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right) \cdot \color{blue}{{x}^{4}} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} \cdot \left(1 + \frac{1}{6} \cdot {y}^{2}\right)\right), \color{blue}{\left({x}^{4}\right)}\right) \]

       4. distribute-rgt-in N/A

          \[\leadsto \mathsf{*.f64}\left(\left(1 \cdot \frac{1}{24} + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{1}{24}\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{24} + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{1}{24}\right), \left({x}^{4}\right)\right) \]

       6. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \frac{1}{24}\right)\right), \left({\color{blue}{x}}^{4}\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\frac{1}{24} \cdot \left(\frac{1}{6} \cdot {y}^{2}\right)\right)\right), \left({x}^{4}\right)\right) \]

       8. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left(\left(\frac{1}{24} \cdot \frac{1}{6}\right) \cdot {y}^{2}\right)\right), \left({x}^{4}\right)\right) \]

       9. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \left({y}^{2} \cdot \left(\frac{1}{24} \cdot \frac{1}{6}\right)\right)\right), \left({x}^{4}\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({y}^{2}\right), \left(\frac{1}{24} \cdot \frac{1}{6}\right)\right)\right), \left({x}^{4}\right)\right) \]

       11. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\frac{1}{24} \cdot \frac{1}{6}\right)\right)\right), \left({x}^{4}\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{24} \cdot \frac{1}{6}\right)\right)\right), \left({x}^{4}\right)\right) \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right)\right), \left({x}^{4}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right)\right), \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       15. pow-sqr N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right)\right), \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

       16. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right)\right), \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       17. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right)\right), \mathsf{*.f64}\left(\left(x \cdot x\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

       18. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

       19. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       20. *-lowering-*.f64 34.6%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{+.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right)\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   11. Simplified 34.6%

       \[\leadsto \color{blue}{\left(0.041666666666666664 + \left(y \cdot y\right) \cdot 0.006944444444444444\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]

   12. Taylor expanded in y around inf

       \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{144} \cdot {y}^{2}\right)}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]
   13. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y}^{2} \cdot \frac{1}{144}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, x\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left({y}^{2}\right), \frac{1}{144}\right), \mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(x, x\right)}, \mathsf{*.f64}\left(x, x\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{144}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

       4. *-lowering-*.f64 34.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{144}\right), \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{x}, x\right), \mathsf{*.f64}\left(x, x\right)\right)\right) \]

   14. Simplified 34.6%

       \[\leadsto \color{blue}{\left(\left(y \cdot y\right) \cdot 0.006944444444444444\right)} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \]

   if 1.65e154 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 100.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 100.0%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right) \]

      6. *-lowering-*.f64 76.7%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right) \]

   8. Simplified 76.7%

      \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot 0.16666666666666666\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 8 \cdot 10^{+27}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+154}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot 0.006944444444444444\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 + y \cdot \left(y \cdot 0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  1.0
  (*
   (* y y)
   (+
    0.16666666666666666
    (* (* y y) (+ 0.008333333333333333 (* (* y y) 0.0001984126984126984)))))))

C

double code(double x, double y) {
	return 1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * y) * (0.16666666666666666d0 + ((y * y) * (0.008333333333333333d0 + ((y * y) * 0.0001984126984126984d0)))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))));
}

Python

def code(x, y):
	return 1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(Float64(y * y) * Float64(0.008333333333333333 + Float64(Float64(y * y) * 0.0001984126984126984))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * y) * (0.16666666666666666 + ((y * y) * (0.008333333333333333 + ((y * y) * 0.0001984126984126984)))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * N[(0.008333333333333333 + N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

   6. sinh-lowering-sinh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 69.5%

   \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 66.2%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

10. Simplified 66.2%

    \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)} \]
11. Add Preprocessing

Alternative 11: 59.5% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+
  1.0
  (*
   (* y y)
   (+ 0.16666666666666666 (* y (* y (* (* y y) 0.0001984126984126984)))))))

C

double code(double x, double y) {
	return 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * ((y * y) * 0.0001984126984126984)))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * y) * (0.16666666666666666d0 + (y * (y * ((y * y) * 0.0001984126984126984d0)))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * ((y * y) * 0.0001984126984126984)))));
}

