Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 56.9% → 90.4%

Time: 21.0s

Alternatives: 19

Speedup: 2.4×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

Initial Program: 56.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (/
  (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t)
  (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
}

Python

def code(x, y, z, t, a, b, c, i):
	return ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i)

Julia

function code(x, y, z, t, a, b, c, i)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{a}{y} + 1\\ t_2 := x \cdot \left(\frac{-1}{-1 - \frac{a}{y}} + \frac{z}{\left(x \cdot y\right) \cdot \left(t\_1 \cdot t\_1\right)}\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+48}:\\ \;\;\;\;\frac{t + \frac{y}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right)}}}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ a y) 1.0))
        (t_2
         (* x (+ (/ -1.0 (- -1.0 (/ a y))) (/ z (* (* x y) (* t_1 t_1)))))))
   (if (<= y -4e+45)
     t_2
     (if (<= y 1.65e+48)
       (/
        (+
         t
         (/ y (/ 1.0 (fma y (fma y (fma x y z) 27464.7644705) 230661.510616))))
        (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
       t_2))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (a / y) + 1.0;
	double t_2 = x * ((-1.0 / (-1.0 - (a / y))) + (z / ((x * y) * (t_1 * t_1))));
	double tmp;
	if (y <= -4e+45) {
		tmp = t_2;
	} else if (y <= 1.65e+48) {
		tmp = (t + (y / (1.0 / fma(y, fma(y, fma(x, y, z), 27464.7644705), 230661.510616)))) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(a / y) + 1.0)
	t_2 = Float64(x * Float64(Float64(-1.0 / Float64(-1.0 - Float64(a / y))) + Float64(z / Float64(Float64(x * y) * Float64(t_1 * t_1)))))
	tmp = 0.0
	if (y <= -4e+45)
		tmp = t_2;
	elseif (y <= 1.65e+48)
		tmp = Float64(Float64(t + Float64(y / Float64(1.0 / fma(y, fma(y, fma(x, y, z), 27464.7644705), 230661.510616)))) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(-1.0 / N[(-1.0 - N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z / N[(N[(x * y), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+45], t$95$2, If[LessEqual[y, 1.65e+48], N[(N[(t + N[(y / N[(1.0 / N[(y * N[(y * N[(x * y + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.9999999999999997e45 or 1.65000000000000011e48 < y

   1. Initial program 1.6%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 1.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 64.1

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 64.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

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

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

      15. /-lowering-/.f64 84.2

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

   10. Simplified 84.2%

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

   if -3.9999999999999997e45 < y < 1.65000000000000011e48

   1. Initial program 96.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. flip3-+ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right)}^{3} + {\frac{28832688827}{125000}}^{3}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) + \left(\frac{28832688827}{125000} \cdot \frac{28832688827}{125000} - \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \frac{28832688827}{125000}\right)}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. clear-num N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) + \left(\frac{28832688827}{125000} \cdot \frac{28832688827}{125000} - \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \frac{28832688827}{125000}\right)}{{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right)}^{3} + {\frac{28832688827}{125000}}^{3}}}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) + \left(\frac{28832688827}{125000} \cdot \frac{28832688827}{125000} - \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \frac{28832688827}{125000}\right)}{{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right)}^{3} + {\frac{28832688827}{125000}}^{3}}}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

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

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) + \left(\frac{28832688827}{125000} \cdot \frac{28832688827}{125000} - \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \frac{28832688827}{125000}\right)}{{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right)}^{3} + {\frac{28832688827}{125000}}^{3}}}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied egg-rr 96.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right)}}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+45}:\\ \;\;\;\;x \cdot \left(\frac{-1}{-1 - \frac{a}{y}} + \frac{z}{\left(x \cdot y\right) \cdot \left(\left(\frac{a}{y} + 1\right) \cdot \left(\frac{a}{y} + 1\right)\right)}\right)\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+48}:\\ \;\;\;\;\frac{t + \frac{y}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right)}}}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{-1 - \frac{a}{y}} + \frac{z}{\left(x \cdot y\right) \cdot \left(\left(\frac{a}{y} + 1\right) \cdot \left(\frac{a}{y} + 1\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)\\ t_2 := \frac{a}{y} + 1\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{t\_1}, \mathsf{fma}\left(x, \frac{{y}^{4}}{t\_1}, \frac{t}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{-1 - \frac{a}{y}} + \frac{z}{\left(x \cdot y\right) \cdot \left(t\_2 \cdot t\_2\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y (fma y (fma y (+ y a) b) c) i)) (t_2 (+ (/ a y) 1.0)))
   (if (<=
        (/
         (+
          (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
          t)
         (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
        INFINITY)
     (fma
      y
      (/ (fma y (fma y z 27464.7644705) 230661.510616) t_1)
      (fma x (/ (pow y 4.0) t_1) (/ t t_1)))
     (* x (+ (/ -1.0 (- -1.0 (/ a y))) (/ z (* (* x y) (* t_2 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, fma(y, fma(y, (y + a), b), c), i);
	double t_2 = (a / y) + 1.0;
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)) <= ((double) INFINITY)) {
		tmp = fma(y, (fma(y, fma(y, z, 27464.7644705), 230661.510616) / t_1), fma(x, (pow(y, 4.0) / t_1), (t / t_1)));
	} else {
		tmp = x * ((-1.0 / (-1.0 - (a / y))) + (z / ((x * y) * (t_2 * t_2))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, fma(y, fma(y, Float64(y + a), b), c), i)
	t_2 = Float64(Float64(a / y) + 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)) <= Inf)
		tmp = fma(y, Float64(fma(y, fma(y, z, 27464.7644705), 230661.510616) / t_1), fma(x, Float64((y ^ 4.0) / t_1), Float64(t / t_1)));
	else
		tmp = Float64(x * Float64(Float64(-1.0 / Float64(-1.0 - Float64(a / y))) + Float64(z / Float64(Float64(x * y) * Float64(t_2 * t_2)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(y * N[(N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x * N[(N[Power[y, 4.0], $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(-1.0 / N[(-1.0 - N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z / N[(N[(x * y), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 89.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right) + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Simplified 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \mathsf{fma}\left(x, \frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\right)} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 0.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 67.8

