Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 55.7% → 90.1%

Time: 25.7s

Alternatives: 16

Speedup: 3.0×

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: 55.7% 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: 90.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{27464.7644705}{z \cdot z}\\ t_2 := \left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1\\ t_3 := \frac{x}{t_2}\\ t_4 := \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{if}\;y \leq -3.25 \cdot 10^{+128}:\\ \;\;\;\;t_4 + \left(t_3 + \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+26}:\\ \;\;\;\;t_4 + \left(t_3 + \frac{y}{\left(\left(\frac{y \cdot y}{z} + \frac{b}{z}\right) + y \cdot \left(\frac{a}{z} - t_1\right)\right) - \mathsf{fma}\left(-27464.7644705, \frac{t_1 - \frac{a}{z}}{z}, \frac{230661.510616}{z \cdot z}\right)}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+18}:\\ \;\;\;\;\frac{t + \left(x \cdot {y}^{4} + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_4 + \left(\frac{z}{y} + x \cdot \frac{1}{t_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (/ 27464.7644705 (* z z)))
        (t_2 (+ (+ (/ b (* y y)) (/ a y)) 1.0))
        (t_3 (/ x t_2))
        (t_4 (/ t (fma y (fma y (fma y (+ y a) b) c) i))))
   (if (<= y -3.25e+128)
     (+ t_4 (+ t_3 (/ z y)))
     (if (<= y -2.6e+26)
       (+
        t_4
        (+
         t_3
         (/
          y
          (-
           (+ (+ (/ (* y y) z) (/ b z)) (* y (- (/ a z) t_1)))
           (fma
            -27464.7644705
            (/ (- t_1 (/ a z)) z)
            (/ 230661.510616 (* z z)))))))
       (if (<= y 1.35e+18)
         (/
          (+
           t
           (+
            (* x (pow y 4.0))
            (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y z)))))))
          (+ i (* y (+ c (* y (+ b (* y (+ y a))))))))
         (+ t_4 (+ (/ z y) (* x (/ 1.0 t_2)))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 27464.7644705 / (z * z);
	double t_2 = ((b / (y * y)) + (a / y)) + 1.0;
	double t_3 = x / t_2;
	double t_4 = t / fma(y, fma(y, fma(y, (y + a), b), c), i);
	double tmp;
	if (y <= -3.25e+128) {
		tmp = t_4 + (t_3 + (z / y));
	} else if (y <= -2.6e+26) {
		tmp = t_4 + (t_3 + (y / (((((y * y) / z) + (b / z)) + (y * ((a / z) - t_1))) - fma(-27464.7644705, ((t_1 - (a / z)) / z), (230661.510616 / (z * z))))));
	} else if (y <= 1.35e+18) {
		tmp = (t + ((x * pow(y, 4.0)) + (y * (230661.510616 + (y * (27464.7644705 + (y * z))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	} else {
		tmp = t_4 + ((z / y) + (x * (1.0 / t_2)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(27464.7644705 / Float64(z * z))
	t_2 = Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0)
	t_3 = Float64(x / t_2)
	t_4 = Float64(t / fma(y, fma(y, fma(y, Float64(y + a), b), c), i))
	tmp = 0.0
	if (y <= -3.25e+128)
		tmp = Float64(t_4 + Float64(t_3 + Float64(z / y)));
	elseif (y <= -2.6e+26)
		tmp = Float64(t_4 + Float64(t_3 + Float64(y / Float64(Float64(Float64(Float64(Float64(y * y) / z) + Float64(b / z)) + Float64(y * Float64(Float64(a / z) - t_1))) - fma(-27464.7644705, Float64(Float64(t_1 - Float64(a / z)) / z), Float64(230661.510616 / Float64(z * z)))))));
	elseif (y <= 1.35e+18)
		tmp = Float64(Float64(t + Float64(Float64(x * (y ^ 4.0)) + Float64(y * Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * z))))))) / Float64(i + Float64(y * Float64(c + Float64(y * Float64(b + Float64(y * Float64(y + a))))))));
	else
		tmp = Float64(t_4 + Float64(Float64(z / y) + Float64(x * Float64(1.0 / t_2))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(27464.7644705 / N[(z * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$3 = N[(x / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.25e+128], N[(t$95$4 + N[(t$95$3 + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2.6e+26], N[(t$95$4 + N[(t$95$3 + N[(y / N[(N[(N[(N[(N[(y * y), $MachinePrecision] / z), $MachinePrecision] + N[(b / z), $MachinePrecision]), $MachinePrecision] + N[(y * N[(N[(a / z), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(-27464.7644705 * N[(N[(t$95$1 - N[(a / z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision] + N[(230661.510616 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.35e+18], N[(N[(t + N[(N[(x * N[Power[y, 4.0], $MachinePrecision]), $MachinePrecision] + N[(y * N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * N[(b + N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$4 + N[(N[(z / y), $MachinePrecision] + N[(x * N[(1.0 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.25000000000000015e128