Python

def code(x, y):
	return 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * ((y * y) * 0.0001984126984126984)))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * y) * Float64(0.16666666666666666 + Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984))))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * y) * (0.16666666666666666 + (y * (y * ((y * y) * 0.0001984126984126984)))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(0.16666666666666666 + N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

   6. sinh-lowering-sinh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 69.5%

   \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 66.2%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

10. Simplified 66.2%

    \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)} \]

11. Taylor expanded in y around inf

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{4}\right)}\right)\right)\right) \]
12. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{5040} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right)\right) \]

    2. pow-sqr N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{5040}} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 66.1%

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

13. Simplified 66.1%

    \[\leadsto 1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \color{blue}{y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)}\right) \]
14. Add Preprocessing

Alternative 12: 59.4% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot y\right) \cdot \left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 1.0 (* (* y y) (* y (* y (* (* y y) 0.0001984126984126984))))))

C

double code(double x, double y) {
	return 1.0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984d0))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984))));
}

Python

def code(x, y):
	return 1.0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * y) * Float64(y * Float64(y * Float64(Float64(y * y) * 0.0001984126984126984)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * y) * (y * (y * ((y * y) * 0.0001984126984126984))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * y), $MachinePrecision] * N[(y * N[(y * N[(N[(y * y), $MachinePrecision] * 0.0001984126984126984), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \cos x \cdot \frac{1}{\color{blue}{\frac{y}{\sinh y}}} \]

   2. un-div-inv N/A

      \[\leadsto \frac{\cos x}{\color{blue}{\frac{y}{\sinh y}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\cos x, \color{blue}{\left(\frac{y}{\sinh y}\right)}\right) \]

   4. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\frac{\color{blue}{y}}{\sinh y}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \color{blue}{\sinh y}\right)\right) \]

   6. sinh-lowering-sinh.f64 100.0%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{\cos x}{\frac{y}{\sinh y}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{1}, \mathsf{/.f64}\left(y, \mathsf{sinh.f64}\left(y\right)\right)\right) \]
6. Step-by-step derivation

7. Simplified 69.5%

   \[\leadsto \frac{\color{blue}{1}}{\frac{y}{\sinh y}} \]

8. Taylor expanded in y around 0

   \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{6} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{6}} + {y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right) \]

   5. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\left(\frac{1}{120} + \frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\color{blue}{\frac{1}{120}} + \frac{1}{5040} \cdot {y}^{2}\right)\right)\right)\right)\right) \]

   9. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right)\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

   13. *-lowering-*.f64 66.2%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{+.f64}\left(\frac{1}{120}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right)\right) \]

10. Simplified 66.2%

    \[\leadsto \color{blue}{1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot \left(0.008333333333333333 + \left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)} \]

11. Taylor expanded in y around inf

    \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \color{blue}{\left(\frac{1}{5040} \cdot {y}^{4}\right)}\right)\right) \]
12. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{5040} \cdot {y}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right)\right) \]

    2. pow-sqr N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\frac{1}{5040} \cdot \left({y}^{2} \cdot \color{blue}{{y}^{2}}\right)\right)\right)\right) \]

    3. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\left(\frac{1}{5040} \cdot {y}^{2}\right) \cdot \color{blue}{{y}^{2}}\right)\right)\right) \]

    4. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right) \]

    5. unpow2 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{5040}} \cdot {y}^{2}\right)\right)\right)\right) \]

    6. associate-*l* N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

    7. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{5040} \cdot {y}^{2}\right)\right)}\right)\right)\right) \]

    8. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{5040} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

    9. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \left({y}^{2} \cdot \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right) \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{5040}}\right)\right)\right)\right)\right) \]

    11. unpow2 N/A

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{5040}\right)\right)\right)\right)\right) \]

    12. *-lowering-*.f64 66.0%

        \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{5040}\right)\right)\right)\right)\right) \]

13. Simplified 66.0%

    \[\leadsto 1 + \left(y \cdot y\right) \cdot \color{blue}{\left(y \cdot \left(y \cdot \left(\left(y \cdot y\right) \cdot 0.0001984126984126984\right)\right)\right)} \]
14. Add Preprocessing