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 67.8%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

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

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

      15. /-lowering-/.f64 88.7

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

   10. Simplified 88.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \mathsf{fma}\left(x, \frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{-1 - \frac{a}{y}} + \frac{z}{\left(x \cdot y\right) \cdot \left(\left(\frac{a}{y} + 1\right) \cdot \left(\frac{a}{y} + 1\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ t_2 := \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 27464.7644705, 230661.510616\right), t\right)}{i}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (/ t (fma y c i)))
        (t_2
         (/
          (+
           (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
           t)
          (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))))
   (if (<= t_2 -2e+76)
     t_1
     (if (<= t_2 5e-127)
       (/ (fma y (fma y 27464.7644705 230661.510616) t) i)
       (if (<= t_2 5e+307) t_1 (/ x (+ (/ a y) 1.0)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = t / fma(y, c, i);
	double t_2 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i);
	double tmp;
	if (t_2 <= -2e+76) {
		tmp = t_1;
	} else if (t_2 <= 5e-127) {
		tmp = fma(y, fma(y, 27464.7644705, 230661.510616), t) / i;
	} else if (t_2 <= 5e+307) {
		tmp = t_1;
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t / fma(y, c, i))
	t_2 = Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i))
	tmp = 0.0
	if (t_2 <= -2e+76)
		tmp = t_1;
	elseif (t_2 <= 5e-127)
		tmp = Float64(fma(y, fma(y, 27464.7644705, 230661.510616), t) / i);
	elseif (t_2 <= 5e+307)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t / N[(y * c + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+76], t$95$1, If[LessEqual[t$95$2, 5e-127], N[(N[(y * N[(y * 27464.7644705 + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$2, 5e+307], t$95$1, N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < -2.0000000000000001e76 or 4.9999999999999997e-127 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5e307

   1. Initial program 92.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + {y}^{2}\right), i\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + {y}^{2}\right) + c}, i\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + {y}^{2}, c\right)}, i\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} + b}, c\right), i\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot y} + b, c\right), i\right)} \]

      17. accelerator-lowering-fma.f64 90.2

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y, b\right)}, c\right), i\right)} \]

   5. Simplified 90.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]
   7. Step-by-step derivation

   8. Simplified 74.9%

      \[\leadsto \frac{\color{blue}{t}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \color{blue}{c}, i\right)} \]
   10. Step-by-step derivation

   11. Simplified 72.9%

       \[\leadsto \frac{t}{\mathsf{fma}\left(y, \color{blue}{c}, i\right)} \]

   if -2.0000000000000001e76 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 4.9999999999999997e-127

   1. Initial program 90.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i} \]

      10. accelerator-lowering-fma.f64 59.2

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, 27464.7644705\right), 230661.510616\right), t\right)}{i} \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{i}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i} \]
   7. Step-by-step derivation

   8. Simplified 56.3%

      \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{27464.7644705}, 230661.510616\right), t\right)}{i} \]

   if 5e307 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 2.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 66.1

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 66.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 78.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 78.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq -2 \cdot 10^{+76}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 27464.7644705, 230661.510616\right), t\right)}{i}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ t_2 := \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 27464.7644705, 230661.510616\right), t\right)}{i}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (/ t (fma y c i)))
        (t_2
         (/
          (+
           (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
           t)
          (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))))
   (if (<= t_2 -2e+76)
     t_1
     (if (<= t_2 5e-127)
       (/ (fma y (fma y 27464.7644705 230661.510616) t) i)
       (if (<= t_2 5e+307) t_1 x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = t / fma(y, c, i);
	double t_2 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i);
	double tmp;
	if (t_2 <= -2e+76) {
		tmp = t_1;
	} else if (t_2 <= 5e-127) {
		tmp = fma(y, fma(y, 27464.7644705, 230661.510616), t) / i;
	} else if (t_2 <= 5e+307) {
		tmp = t_1;
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(t / fma(y, c, i))
	t_2 = Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i))
	tmp = 0.0
	if (t_2 <= -2e+76)
		tmp = t_1;
	elseif (t_2 <= 5e-127)
		tmp = Float64(fma(y, fma(y, 27464.7644705, 230661.510616), t) / i);
	elseif (t_2 <= 5e+307)
		tmp = t_1;
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(t / N[(y * c + i), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+76], t$95$1, If[LessEqual[t$95$2, 5e-127], N[(N[(y * N[(y * 27464.7644705 + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / i), $MachinePrecision], If[LessEqual[t$95$2, 5e+307], t$95$1, x]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < -2.0000000000000001e76 or 4.9999999999999997e-127 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5e307

   1. Initial program 92.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + {y}^{2}\right), i\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + {y}^{2}\right) + c}, i\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + {y}^{2}, c\right)}, i\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} + b}, c\right), i\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot y} + b, c\right), i\right)} \]

      17. accelerator-lowering-fma.f64 90.2

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y, b\right)}, c\right), i\right)} \]

   5. Simplified 90.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]
   7. Step-by-step derivation

   8. Simplified 74.9%

      \[\leadsto \frac{\color{blue}{t}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \color{blue}{c}, i\right)} \]
   10. Step-by-step derivation

   11. Simplified 72.9%

       \[\leadsto \frac{t}{\mathsf{fma}\left(y, \color{blue}{c}, i\right)} \]

   if -2.0000000000000001e76 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 4.9999999999999997e-127