   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. Taylor expanded in x around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. +-commutative 0.0%

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

      3. +-commutative 0.0%

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

      4. +-commutative 0.0%

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

      5. fma-udef 0.0%

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

      6. fma-udef 0.0%

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

      7. fma-udef 0.0%

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

   4. Simplified 0.0%

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

   5. Taylor expanded in y around inf 33.3%

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\color{blue}{1 + \left(\frac{a}{y} + \frac{b}{{y}^{2}}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative 33.3%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 33.3%

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

   7. Simplified 33.3%

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

   8. Taylor expanded in y around inf 97.0%

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

   if -3.25000000000000015e128 < y < -2.60000000000000002e26

   1. Initial program 12.1%

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

   2. Taylor expanded in x around 0 11.6%

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

      1. +-commutative 11.6%

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

      2. +-commutative 11.6%

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

      3. +-commutative 11.6%

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

      4. +-commutative 11.6%

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

      5. fma-udef 11.6%

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

      6. fma-udef 11.6%

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

      7. fma-udef 11.6%

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

   4. Simplified 21.1%

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

   5. Taylor expanded in y around inf 31.7%

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\color{blue}{1 + \left(\frac{a}{y} + \frac{b}{{y}^{2}}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative 31.7%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 31.7%

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

   7. Simplified 31.7%

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

   8. Taylor expanded in y around -inf 70.2%

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \left(\frac{b}{y \cdot y} + \frac{a}{y}\right)} + \frac{y}{\color{blue}{\left(-1 \cdot \left(y \cdot \left(-1 \cdot \frac{a}{z} + 27464.7644705 \cdot \frac{1}{{z}^{2}}\right)\right) + \left(\frac{b}{z} + \frac{{y}^{2}}{z}\right)\right) - \left(-27464.7644705 \cdot \frac{-1 \cdot \frac{a}{z} + 27464.7644705 \cdot \frac{1}{{z}^{2}}}{z} + 230661.510616 \cdot \frac{1}{{z}^{2}}\right)}}\right) \]
   9. Step-by-step derivation

   10. Simplified 70.2%

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

   if -2.60000000000000002e26 < y < 1.35e18

   1. Initial program 99.7%

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

   2. Taylor expanded in x around 0 99.7%

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

   if 1.35e18 < y

   1. Initial program 4.5%

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

   2. Taylor expanded in x around 0 4.4%

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

      1. +-commutative 4.4%

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

      2. +-commutative 4.4%

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

      3. +-commutative 4.4%

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

      4. +-commutative 4.4%

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

      5. fma-udef 4.4%

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

      6. fma-udef 4.4%

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

      7. fma-udef 4.4%

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

   4. Simplified 12.7%

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

   5. Taylor expanded in y around inf 35.4%

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\color{blue}{1 + \left(\frac{a}{y} + \frac{b}{{y}^{2}}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative 35.4%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.4%

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

   7. Simplified 35.4%

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

      1. div-inv 35.4%

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

   9. Applied egg-rr 35.4%

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

   10. Taylor expanded in y around inf 76.6%

       \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(x \cdot \frac{1}{1 + \left(\frac{b}{y \cdot y} + \frac{a}{y}\right)} + \color{blue}{\frac{z}{y}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.25 \cdot 10^{+128}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1} + \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+26}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1} + \frac{y}{\left(\left(\frac{y \cdot y}{z} + \frac{b}{z}\right) + y \cdot \left(\frac{a}{z} - \frac{27464.7644705}{z \cdot z}\right)\right) - \mathsf{fma}\left(-27464.7644705, \frac{\frac{27464.7644705}{z \cdot z} - \frac{a}{z}}{z}, \frac{230661.510616}{z \cdot z}\right)}\right)\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+18}:\\ \;\;\;\;\frac{t + \left(x \cdot {y}^{4} + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{z}{y} + x \cdot \frac{1}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\right)\\ \end{array} \]