Alternative 13: 48.1% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+130}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 4.5e+130)
   (+ 1.0 (* (* y y) 0.16666666666666666))
   (* (* (* x x) (* x x)) 0.041666666666666664)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4.5e+130) {
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	} else {
		tmp = ((x * x) * (x * x)) * 0.041666666666666664;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 4.5d+130) then
        tmp = 1.0d0 + ((y * y) * 0.16666666666666666d0)
    else
        tmp = ((x * x) * (x * x)) * 0.041666666666666664d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4.5e+130) {
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	} else {
		tmp = ((x * x) * (x * x)) * 0.041666666666666664;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4.5e+130:
		tmp = 1.0 + ((y * y) * 0.16666666666666666)
	else:
		tmp = ((x * x) * (x * x)) * 0.041666666666666664
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4.5e+130)
		tmp = Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666));
	else
		tmp = Float64(Float64(Float64(x * x) * Float64(x * x)) * 0.041666666666666664);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4.5e+130)
		tmp = 1.0 + ((y * y) * 0.16666666666666666);
	else
		tmp = ((x * x) * (x * x)) * 0.041666666666666664;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4.5e+130], N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.041666666666666664), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.50000000000000039e130

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 82.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 82.9%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cos x}{1}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1}{\cos x}}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\cos x}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \cos x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 82.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   7. Applied egg-rr 82.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\cos x}}} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      4. *-lowering-*.f64 62.0%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   10. Simplified 62.0%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   if 4.50000000000000039e130 < x

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 58.1%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 58.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)} \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{24} \cdot {x}^{2} + \frac{-1}{2}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{2} + \color{blue}{\frac{1}{24} \cdot {x}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \left({x}^{2} \cdot \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{24}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{24}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 28.3%

          \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{24}\right)\right)\right)\right) \]

   8. Simplified 28.3%

      \[\leadsto \color{blue}{1 + \left(x \cdot x\right) \cdot \left(-0.5 + \left(x \cdot x\right) \cdot 0.041666666666666664\right)} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot {x}^{4}} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \color{blue}{\left({x}^{4}\right)}\right) \]

       2. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({x}^{\left(2 \cdot \color{blue}{2}\right)}\right)\right) \]

       3. pow-sqr N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left({x}^{2}\right)}\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left({\color{blue}{x}}^{2}\right)\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(x \cdot \color{blue}{x}\right)\right)\right) \]

       8. *-lowering-*.f64 28.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   11. Simplified 28.3%

       \[\leadsto \color{blue}{0.041666666666666664 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5 \cdot 10^{+130}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot 0.16666666666666666\\ \mathbf{else}:\\ \;\;\;\;\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot 0.041666666666666664\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.5% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ 1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (+ 1.0 (* y (* y (+ 0.16666666666666666 (* (* y y) 0.008333333333333333))))))

C

double code(double x, double y) {
	return 1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (y * (y * (0.16666666666666666d0 + ((y * y) * 0.008333333333333333d0))))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
}

Python

def code(x, y):
	return 1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))))

Julia

function code(x, y)
	return Float64(1.0 + Float64(y * Float64(y * Float64(0.16666666666666666 + Float64(Float64(y * y) * 0.008333333333333333)))))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (y * (y * (0.16666666666666666 + ((y * y) * 0.008333333333333333))));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(y * N[(y * N[(0.16666666666666666 + N[(N[(y * y), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x + {y}^{2} \cdot \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \cos x + \left(\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2} + \color{blue}{\left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}}\right) \]

   2. *-rgt-identity N/A

      \[\leadsto \cos x \cdot 1 + \left(\color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right)\right) \cdot {y}^{2}} + \left(\frac{1}{6} \cdot \cos x\right) \cdot {y}^{2}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \cos x \cdot 1 + {y}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right)} \]

   4. *-commutative N/A

      \[\leadsto \cos x \cdot 1 + \left(\frac{1}{120} \cdot \left({y}^{2} \cdot \cos x\right) + \frac{1}{6} \cdot \cos x\right) \cdot \color{blue}{{y}^{2}} \]