   1. Initial program 90.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in i around inf

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i} \]

      10. accelerator-lowering-fma.f64 59.2

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, 27464.7644705\right), 230661.510616\right), t\right)}{i} \]

   5. Simplified 59.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{i}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i} \]
   7. Step-by-step derivation

   8. Simplified 56.3%

      \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{27464.7644705}, 230661.510616\right), t\right)}{i} \]

   if 5e307 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 2.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 62.3%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq -2 \cdot 10^{+76}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 27464.7644705, 230661.510616\right), t\right)}{i}\\ \mathbf{elif}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)\\ t_2 := \frac{a}{y} + 1\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{t\_1}, y, \mathsf{fma}\left(x, \frac{y \cdot \left(y \cdot \left(y \cdot y\right)\right)}{t\_1}, \frac{t}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{-1 - \frac{a}{y}} + \frac{z}{\left(x \cdot y\right) \cdot \left(t\_2 \cdot t\_2\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y (fma y (fma y (+ y a) b) c) i)) (t_2 (+ (/ a y) 1.0)))
   (if (<=
        (/
         (+
          (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
          t)
         (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
        INFINITY)
     (fma
      (/ (fma y (fma y z 27464.7644705) 230661.510616) t_1)
      y
      (fma x (/ (* y (* y (* y y))) t_1) (/ t t_1)))
     (* x (+ (/ -1.0 (- -1.0 (/ a y))) (/ z (* (* x y) (* t_2 t_2))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, fma(y, fma(y, (y + a), b), c), i);
	double t_2 = (a / y) + 1.0;
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)) <= ((double) INFINITY)) {
		tmp = fma((fma(y, fma(y, z, 27464.7644705), 230661.510616) / t_1), y, fma(x, ((y * (y * (y * y))) / t_1), (t / t_1)));
	} else {
		tmp = x * ((-1.0 / (-1.0 - (a / y))) + (z / ((x * y) * (t_2 * t_2))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, fma(y, fma(y, Float64(y + a), b), c), i)
	t_2 = Float64(Float64(a / y) + 1.0)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)) <= Inf)
		tmp = fma(Float64(fma(y, fma(y, z, 27464.7644705), 230661.510616) / t_1), y, fma(x, Float64(Float64(y * Float64(y * Float64(y * y))) / t_1), Float64(t / t_1)));
	else
		tmp = Float64(x * Float64(Float64(-1.0 / Float64(-1.0 - Float64(a / y))) + Float64(z / Float64(Float64(x * y) * Float64(t_2 * t_2)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] * y + N[(x * N[(N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(-1.0 / N[(-1.0 - N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(z / N[(N[(x * y), $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < +inf.0

   1. Initial program 89.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right) + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      3. associate-+l+ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \left(\frac{x \cdot {y}^{4}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{t}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}\right)} \]

   5. Simplified 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \mathsf{fma}\left(x, \frac{{y}^{4}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(y \cdot z + \frac{54929528941}{2000000}\right) + \frac{28832688827}{125000}}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \cdot y} + \left(x \cdot \frac{{y}^{4}}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} + \frac{t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\right) \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y \cdot \left(y \cdot z + \frac{54929528941}{2000000}\right) + \frac{28832688827}{125000}}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}, y, x \cdot \frac{{y}^{4}}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} + \frac{t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\right)} \]

   7. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, y, \mathsf{fma}\left(x, \frac{y \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\right)} \]

   if +inf.0 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 0.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 0.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 67.8

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 67.8%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. associate-*r* N/A

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

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

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

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

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

      10. unpow2 N/A

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

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

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

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

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

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

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

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

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

      15. /-lowering-/.f64 88.7

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

   10. Simplified 88.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, y, \mathsf{fma}\left(x, \frac{y \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{-1 - \frac{a}{y}} + \frac{z}{\left(x \cdot y\right) \cdot \left(\left(\frac{a}{y} + 1\right) \cdot \left(\frac{a}{y} + 1\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{t\_1} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{t + \frac{y}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right)}}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i)))
   (if (<=
        (/
         (+
          (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
          t)
         t_1)
        5e+307)
     (/
      (+
       t
       (/ y (/ 1.0 (fma y (fma y (fma x y z) 27464.7644705) 230661.510616))))
      t_1)
     (/ x (+ (/ a y) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * ((y * ((y * (y + a)) + b)) + c)) + i;
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1) <= 5e+307) {
		tmp = (t + (y / (1.0 / fma(y, fma(y, fma(x, y, z), 27464.7644705), 230661.510616)))) / t_1;
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1) <= 5e+307)
		tmp = Float64(Float64(t + Float64(y / Float64(1.0 / fma(y, fma(y, fma(x, y, z), 27464.7644705), 230661.510616)))) / t_1);
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / t$95$1), $MachinePrecision], 5e+307], N[(N[(t + N[(y / N[(1.0 / N[(y * N[(y * N[(x * y + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5e307

   1. Initial program 91.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. flip3-+ N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right)}^{3} + {\frac{28832688827}{125000}}^{3}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) + \left(\frac{28832688827}{125000} \cdot \frac{28832688827}{125000} - \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \frac{28832688827}{125000}\right)}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. clear-num N/A

         \[\leadsto \frac{y \cdot \color{blue}{\frac{1}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) + \left(\frac{28832688827}{125000} \cdot \frac{28832688827}{125000} - \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \frac{28832688827}{125000}\right)}{{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right)}^{3} + {\frac{28832688827}{125000}}^{3}}}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      4. un-div-inv N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) + \left(\frac{28832688827}{125000} \cdot \frac{28832688827}{125000} - \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \frac{28832688827}{125000}\right)}{{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right)}^{3} + {\frac{28832688827}{125000}}^{3}}}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