Alternative 2: 90.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{z}{y} + x \cdot \frac{1}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (/
          (+
           t
           (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* y x))))))))
          (+ i (* y (+ c (* y (+ b (* y (+ y a))))))))))
   (if (<= t_1 5e+284)
     t_1
     (+
      (/ t (fma y (fma y (fma y (+ y a) b) c) i))
      (+ (/ z y) (* x (/ 1.0 (+ (+ (/ b (* y y)) (/ 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 + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	double tmp;
	if (t_1 <= 5e+284) {
		tmp = t_1;
	} else {
		tmp = (t / fma(y, fma(y, fma(y, (y + a), b), c), i)) + ((z / y) + (x * (1.0 / (((b / (y * y)) + (a / y)) + 1.0))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(y * x)))))))) / Float64(i + Float64(y * Float64(c + Float64(y * Float64(b + Float64(y * Float64(y + a))))))))
	tmp = 0.0
	if (t_1 <= 5e+284)
		tmp = t_1;
	else
		tmp = Float64(Float64(t / fma(y, fma(y, fma(y, Float64(y + a), b), c), i)) + Float64(Float64(z / y) + Float64(x * Float64(1.0 / Float64(Float64(Float64(b / Float64(y * y)) + 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[(t + N[(y * N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * N[(b + N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+284], t$95$1, N[(N[(t / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] + N[(N[(z / y), $MachinePrecision] + N[(x * N[(1.0 / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $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) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 4.9999999999999999e284

   1. Initial program 89.9%

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

   if 4.9999999999999999e284 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 1.5%

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

   2. Taylor expanded in x around 0 1.5%

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

      1. +-commutative 1.5%

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

      2. +-commutative 1.5%

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

      3. +-commutative 1.5%

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

      4. +-commutative 1.5%

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

      5. fma-udef 1.5%

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

      6. fma-udef 1.5%

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

      7. fma-udef 1.5%

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

   4. Simplified 7.2%

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

   5. Taylor expanded in y around inf 35.3%

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\color{blue}{1 + \left(\frac{a}{y} + \frac{b}{{y}^{2}}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative 35.3%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.3%

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

   7. Simplified 35.3%

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

      1. div-inv 35.3%

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

   9. Applied egg-rr 35.3%

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

   10. Taylor expanded in y around inf 82.7%

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

4. Final simplification 87.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)} \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{z}{y} + x \cdot \frac{1}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\right)\\ \end{array} \]

Alternative 3: 90.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{if}\;t_1 \leq 5 \cdot 10^{+284}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1} + \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (/
          (+
           t
           (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* y x))))))))
          (+ i (* y (+ c (* y (+ b (* y (+ y a))))))))))
   (if (<= t_1 5e+284)
     t_1
     (+
      (/ t (fma y (fma y (fma y (+ y a) b) c) i))
      (+ (/ x (+ (+ (/ b (* y y)) (/ a y)) 1.0)) (/ z y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	double tmp;
	if (t_1 <= 5e+284) {
		tmp = t_1;
	} else {
		tmp = (t / fma(y, fma(y, fma(y, (y + a), b), c), i)) + ((x / (((b / (y * y)) + (a / y)) + 1.0)) + (z / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(y * x)))))))) / Float64(i + Float64(y * Float64(c + Float64(y * Float64(b + Float64(y * Float64(y + a))))))))
	tmp = 0.0
	if (t_1 <= 5e+284)
		tmp = t_1;
	else
		tmp = Float64(Float64(t / fma(y, fma(y, fma(y, Float64(y + a), b), c), i)) + Float64(Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0)) + Float64(z / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(t + N[(y * N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * N[(b + N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+284], t$95$1, N[(N[(t / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision] + N[(N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + N[(z / y), $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) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 4.9999999999999999e284

   1. Initial program 89.9%

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

   if 4.9999999999999999e284 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 1.5%

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

   2. Taylor expanded in x around 0 1.5%

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

      1. +-commutative 1.5%

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

      2. +-commutative 1.5%

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

      3. +-commutative 1.5%

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

      4. +-commutative 1.5%

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

      5. fma-udef 1.5%

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

      6. fma-udef 1.5%

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

      7. fma-udef 1.5%

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

   4. Simplified 7.2%

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

   5. Taylor expanded in y around inf 35.3%

      \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\color{blue}{1 + \left(\frac{a}{y} + \frac{b}{{y}^{2}}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]
   6. Step-by-step derivation