   5. associate-*r* N/A

      \[\leadsto \cos x \cdot 1 + \left(\left(\frac{1}{120} \cdot {y}^{2}\right) \cdot \cos x + \frac{1}{6} \cdot \cos x\right) \cdot {y}^{2} \]

   6. distribute-rgt-out N/A

      \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{120} \cdot {y}^{2} + \frac{1}{6}\right)\right) \cdot {\color{blue}{y}}^{2} \]

   7. +-commutative N/A

      \[\leadsto \cos x \cdot 1 + \left(\cos x \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right) \cdot {y}^{2} \]

   8. associate-*l* N/A

      \[\leadsto \cos x \cdot 1 + \cos x \cdot \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right) \cdot {y}^{2}\right)} \]

   9. *-commutative N/A

      \[\leadsto \cos x \cdot 1 + \cos x \cdot \left({y}^{2} \cdot \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right) \]

   10. distribute-lft-in N/A

       \[\leadsto \cos x \cdot \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)} \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

   12. cos-lowering-cos.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

   13. +-lowering-+.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

5. Simplified 92.2%

   \[\leadsto \color{blue}{\cos x \cdot \left(1 + \left(y \cdot y\right) \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + {y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left({y}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \left(\color{blue}{\frac{1}{6}} + \frac{1}{120} \cdot {y}^{2}\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \color{blue}{\left(\frac{1}{120} \cdot {y}^{2}\right)}\right)\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \left({y}^{2} \cdot \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left({y}^{2}\right), \color{blue}{\frac{1}{120}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\left(y \cdot y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

   10. *-lowering-*.f64 63.6%

       \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \mathsf{+.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(y, y\right), \frac{1}{120}\right)\right)\right)\right)\right) \]

8. Simplified 63.6%

   \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot \left(0.16666666666666666 + \left(y \cdot y\right) \cdot 0.008333333333333333\right)\right)} \]
9. Add Preprocessing

Alternative 15: 37.5% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.007:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.007) 1.0 (* (* y y) 0.16666666666666666)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.007) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * 0.16666666666666666;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.007d0) then
        tmp = 1.0d0
    else
        tmp = (y * y) * 0.16666666666666666d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.007) {
		tmp = 1.0;
	} else {
		tmp = (y * y) * 0.16666666666666666;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.007:
		tmp = 1.0
	else:
		tmp = (y * y) * 0.16666666666666666
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.007)
		tmp = 1.0;
	else
		tmp = Float64(Float64(y * y) * 0.16666666666666666);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.007)
		tmp = 1.0;
	else
		tmp = (y * y) * 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.007], 1.0, N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00700000000000000015

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x} \]
   4. Step-by-step derivation

      1. cos-lowering-cos.f64 69.7%

         \[\leadsto \mathsf{cos.f64}\left(x\right) \]

   5. Simplified 69.7%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 42.3%

      \[\leadsto \color{blue}{1} \]

   if 0.00700000000000000015 < y

   1. Initial program 100.0%

      \[\cos x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

      2. *-lft-identity N/A

         \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

      3. distribute-rgt-in N/A

         \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      5. cos-lowering-cos.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 57.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
   6. Step-by-step derivation

      1. /-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cos x}{1}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1}{\cos x}}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\cos x}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \cos x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

      5. cos-lowering-cos.f64 57.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   7. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\cos x}}} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
   9. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

      4. *-lowering-*.f64 43.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

   10. Simplified 43.9%

       \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

   11. Taylor expanded in y around inf

       \[\leadsto \color{blue}{\frac{1}{6} \cdot {y}^{2}} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right) \]

       3. *-lowering-*.f64 43.9%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right) \]

   13. Simplified 43.9%

       \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(y \cdot y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.007:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot y\right) \cdot 0.16666666666666666\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 47.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + y \cdot \left(y \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* y (* y 0.16666666666666666))))

C

double code(double x, double y) {
	return 1.0 + (y * (y * 0.16666666666666666));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (y * (y * 0.16666666666666666d0))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (y * (y * 0.16666666666666666));
}

Python

def code(x, y):
	return 1.0 + (y * (y * 0.16666666666666666))