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

         \[\leadsto \frac{\color{blue}{\frac{y}{\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) + \left(\frac{28832688827}{125000} \cdot \frac{28832688827}{125000} - \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot \frac{28832688827}{125000}\right)}{{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right)}^{3} + {\frac{28832688827}{125000}}^{3}}}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied egg-rr 91.7%

      \[\leadsto \frac{\color{blue}{\frac{y}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right)}}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   if 5e307 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 2.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 66.1

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 66.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 78.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 78.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{t + \frac{y}{\frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right)}}}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (/
          (+
           (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
           t)
          (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))))
   (if (<= t_1 5e+307) t_1 (/ x (+ (/ a y) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i);
	double tmp;
	if (t_1 <= 5e+307) {
		tmp = t_1;
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705d0)) + 230661.510616d0)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)
    if (t_1 <= 5d+307) then
        tmp = t_1
    else
        tmp = x / ((a / y) + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i);
	double tmp;
	if (t_1 <= 5e+307) {
		tmp = t_1;
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)
	tmp = 0
	if t_1 <= 5e+307:
		tmp = t_1
	else:
		tmp = x / ((a / y) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i))
	tmp = 0.0
	if (t_1 <= 5e+307)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i);
	tmp = 0.0;
	if (t_1 <= 5e+307)
		tmp = t_1;
	else
		tmp = x / ((a / y) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+307], t$95$1, N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5e307

   1. Initial program 91.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   if 5e307 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 2.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 66.1

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 66.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 78.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 78.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 83.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i\\ \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{t\_1} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{t + \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), y, y \cdot 230661.510616\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i)))
   (if (<=
        (/
         (+
          (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
          t)
         t_1)
        5e+307)
     (/
      (+ t (fma (* y (fma y (fma x y z) 27464.7644705)) y (* y 230661.510616)))
      t_1)
     (/ x (+ (/ a y) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (y * ((y * ((y * (y + a)) + b)) + c)) + i;
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1) <= 5e+307) {
		tmp = (t + fma((y * fma(y, fma(x, y, z), 27464.7644705)), y, (y * 230661.510616))) / t_1;
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / t_1) <= 5e+307)
		tmp = Float64(Float64(t + fma(Float64(y * fma(y, fma(x, y, z), 27464.7644705)), y, Float64(y * 230661.510616))) / t_1);
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]}, If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / t$95$1), $MachinePrecision], 5e+307], N[(N[(t + N[(N[(y * N[(y * N[(x * y + z), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] * y + N[(y * 230661.510616), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5e307

   1. Initial program 91.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y\right) \cdot y + \frac{28832688827}{125000} \cdot y\right)} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y, y, \frac{28832688827}{125000} \cdot y\right)} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      4. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right)}, y, \frac{28832688827}{125000} \cdot y\right) + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{y \cdot \left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right)}, y, \frac{28832688827}{125000} \cdot y\right) + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \left(\color{blue}{y \cdot \left(x \cdot y + z\right)} + \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000} \cdot y\right) + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \color{blue}{\mathsf{fma}\left(y, x \cdot y + z, \frac{54929528941}{2000000}\right)}, y, \frac{28832688827}{125000} \cdot y\right) + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(x, y, z\right)}, \frac{54929528941}{2000000}\right), y, \frac{28832688827}{125000} \cdot y\right) + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      9. *-lowering-*.f64 91.7

         \[\leadsto \frac{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), y, \color{blue}{230661.510616 \cdot y}\right) + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   4. Applied egg-rr 91.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), y, 230661.510616 \cdot y\right)} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   if 5e307 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 2.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 66.1

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 66.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 78.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 78.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{t + \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), y, y \cdot 230661.510616\right)}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 83.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
        t)
       (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
      5e+307)
   (*
    (fma y (fma y (fma y (fma x y z) 27464.7644705) 230661.510616) t)
    (/ 1.0 (fma y (fma y (fma y (+ y a) b) c) i)))
   (/ x (+ (/ a y) 1.0))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)) <= 5e+307) {
		tmp = fma(y, fma(y, fma(y, fma(x, y, z), 27464.7644705), 230661.510616), t) * (1.0 / fma(y, fma(y, fma(y, (y + a), b), c), i));
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)) <= 5e+307)
		tmp = Float64(fma(y, fma(y, fma(y, fma(x, y, z), 27464.7644705), 230661.510616), t) * Float64(1.0 / fma(y, fma(y, fma(y, Float64(y + a), b), c), i)));
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], 5e+307], N[(N[(y * N[(y * N[(y * N[(x * y + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] * N[(1.0 / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5e307

   1. Initial program 91.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

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

         \[\leadsto \color{blue}{\left(\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

      3. *-commutative N/A

         \[\leadsto \left(\color{blue}{y \cdot \left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right)} + t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

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

         \[\leadsto \color{blue}{\mathsf{fma}\left(y, \left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}, t\right)} \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right)} + \frac{28832688827}{125000}, t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}, \frac{28832688827}{125000}\right)}, t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(x \cdot y + z\right)} + \frac{54929528941}{2000000}, \frac{28832688827}{125000}\right), t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x \cdot y + z, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(x, y, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right) \cdot \frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

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

          \[\leadsto \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right) \cdot \color{blue}{\frac{1}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}} \]

   4. Applied egg-rr 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}} \]

   if 5e307 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 2.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 66.1