      1. +-commutative 35.3%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.3%

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

   7. Simplified 35.3%

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

   8. Taylor expanded in y around inf 82.7%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)} \leq 5 \cdot 10^{+284}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1} + \frac{z}{y}\right)\\ \end{array} \]

Alternative 4: 84.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{if}\;t_1 \leq 2 \cdot 10^{+300}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (/
          (+
           t
           (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* y x))))))))
          (+ i (* y (+ c (* y (+ b (* y (+ y a))))))))))
   (if (<= t_1 2e+300) t_1 (/ x (+ (+ (/ b (* y y)) (/ 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 + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	double tmp;
	if (t_1 <= 2e+300) {
		tmp = t_1;
	} else {
		tmp = x / (((b / (y * y)) + (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 = (t + (y * (230661.510616d0 + (y * (27464.7644705d0 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))))
    if (t_1 <= 2d+300) then
        tmp = t_1
    else
        tmp = x / (((b / (y * y)) + (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 = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	double tmp;
	if (t_1 <= 2e+300) {
		tmp = t_1;
	} else {
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))))
	tmp = 0
	if t_1 <= 2e+300:
		tmp = t_1
	else:
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(y * x)))))))) / Float64(i + Float64(y * Float64(c + Float64(y * Float64(b + Float64(y * Float64(y + a))))))))
	tmp = 0.0
	if (t_1 <= 2e+300)
		tmp = t_1;
	else
		tmp = Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	tmp = 0.0;
	if (t_1 <= 2e+300)
		tmp = t_1;
	else
		tmp = x / (((b / (y * y)) + (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[(t + N[(y * N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * N[(b + N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+300], t$95$1, N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $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) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 2.0000000000000001e300

   1. Initial program 89.9%

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

   if 2.0000000000000001e300 < (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) 54929528941/2000000) y) 28832688827/125000) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i))

   1. Initial program 0.5%

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

   2. Taylor expanded in x around inf 0.4%

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

      1. associate-/l* 4.3%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 4.3%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 4.3%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 4.3%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 4.3%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 4.3%

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

      7. fma-udef 4.3%

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

      8. fma-udef 4.3%

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

   4. Simplified 4.3%

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

   5. Taylor expanded in y around inf 72.0%

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

      1. +-commutative 34.6%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 34.6%

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

   7. Simplified 72.0%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)} \leq 2 \cdot 10^{+300}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \]

Alternative 5: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+17}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(y \cdot y\right) \cdot z\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -8e+40)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 4.4e+17)
     (/
      (+ t (* y (+ 230661.510616 (* (* y y) z))))
      (+ i (* y (+ c (* y (+ b (* y (+ y a))))))))
     (/ x (+ (+ (/ b (* y y)) (/ 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 <= -8e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 4.4e+17) {
		tmp = (t + (y * (230661.510616 + ((y * y) * z)))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	} else {
		tmp = x / (((b / (y * y)) + (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) :: tmp
    if (y <= (-8d+40)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 4.4d+17) then
        tmp = (t + (y * (230661.510616d0 + ((y * y) * z)))) / (i + (y * (c + (y * (b + (y * (y + a)))))))
    else
        tmp = x / (((b / (y * y)) + (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 tmp;
	if (y <= -8e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 4.4e+17) {
		tmp = (t + (y * (230661.510616 + ((y * y) * z)))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	} else {
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -8e+40:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 4.4e+17:
		tmp = (t + (y * (230661.510616 + ((y * y) * z)))) / (i + (y * (c + (y * (b + (y * (y + a)))))))
	else:
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -8e+40)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 4.4e+17)
		tmp = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(Float64(y * y) * z)))) / Float64(i + Float64(y * Float64(c + Float64(y * Float64(b + Float64(y * Float64(y + a))))))));
	else
		tmp = Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -8e+40)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 4.4e+17)
		tmp = (t + (y * (230661.510616 + ((y * y) * z)))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	else
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -8e+40], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+17], N[(N[(t + N[(y * N[(230661.510616 + N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * N[(b + N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000024e40