Julia

function code(x, y)
	return Float64(1.0 + Float64(y * Float64(y * 0.16666666666666666)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (y * (y * 0.16666666666666666));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(y * N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

   2. *-lft-identity N/A

      \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

   3. distribute-rgt-in N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 82.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

5. Simplified 82.5%

   \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
7. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left({y}^{2} \cdot \color{blue}{\frac{1}{6}}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(\left(y \cdot y\right) \cdot \frac{1}{6}\right)\right) \]

   4. associate-*l* N/A

      \[\leadsto \mathsf{+.f64}\left(1, \left(y \cdot \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \color{blue}{\left(y \cdot \frac{1}{6}\right)}\right)\right) \]

   6. *-lowering-*.f64 55.8%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(y, \color{blue}{\frac{1}{6}}\right)\right)\right) \]

8. Simplified 55.8%

   \[\leadsto \color{blue}{1 + y \cdot \left(y \cdot 0.16666666666666666\right)} \]
9. Add Preprocessing

Alternative 17: 47.1% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + \left(y \cdot y\right) \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* (* y y) 0.16666666666666666)))

C

double code(double x, double y) {
	return 1.0 + ((y * y) * 0.16666666666666666);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + ((y * y) * 0.16666666666666666d0)
end function

Java

public static double code(double x, double y) {
	return 1.0 + ((y * y) * 0.16666666666666666);
}

Python

def code(x, y):
	return 1.0 + ((y * y) * 0.16666666666666666)

Julia

function code(x, y)
	return Float64(1.0 + Float64(Float64(y * y) * 0.16666666666666666))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + ((y * y) * 0.16666666666666666);
end

Wolfram

code[x_, y_] := N[(1.0 + N[(N[(y * y), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x + \frac{1}{6} \cdot \left({y}^{2} \cdot \cos x\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \cos x + \left(\frac{1}{6} \cdot {y}^{2}\right) \cdot \color{blue}{\cos x} \]

   2. *-lft-identity N/A

      \[\leadsto 1 \cdot \cos x + \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)} \cdot \cos x \]

   3. distribute-rgt-in N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\cos x, \color{blue}{\left(1 + \frac{1}{6} \cdot {y}^{2}\right)}\right) \]

   5. cos-lowering-cos.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \left(\color{blue}{1} + \frac{1}{6} \cdot {y}^{2}\right)\right) \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right)\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right)\right) \]

   9. *-lowering-*.f64 82.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(x\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right)\right) \]

5. Simplified 82.5%

   \[\leadsto \color{blue}{\cos x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
6. Step-by-step derivation

   1. /-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{\cos x}{1}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   2. clear-num N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\frac{1}{\frac{1}{\cos x}}\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \left(\frac{1}{\cos x}\right)\right), \mathsf{+.f64}\left(\color{blue}{1}, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \cos x\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

   5. cos-lowering-cos.f64 82.4%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(1, \mathsf{/.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right)\right), \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, y\right)\right)\right)\right) \]

7. Applied egg-rr 82.4%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\cos x}}} \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{1}{6} \cdot {y}^{2}} \]
9. Step-by-step derivation

   1. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(\frac{1}{6} \cdot {y}^{2}\right)}\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \color{blue}{\left({y}^{2}\right)}\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \left(y \cdot \color{blue}{y}\right)\right)\right) \]

   4. *-lowering-*.f64 55.8%

      \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\frac{1}{6}, \mathsf{*.f64}\left(y, \color{blue}{y}\right)\right)\right) \]

10. Simplified 55.8%

    \[\leadsto \color{blue}{1 + 0.16666666666666666 \cdot \left(y \cdot y\right)} \]

11. Final simplification 55.8%

    \[\leadsto 1 + \left(y \cdot y\right) \cdot 0.16666666666666666 \]
12. Add Preprocessing

Alternative 18: 28.2% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 100.0%

   \[\cos x \cdot \frac{\sinh y}{y} \]
2. Add Preprocessing

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{\cos x} \]
4. Step-by-step derivation

   1. cos-lowering-cos.f64 55.4%

      \[\leadsto \mathsf{cos.f64}\left(x\right) \]

5. Simplified 55.4%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 33.8%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024161 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))