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 66.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 78.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 78.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), y \cdot y, \mathsf{fma}\left(y, 230661.510616, t\right)\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
        t)
       (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
      5e+307)
   (/
    (fma (fma y (fma y x z) 27464.7644705) (* y y) (fma y 230661.510616 t))
    (fma y (fma y (fma y y b) c) i))
   (/ x (+ (/ a y) 1.0))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)) <= 5e+307) {
		tmp = fma(fma(y, fma(y, x, z), 27464.7644705), (y * y), fma(y, 230661.510616, t)) / fma(y, fma(y, fma(y, y, b), c), i);
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)) <= 5e+307)
		tmp = Float64(fma(fma(y, fma(y, x, z), 27464.7644705), Float64(y * y), fma(y, 230661.510616, t)) / fma(y, fma(y, fma(y, y, b), c), i));
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], 5e+307], N[(N[(N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] * N[(y * y), $MachinePrecision] + N[(y * 230661.510616 + t), $MachinePrecision]), $MachinePrecision] / N[(y * N[(y * N[(y * y + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5e307

   1. Initial program 91.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + {y}^{2}\right), i\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + {y}^{2}\right) + c}, i\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + {y}^{2}, c\right)}, i\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} + b}, c\right), i\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot y} + b, c\right), i\right)} \]

      17. accelerator-lowering-fma.f64 88.8

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y, b\right)}, c\right), i\right)} \]

   5. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y \cdot \left(y \cdot \left(y \cdot x + z\right) + \frac{54929528941}{2000000}\right)\right) \cdot y + \frac{28832688827}{125000} \cdot y\right)} + t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(y \cdot \left(y \cdot x + z\right) + \frac{54929528941}{2000000}\right)\right) \cdot y + \left(\frac{28832688827}{125000} \cdot y + t\right)}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\left(y \cdot \left(y \cdot x + z\right) + \frac{54929528941}{2000000}\right) \cdot y\right)} \cdot y + \left(\frac{28832688827}{125000} \cdot y + t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

      4. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \left(y \cdot x + z\right) + \frac{54929528941}{2000000}\right) \cdot \left(y \cdot y\right)} + \left(\frac{28832688827}{125000} \cdot y + t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{\left(y \cdot \left(y \cdot x + z\right) + \frac{54929528941}{2000000}\right) \cdot \left(y \cdot y\right) + \left(\color{blue}{y \cdot \frac{28832688827}{125000}} + t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y \cdot \left(y \cdot x + z\right) + \frac{54929528941}{2000000}, y \cdot y, y \cdot \frac{28832688827}{125000} + t\right)}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(y, y \cdot x + z, \frac{54929528941}{2000000}\right)}, y \cdot y, y \cdot \frac{28832688827}{125000} + t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), y \cdot y, y \cdot \frac{28832688827}{125000} + t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \color{blue}{y \cdot y}, y \cdot \frac{28832688827}{125000} + t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

      10. accelerator-lowering-fma.f64 88.8

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), y \cdot y, \color{blue}{\mathsf{fma}\left(y, 230661.510616, t\right)}\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

   7. Applied egg-rr 88.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), y \cdot y, \mathsf{fma}\left(y, 230661.510616, t\right)\right)}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

   if 5e307 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 2.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 66.1

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 66.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 78.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 78.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), y \cdot y, \mathsf{fma}\left(y, 230661.510616, t\right)\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* y (+ (* y (+ (* y (+ (* x y) z)) 27464.7644705)) 230661.510616))
        t)
       (+ (* y (+ (* y (+ (* y (+ y a)) b)) c)) i))
      5e+307)
   (/
    (fma y (fma y (fma y (fma y x z) 27464.7644705) 230661.510616) t)
    (fma y (fma y (fma y y b) c) i))
   (/ x (+ (/ a y) 1.0))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((y * ((y * ((y * ((x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / ((y * ((y * ((y * (y + a)) + b)) + c)) + i)) <= 5e+307) {
		tmp = fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, y, b), c), i);
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (Float64(Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(Float64(x * y) + z)) + 27464.7644705)) + 230661.510616)) + t) / Float64(Float64(y * Float64(Float64(y * Float64(Float64(y * Float64(y + a)) + b)) + c)) + i)) <= 5e+307)
		tmp = Float64(fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, y, b), c), i));
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[N[(N[(N[(y * N[(N[(y * N[(N[(y * N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision] + 27464.7644705), $MachinePrecision]), $MachinePrecision] + 230661.510616), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(N[(y * N[(N[(y * N[(N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], 5e+307], N[(N[(y * N[(y * N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * y + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5e307

   1. Initial program 91.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + {y}^{2}\right), i\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + {y}^{2}\right) + c}, i\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + {y}^{2}, c\right)}, i\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} + b}, c\right), i\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot y} + b, c\right), i\right)} \]

      17. accelerator-lowering-fma.f64 88.8

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y, b\right)}, c\right), i\right)} \]

   5. Simplified 88.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}} \]

   if 5e307 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 2.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 2.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 66.1

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 66.1%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 78.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 78.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(y \cdot \left(y \cdot \left(x \cdot y + z\right) + 27464.7644705\right) + 230661.510616\right) + t}{y \cdot \left(y \cdot \left(y \cdot \left(y + a\right) + b\right) + c\right) + i} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 80.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+46}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.85e+46)
   (- x (/ (- (* x a) z) y))
   (if (<= y 6.8e+34)
     (/
      (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
      (fma y (fma y (fma y (+ y a) b) c) i))
     (/ x (+ (/ a y) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.85e+46) {
		tmp = x - (((x * a) - z) / y);
	} else if (y <= 6.8e+34) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.85e+46)
		tmp = Float64(x - Float64(Float64(Float64(x * a) - z) / y));
	elseif (y <= 6.8e+34)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.85e+46], N[(x - N[(N[(N[(x * a), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e+34], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.84999999999999995e46

   1. Initial program 3.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto x - \color{blue}{\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot x\right)}}{y} \]

      6. metadata-eval N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{1} \cdot \left(a \cdot x\right)}{y} \]