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

   if -8.00000000000000024e40 < y < 4.4e17

   1. Initial program 97.6%

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

   2. Taylor expanded in z around inf 91.6%

      \[\leadsto \frac{\left(\color{blue}{{y}^{2} \cdot z} + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. *-commutative 91.6%

         \[\leadsto \frac{\left(\color{blue}{z \cdot {y}^{2}} + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

      2. unpow2 91.6%

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

   4. Simplified 91.6%

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

   if 4.4e17 < y

   1. Initial program 4.5%

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

   2. Taylor expanded in x around inf 2.6%

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

      1. associate-/l* 9.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 9.2%

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

      7. fma-udef 9.2%

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

      8. fma-udef 9.2%

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

   4. Simplified 9.2%

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

   5. Taylor expanded in y around inf 70.3%

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

      1. +-commutative 35.4%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.4%

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

   7. Simplified 70.3%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+17}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + \left(y \cdot y\right) \cdot z\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \]

Alternative 6: 80.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{+25}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -7.5e+40)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 3.75e+25)
     (/
      (+ t (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* y x))))))))
      (+ i (* y (+ c (* y b)))))
     (/ x (+ (+ (/ b (* y y)) (/ 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 <= -7.5e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 3.75e+25) {
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (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) :: tmp
    if (y <= (-7.5d+40)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 3.75d+25) then
        tmp = (t + (y * (230661.510616d0 + (y * (27464.7644705d0 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * b))))
    else
        tmp = x / (((b / (y * y)) + (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 tmp;
	if (y <= -7.5e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 3.75e+25) {
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -7.5e+40:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 3.75e+25:
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * b))))
	else:
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -7.5e+40)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 3.75e+25)
		tmp = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(y * x)))))))) / Float64(i + Float64(y * Float64(c + Float64(y * b)))));
	else
		tmp = Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -7.5e+40)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 3.75e+25)
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * b))));
	else
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -7.5e+40], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.75e+25], N[(N[(t + N[(y * N[(230661.510616 + N[(y * N[(27464.7644705 + N[(y * N[(z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4999999999999996e40

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

   if -7.4999999999999996e40 < y < 3.74999999999999996e25

   1. Initial program 95.6%

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

   2. Taylor expanded in y around 0 92.4%

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

      1. *-commutative 92.4%

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

   4. Simplified 92.4%

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

   if 3.74999999999999996e25 < y

   1. Initial program 4.5%

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

   2. Taylor expanded in x around inf 2.6%

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

      1. associate-/l* 7.8%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 7.8%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 7.8%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 7.8%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 7.8%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 7.8%

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

      7. fma-udef 7.8%

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

      8. fma-udef 7.8%

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

   4. Simplified 7.8%

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

   5. Taylor expanded in y around inf 73.9%

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

      1. +-commutative 36.1%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 36.1%

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

   7. Simplified 73.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{+25}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \]

Alternative 7: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+18}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -8e+40)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 1.12e+18)
     (/ (+ t (* y 230661.510616)) (+ i (* y (+ c (* y (+ b (* y (+ y a))))))))
     (/ x (+ (+ (/ b (* y y)) (/ 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 <= -8e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 1.12e+18) {
		tmp = (t + (y * 230661.510616)) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	} else {
		tmp = x / (((b / (y * y)) + (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) :: tmp
    if (y <= (-8d+40)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 1.12d+18) then
        tmp = (t + (y * 230661.510616d0)) / (i + (y * (c + (y * (b + (y * (y + a)))))))
    else
        tmp = x / (((b / (y * y)) + (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 tmp;
	if (y <= -8e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 1.12e+18) {
		tmp = (t + (y * 230661.510616)) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	} else {
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -8e+40:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 1.12e+18:
		tmp = (t + (y * 230661.510616)) / (i + (y * (c + (y * (b + (y * (y + a)))))))
	else:
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -8e+40)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 1.12e+18)
		tmp = Float64(Float64(t + Float64(y * 230661.510616)) / Float64(i + Float64(y * Float64(c + Float64(y * Float64(b + Float64(y * Float64(y + a))))))));
	else
		tmp = Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -8e+40)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 1.12e+18)
		tmp = (t + (y * 230661.510616)) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	else
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -8e+40], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+18], N[(N[(t + N[(y * 230661.510616), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * N[(b + N[(y * N[(y + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.00000000000000024e40

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

   if -8.00000000000000024e40 < y < 1.12e18

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 84.0%

      \[\leadsto \frac{\color{blue}{230661.510616 \cdot y} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   3. Step-by-step derivation

      1. *-commutative 82.1%

         \[\leadsto \frac{\color{blue}{y \cdot 230661.510616} + t}{\left(y \cdot b + c\right) \cdot y + i} \]

   4. Simplified 84.0%

      \[\leadsto \frac{\color{blue}{y \cdot 230661.510616} + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]

   if 1.12e18 < y

   1. Initial program 4.5%

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

   2. Taylor expanded in x around inf 2.6%

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

      1. associate-/l* 9.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 9.2%

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

      7. fma-udef 9.2%

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

      8. fma-udef 9.2%

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

   4. Simplified 9.2%

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

   5. Taylor expanded in y around inf 70.3%

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

      1. +-commutative 35.4%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.4%

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

   7. Simplified 70.3%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+18}:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \]

Alternative 8: 77.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 61000:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -8.5e+40)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 61000.0)
     (/ (+ t (* y (+ 230661.510616 (* y (* y z))))) (+ i (* y (+ c (* y b)))))
     (/ x (+ (+ (/ b (* y y)) (/ 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 <= -8.5e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 61000.0) {
		tmp = (t + (y * (230661.510616 + (y * (y * z))))) / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (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) :: tmp
    if (y <= (-8.5d+40)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 61000.0d0) then
        tmp = (t + (y * (230661.510616d0 + (y * (y * z))))) / (i + (y * (c + (y * b))))
    else
        tmp = x / (((b / (y * y)) + (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 tmp;
	if (y <= -8.5e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 61000.0) {
		tmp = (t + (y * (230661.510616 + (y * (y * z))))) / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -8.5e+40:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 61000.0:
		tmp = (t + (y * (230661.510616 + (y * (y * z))))) / (i + (y * (c + (y * b))))
	else:
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -8.5e+40)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 61000.0)
		tmp = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(y * Float64(y * z))))) / Float64(i + Float64(y * Float64(c + Float64(y * b)))));
	else
		tmp = Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -8.5e+40)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 61000.0)
		tmp = (t + (y * (230661.510616 + (y * (y * z))))) / (i + (y * (c + (y * b))));
	else
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -8.5e+40], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 61000.0], N[(N[(t + N[(y * N[(230661.510616 + N[(y * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.49999999999999996e40

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

   if -8.49999999999999996e40 < y < 61000

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 94.9%

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

      1. *-commutative 94.9%

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

   4. Simplified 94.9%

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

   5. Taylor expanded in z around inf 90.2%

      \[\leadsto \frac{\left(\color{blue}{{y}^{2} \cdot z} + 230661.510616\right) \cdot y + t}{\left(y \cdot b + c\right) \cdot y + i} \]
   6. Step-by-step derivation

      1. unpow2 90.2%

         \[\leadsto \frac{\left(\color{blue}{\left(y \cdot y\right)} \cdot z + 230661.510616\right) \cdot y + t}{\left(y \cdot b + c\right) \cdot y + i} \]

      2. associate-*l* 90.2%

         \[\leadsto \frac{\left(\color{blue}{y \cdot \left(y \cdot z\right)} + 230661.510616\right) \cdot y + t}{\left(y \cdot b + c\right) \cdot y + i} \]

   7. Simplified 90.2%

      \[\leadsto \frac{\left(\color{blue}{y \cdot \left(y \cdot z\right)} + 230661.510616\right) \cdot y + t}{\left(y \cdot b + c\right) \cdot y + i} \]

   if 61000 < y

   1. Initial program 7.6%

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

   2. Taylor expanded in x around inf 4.3%

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

      1. associate-/l* 10.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 10.7%

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

      7. fma-udef 10.7%

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

      8. fma-udef 10.7%

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

   4. Simplified 10.7%

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

   5. Taylor expanded in y around inf 68.0%

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

      1. +-commutative 35.9%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.9%

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

   7. Simplified 68.0%

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

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 61000:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \]