      7. *-lft-identity N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{a \cdot x}}{y} \]

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

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + a \cdot x}}{y} \]

      9. mul-1-neg N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

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

          \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{\left(\mathsf{neg}\left(z\right)\right) + \color{blue}{x \cdot a}}{y} \]

      12. *-lowering-*.f64 69.2

          \[\leadsto x - \frac{\left(-z\right) + \color{blue}{x \cdot a}}{y} \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{x - \frac{\left(-z\right) + x \cdot a}{y}} \]

   if -1.84999999999999995e46 < y < 6.7999999999999999e34

   1. Initial program 98.4%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot z, \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot z + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + y \cdot \left(a + y\right)\right), i\right)}} \]

      10. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + y \cdot \left(a + y\right)\right) + c}, i\right)} \]

      11. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + y \cdot \left(a + y\right), c\right)}, i\right)} \]

      12. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(a + y\right) + b}, c\right), i\right)} \]

      13. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, a + y, b\right)}, c\right), i\right)} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)} \]

      15. +-lowering-+.f64 90.7

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)} \]

   5. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}} \]

   if 6.7999999999999999e34 < y

   1. Initial program 2.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 2.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 69.3

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 69.3%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 76.4

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 76.4%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+46}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 74.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 2700000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 27464.7644705, 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -9.5e+40)
   (- x (/ (- (* x a) z) y))
   (if (<= y 2700000000000.0)
     (/
      (fma y (fma y 27464.7644705 230661.510616) t)
      (fma y (fma y (fma y y b) c) i))
     (/ x (+ (/ a y) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -9.5e+40) {
		tmp = x - (((x * a) - z) / y);
	} else if (y <= 2700000000000.0) {
		tmp = fma(y, fma(y, 27464.7644705, 230661.510616), t) / fma(y, fma(y, fma(y, y, b), c), i);
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -9.5e+40)
		tmp = Float64(x - Float64(Float64(Float64(x * a) - z) / y));
	elseif (y <= 2700000000000.0)
		tmp = Float64(fma(y, fma(y, 27464.7644705, 230661.510616), t) / fma(y, fma(y, fma(y, y, b), c), i));
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -9.5e+40], N[(x - N[(N[(N[(x * a), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2700000000000.0], N[(N[(y * N[(y * 27464.7644705 + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * y + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.5000000000000003e40

   1. Initial program 5.5%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto x - \color{blue}{\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot x\right)}}{y} \]

      6. metadata-eval N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{1} \cdot \left(a \cdot x\right)}{y} \]

      7. *-lft-identity N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{a \cdot x}}{y} \]

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

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + a \cdot x}}{y} \]

      9. mul-1-neg N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

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

          \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{\left(\mathsf{neg}\left(z\right)\right) + \color{blue}{x \cdot a}}{y} \]

      12. *-lowering-*.f64 67.8

          \[\leadsto x - \frac{\left(-z\right) + \color{blue}{x \cdot a}}{y} \]

   5. Simplified 67.8%

      \[\leadsto \color{blue}{x - \frac{\left(-z\right) + x \cdot a}{y}} \]

   if -9.5000000000000003e40 < y < 2.7e12

   1. Initial program 99.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + {y}^{2}\right), i\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + {y}^{2}\right) + c}, i\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + {y}^{2}, c\right)}, i\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} + b}, c\right), i\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot y} + b, c\right), i\right)} \]

      17. accelerator-lowering-fma.f64 96.4

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y, b\right)}, c\right), i\right)} \]

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]
   7. Step-by-step derivation

   8. Simplified 85.6%

      \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{27464.7644705}, 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

   if 2.7e12 < y

   1. Initial program 11.2%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 11.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 63.7

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 63.7%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 70.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 70.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+40}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 2700000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, 27464.7644705, 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+44}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 30000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 230661.510616, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -3.5e+44)
   (- x (/ (- (* x a) z) y))
   (if (<= y 30000000000000.0)
     (/ (fma y 230661.510616 t) (fma y (fma y (fma y y b) c) i))
     (/ x (+ (/ a y) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -3.5e+44) {
		tmp = x - (((x * a) - z) / y);
	} else if (y <= 30000000000000.0) {
		tmp = fma(y, 230661.510616, t) / fma(y, fma(y, fma(y, y, b), c), i);
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -3.5e+44)
		tmp = Float64(x - Float64(Float64(Float64(x * a) - z) / y));
	elseif (y <= 30000000000000.0)
		tmp = Float64(fma(y, 230661.510616, t) / fma(y, fma(y, fma(y, y, b), c), i));
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -3.5e+44], N[(x - N[(N[(N[(x * a), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 30000000000000.0], N[(N[(y * 230661.510616 + t), $MachinePrecision] / N[(y * N[(y * N[(y * y + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4999999999999999e44

   1. Initial program 3.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto x - \color{blue}{\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot x\right)}}{y} \]

      6. metadata-eval N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{1} \cdot \left(a \cdot x\right)}{y} \]

      7. *-lft-identity N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{a \cdot x}}{y} \]

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

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + a \cdot x}}{y} \]

      9. mul-1-neg N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

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

          \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{\left(\mathsf{neg}\left(z\right)\right) + \color{blue}{x \cdot a}}{y} \]

      12. *-lowering-*.f64 69.2

          \[\leadsto x - \frac{\left(-z\right) + \color{blue}{x \cdot a}}{y} \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{x - \frac{\left(-z\right) + x \cdot a}{y}} \]

   if -3.4999999999999999e44 < y < 3e13

   1. Initial program 99.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + {y}^{2}\right), i\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + {y}^{2}\right) + c}, i\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + {y}^{2}, c\right)}, i\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} + b}, c\right), i\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot y} + b, c\right), i\right)} \]