Alternative 9: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -7.5e+40)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 1.25)
     (/
      (+ t (* y (+ 230661.510616 (* y 27464.7644705))))
      (+ i (* y (+ c (* y b)))))
     (/ x (+ (+ (/ b (* y y)) (/ 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 <= -7.5e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 1.25) {
		tmp = (t + (y * (230661.510616 + (y * 27464.7644705)))) / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (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) :: tmp
    if (y <= (-7.5d+40)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 1.25d0) then
        tmp = (t + (y * (230661.510616d0 + (y * 27464.7644705d0)))) / (i + (y * (c + (y * b))))
    else
        tmp = x / (((b / (y * y)) + (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 tmp;
	if (y <= -7.5e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 1.25) {
		tmp = (t + (y * (230661.510616 + (y * 27464.7644705)))) / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -7.5e+40:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 1.25:
		tmp = (t + (y * (230661.510616 + (y * 27464.7644705)))) / (i + (y * (c + (y * b))))
	else:
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -7.5e+40)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 1.25)
		tmp = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(y * 27464.7644705)))) / Float64(i + Float64(y * Float64(c + Float64(y * b)))));
	else
		tmp = Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -7.5e+40)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 1.25)
		tmp = (t + (y * (230661.510616 + (y * 27464.7644705)))) / (i + (y * (c + (y * b))));
	else
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -7.5e+40], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25], N[(N[(t + N[(y * N[(230661.510616 + N[(y * 27464.7644705), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.4999999999999996e40

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

   if -7.4999999999999996e40 < y < 1.25

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 94.9%

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

      1. *-commutative 94.9%

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

   4. Simplified 94.9%

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

   5. Taylor expanded in y around 0 83.7%

      \[\leadsto \frac{\left(\color{blue}{27464.7644705 \cdot y} + 230661.510616\right) \cdot y + t}{\left(y \cdot b + c\right) \cdot y + i} \]
   6. Step-by-step derivation

      1. *-commutative 83.7%

         \[\leadsto \frac{\left(\color{blue}{y \cdot 27464.7644705} + 230661.510616\right) \cdot y + t}{\left(y \cdot b + c\right) \cdot y + i} \]

   7. Simplified 83.7%

      \[\leadsto \frac{\left(\color{blue}{y \cdot 27464.7644705} + 230661.510616\right) \cdot y + t}{\left(y \cdot b + c\right) \cdot y + i} \]

   if 1.25 < y

   1. Initial program 7.6%

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

   2. Taylor expanded in x around inf 4.3%

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

      1. associate-/l* 10.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 10.7%

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

      7. fma-udef 10.7%

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

      8. fma-udef 10.7%

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

   4. Simplified 10.7%

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

   5. Taylor expanded in y around inf 68.0%

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

      1. +-commutative 35.9%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.9%

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

   7. Simplified 68.0%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 1.25:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot 27464.7644705\right)}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \]

Alternative 10: 75.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 8600:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -7.2e+40)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 8600.0)
     (/ (+ t (* y 230661.510616)) (+ i (* y (+ c (* y b)))))
     (/ x (+ (+ (/ b (* y y)) (/ 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 <= -7.2e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 8600.0) {
		tmp = (t + (y * 230661.510616)) / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (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) :: tmp
    if (y <= (-7.2d+40)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 8600.0d0) then
        tmp = (t + (y * 230661.510616d0)) / (i + (y * (c + (y * b))))
    else
        tmp = x / (((b / (y * y)) + (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 tmp;
	if (y <= -7.2e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 8600.0) {
		tmp = (t + (y * 230661.510616)) / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -7.2e+40:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 8600.0:
		tmp = (t + (y * 230661.510616)) / (i + (y * (c + (y * b))))
	else:
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -7.2e+40)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 8600.0)
		tmp = Float64(Float64(t + Float64(y * 230661.510616)) / Float64(i + Float64(y * Float64(c + Float64(y * b)))));
	else
		tmp = Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -7.2e+40)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 8600.0)
		tmp = (t + (y * 230661.510616)) / (i + (y * (c + (y * b))));
	else
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -7.2e+40], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8600.0], N[(N[(t + N[(y * 230661.510616), $MachinePrecision]), $MachinePrecision] / N[(i + N[(y * N[(c + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999993e40

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

   if -7.19999999999999993e40 < y < 8600

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 94.9%

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

      1. *-commutative 94.9%

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

   4. Simplified 94.9%

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

   5. Taylor expanded in y around 0 83.3%

      \[\leadsto \frac{\color{blue}{230661.510616 \cdot y} + t}{\left(y \cdot b + c\right) \cdot y + i} \]
   6. Step-by-step derivation

      1. *-commutative 83.3%

         \[\leadsto \frac{\color{blue}{y \cdot 230661.510616} + t}{\left(y \cdot b + c\right) \cdot y + i} \]