      17. accelerator-lowering-fma.f64 95.8

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y, b\right)}, c\right), i\right)} \]

   5. Simplified 95.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\frac{28832688827}{125000}}, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]
   7. Step-by-step derivation

   8. Simplified 83.0%

      \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{230661.510616}, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

   if 3e13 < y

   1. Initial program 11.2%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 11.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 63.7

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 63.7%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 70.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 70.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+44}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 30000000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 230661.510616, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+44}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 20500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 230661.510616, t\right)}{i + y \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -3.5e+44)
   (- x (/ (- (* x a) z) y))
   (if (<= y 20500000000000.0)
     (/ (fma y 230661.510616 t) (+ i (* y c)))
     (/ x (+ (/ a y) 1.0)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -3.5e+44) {
		tmp = x - (((x * a) - z) / y);
	} else if (y <= 20500000000000.0) {
		tmp = fma(y, 230661.510616, t) / (i + (y * c));
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -3.5e+44)
		tmp = Float64(x - Float64(Float64(Float64(x * a) - z) / y));
	elseif (y <= 20500000000000.0)
		tmp = Float64(fma(y, 230661.510616, t) / Float64(i + Float64(y * c)));
	else
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -3.5e+44], N[(x - N[(N[(N[(x * a), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 20500000000000.0], N[(N[(y * 230661.510616 + t), $MachinePrecision] / N[(i + N[(y * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.4999999999999999e44

   1. Initial program 3.3%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto x - \color{blue}{\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot x\right)}}{y} \]

      6. metadata-eval N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{1} \cdot \left(a \cdot x\right)}{y} \]

      7. *-lft-identity N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{a \cdot x}}{y} \]

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

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + a \cdot x}}{y} \]

      9. mul-1-neg N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

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

          \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{\left(\mathsf{neg}\left(z\right)\right) + \color{blue}{x \cdot a}}{y} \]

      12. *-lowering-*.f64 69.2

          \[\leadsto x - \frac{\left(-z\right) + \color{blue}{x \cdot a}}{y} \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{x - \frac{\left(-z\right) + x \cdot a}{y}} \]

   if -3.4999999999999999e44 < y < 2.05e13

   1. Initial program 99.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t + \frac{28832688827}{125000} \cdot y}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{28832688827}{125000} \cdot y + t}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \frac{28832688827}{125000}} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      3. accelerator-lowering-fma.f64 84.7

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 230661.510616, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   5. Simplified 84.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, 230661.510616, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{28832688827}{125000}, t\right)}{\color{blue}{c} \cdot y + i} \]
   7. Step-by-step derivation

   8. Simplified 79.7%

      \[\leadsto \frac{\mathsf{fma}\left(y, 230661.510616, t\right)}{\color{blue}{c} \cdot y + i} \]

   if 2.05e13 < y

   1. Initial program 11.2%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

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

         \[\leadsto \frac{1}{\color{blue}{\frac{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\color{blue}{y \cdot \left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right)} + i}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

         \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(y, \left(\left(y + a\right) \cdot y + b\right) \cdot y + c, i\right)}}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      6. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\left(y + a\right) \cdot y + b\right)} + c, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \left(y + a\right) \cdot y + b, c\right)}, i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      8. *-commutative N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(y + a\right)} + b, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

      9. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y + a, b\right)}, c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

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

          \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y + a}, b\right), c\right), i\right)}{\left(\left(\left(x \cdot y + z\right) \cdot y + \frac{54929528941}{2000000}\right) \cdot y + \frac{28832688827}{125000}\right) \cdot y + t}} \]

   4. Applied egg-rr 11.3%

      \[\leadsto \color{blue}{\frac{1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right)}}} \]

   5. Taylor expanded in y around inf

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

      9. *-lowering-*.f64 63.7

         \[\leadsto \frac{1}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\color{blue}{\left(x \cdot x\right)} \cdot y}} \]

   7. Simplified 63.7%

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{x} + \frac{a}{x \cdot y}\right) - \frac{z}{\left(x \cdot x\right) \cdot y}}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
   9. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]

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

         \[\leadsto \frac{x}{\color{blue}{1 + \frac{a}{y}}} \]

      3. /-lowering-/.f64 70.1

         \[\leadsto \frac{x}{1 + \color{blue}{\frac{a}{y}}} \]

   10. Simplified 70.1%

       \[\leadsto \color{blue}{\frac{x}{1 + \frac{a}{y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+44}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 20500000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 230661.510616, t\right)}{i + y \cdot c}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 63.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{x \cdot a - z}{y}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1900000000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- x (/ (- (* x a) z) y))))
   (if (<= y -2e+30) t_1 (if (<= y 1900000000000.0) (/ t (fma y c i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x - (((x * a) - z) / y);
	double tmp;
	if (y <= -2e+30) {
		tmp = t_1;
	} else if (y <= 1900000000000.0) {
		tmp = t / fma(y, c, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x - Float64(Float64(Float64(x * a) - z) / y))
	tmp = 0.0
	if (y <= -2e+30)
		tmp = t_1;
	elseif (y <= 1900000000000.0)
		tmp = Float64(t / fma(y, c, i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x - N[(N[(N[(x * a), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+30], t$95$1, If[LessEqual[y, 1900000000000.0], N[(t / N[(y * c + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2e30 or 1.9e12 < y

   1. Initial program 9.7%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto \color{blue}{x - \frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

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

         \[\leadsto x - \color{blue}{\frac{-1 \cdot z - -1 \cdot \left(a \cdot x\right)}{y}} \]

      5. cancel-sign-sub-inv N/A

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + \left(\mathsf{neg}\left(-1\right)\right) \cdot \left(a \cdot x\right)}}{y} \]

      6. metadata-eval N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{1} \cdot \left(a \cdot x\right)}{y} \]

      7. *-lft-identity N/A

         \[\leadsto x - \frac{-1 \cdot z + \color{blue}{a \cdot x}}{y} \]

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

         \[\leadsto x - \frac{\color{blue}{-1 \cdot z + a \cdot x}}{y} \]

      9. mul-1-neg N/A

         \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

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

          \[\leadsto x - \frac{\color{blue}{\left(\mathsf{neg}\left(z\right)\right)} + a \cdot x}{y} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{\left(\mathsf{neg}\left(z\right)\right) + \color{blue}{x \cdot a}}{y} \]

      12. *-lowering-*.f64 62.2

          \[\leadsto x - \frac{\left(-z\right) + \color{blue}{x \cdot a}}{y} \]

   5. Simplified 62.2%

      \[\leadsto \color{blue}{x - \frac{\left(-z\right) + x \cdot a}{y}} \]

   if -2e30 < y < 1.9e12

   1. Initial program 99.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + {y}^{2}\right), i\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + {y}^{2}\right) + c}, i\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + {y}^{2}, c\right)}, i\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} + b}, c\right), i\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot y} + b, c\right), i\right)} \]

      17. accelerator-lowering-fma.f64 96.4

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y, b\right)}, c\right), i\right)} \]

   5. Simplified 96.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]
   7. Step-by-step derivation

   8. Simplified 67.6%

      \[\leadsto \frac{\color{blue}{t}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \color{blue}{c}, i\right)} \]
   10. Step-by-step derivation

   11. Simplified 66.4%

       \[\leadsto \frac{t}{\mathsf{fma}\left(y, \color{blue}{c}, i\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+30}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \mathbf{elif}\;y \leq 1900000000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x \cdot a - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 58.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 620000000000:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, c, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -5000.0) x (if (<= y 620000000000.0) (/ t (fma y c i)) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -5000.0) {
		tmp = x;
	} else if (y <= 620000000000.0) {
		tmp = t / fma(y, c, i);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -5000.0)
		tmp = x;
	elseif (y <= 620000000000.0)
		tmp = Float64(t / fma(y, c, i));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -5000.0], x, If[LessEqual[y, 620000000000.0], N[(t / N[(y * c + i), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -5e3 or 6.2e11 < y

   1. Initial program 15.1%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 52.6%

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

   if -5e3 < y < 6.2e11

   1. Initial program 99.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]
   4. Step-by-step derivation

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

         \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right)\right) + t}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(y, \frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right), t\right)}}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)\right) + \frac{28832688827}{125000}}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

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

         \[\leadsto \frac{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, \frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right), \frac{28832688827}{125000}\right)}, t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot \left(z + x \cdot y\right) + \frac{54929528941}{2000000}}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, z + x \cdot y, \frac{54929528941}{2000000}\right)}, \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{x \cdot y + z}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      9. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot x} + z, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, x, z\right)}, \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{i + y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{y \cdot \left(c + y \cdot \left(b + {y}^{2}\right)\right) + i}} \]

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\color{blue}{\mathsf{fma}\left(y, c + y \cdot \left(b + {y}^{2}\right), i\right)}} \]

      13. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{y \cdot \left(b + {y}^{2}\right) + c}, i\right)} \]

      14. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, b + {y}^{2}, c\right)}, i\right)} \]

      15. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{{y}^{2} + b}, c\right), i\right)} \]

      16. unpow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), \frac{54929528941}{2000000}\right), \frac{28832688827}{125000}\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{y \cdot y} + b, c\right), i\right)} \]

      17. accelerator-lowering-fma.f64 97.0

          \[\leadsto \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \color{blue}{\mathsf{fma}\left(y, y, b\right)}, c\right), i\right)} \]

   5. Simplified 97.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\color{blue}{t}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]
   7. Step-by-step derivation

   8. Simplified 70.6%

      \[\leadsto \frac{\color{blue}{t}}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)} \]

   9. Taylor expanded in y around 0

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \color{blue}{c}, i\right)} \]
   10. Step-by-step derivation

   11. Simplified 69.5%

       \[\leadsto \frac{t}{\mathsf{fma}\left(y, \color{blue}{c}, i\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 51.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -750:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 5600000000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -750.0) x (if (<= y 5600000000.0) (/ t i) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -750.0) {
		tmp = x;
	} else if (y <= 5600000000.0) {
		tmp = t / i;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-750.0d0)) then
        tmp = x
    else if (y <= 5600000000.0d0) then
        tmp = t / i
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -750.0) {
		tmp = x;
	} else if (y <= 5600000000.0) {
		tmp = t / i;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -750.0:
		tmp = x
	elif y <= 5600000000.0:
		tmp = t / i
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -750.0)
		tmp = x;
	elseif (y <= 5600000000.0)
		tmp = Float64(t / i);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -750.0)
		tmp = x;
	elseif (y <= 5600000000.0)
		tmp = t / i;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -750.0], x, If[LessEqual[y, 5600000000.0], N[(t / i), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -750 or 5.6e9 < y

   1. Initial program 15.1%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Simplified 52.6%

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

   if -750 < y < 5.6e9

   1. Initial program 99.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t}{i}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 55.4

         \[\leadsto \color{blue}{\frac{t}{i}} \]

   5. Simplified 55.4%

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

Alternative 19: 25.9% accurate, 71.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return x;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return x;
}

Python

def code(x, y, z, t, a, b, c, i):
	return x

Julia

function code(x, y, z, t, a, b, c, i)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := x

Derivation

1. Initial program 60.7%

   \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Simplified 25.7%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(FPCore (x y z t a b c i)
  :name "Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2"
  :precision binary64
  (/ (+ (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y) t) (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i)))