   7. Simplified 83.3%

      \[\leadsto \frac{\color{blue}{y \cdot 230661.510616} + t}{\left(y \cdot b + c\right) \cdot y + i} \]

   if 8600 < y

   1. Initial program 7.6%

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

   2. Taylor expanded in x around inf 4.3%

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

      1. associate-/l* 10.7%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 10.7%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 10.7%

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

      7. fma-udef 10.7%

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

      8. fma-udef 10.7%

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

   4. Simplified 10.7%

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

   5. Taylor expanded in y around inf 68.0%

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

      1. +-commutative 35.9%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.9%

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

   7. Simplified 68.0%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 8600:\\ \;\;\;\;\frac{t + y \cdot 230661.510616}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \]

Alternative 11: 69.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -7.2e+40)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 3.9e+14)
     (/ t (+ i (* y (+ c (* y b)))))
     (/ x (+ (+ (/ b (* y y)) (/ 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 <= -7.2e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 3.9e+14) {
		tmp = t / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (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) :: tmp
    if (y <= (-7.2d+40)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 3.9d+14) then
        tmp = t / (i + (y * (c + (y * b))))
    else
        tmp = x / (((b / (y * y)) + (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 tmp;
	if (y <= -7.2e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 3.9e+14) {
		tmp = t / (i + (y * (c + (y * b))));
	} else {
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -7.2e+40:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 3.9e+14:
		tmp = t / (i + (y * (c + (y * b))))
	else:
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -7.2e+40)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 3.9e+14)
		tmp = Float64(t / Float64(i + Float64(y * Float64(c + Float64(y * b)))));
	else
		tmp = Float64(x / Float64(Float64(Float64(b / Float64(y * y)) + Float64(a / y)) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -7.2e+40)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 3.9e+14)
		tmp = t / (i + (y * (c + (y * b))));
	else
		tmp = x / (((b / (y * y)) + (a / y)) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -7.2e+40], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.9e+14], N[(t / N[(i + N[(y * N[(c + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(N[(b / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(a / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.19999999999999993e40

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

   if -7.19999999999999993e40 < y < 3.9e14

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 94.2%

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

      1. *-commutative 94.2%

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

   4. Simplified 94.2%

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

   5. Taylor expanded in t around inf 71.7%

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + b \cdot y\right)}} \]

   if 3.9e14 < y

   1. Initial program 4.5%

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

   2. Taylor expanded in x around inf 2.6%

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

      1. associate-/l* 9.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 9.2%

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

      7. fma-udef 9.2%

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

      8. fma-udef 9.2%

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

   4. Simplified 9.2%

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

   5. Taylor expanded in y around inf 70.3%

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

      1. +-commutative 35.4%

         \[\leadsto \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} + \left(\frac{x}{1 + \color{blue}{\left(\frac{b}{{y}^{2}} + \frac{a}{y}\right)}} + \frac{y}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}{230661.510616 + y \cdot \left(27464.7644705 + y \cdot z\right)}}\right) \]

      2. unpow2 35.4%

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

   7. Simplified 70.3%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+14}:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(\frac{b}{y \cdot y} + \frac{a}{y}\right) + 1}\\ \end{array} \]

Alternative 12: 68.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 7600000000000:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -8.2e+40)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 7600000000000.0)
     (/ t (+ i (* y (+ c (* y b)))))
     (/ 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 <= -8.2e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 7600000000000.0) {
		tmp = t / (i + (y * (c + (y * b))));
	} 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) :: tmp
    if (y <= (-8.2d+40)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 7600000000000.0d0) then
        tmp = t / (i + (y * (c + (y * b))))
    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 tmp;
	if (y <= -8.2e+40) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 7600000000000.0) {
		tmp = t / (i + (y * (c + (y * b))));
	} else {
		tmp = x / ((a / y) + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -8.2e+40:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 7600000000000.0:
		tmp = t / (i + (y * (c + (y * b))))
	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 <= -8.2e+40)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 7600000000000.0)
		tmp = Float64(t / Float64(i + Float64(y * Float64(c + Float64(y * b)))));
	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)
	tmp = 0.0;
	if (y <= -8.2e+40)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 7600000000000.0)
		tmp = t / (i + (y * (c + (y * b))));
	else
		tmp = x / ((a / y) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -8.2e+40], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7600000000000.0], N[(t / N[(i + N[(y * N[(c + N[(y * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.2000000000000003e40

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 56.4%

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

      1. associate--l+ 56.4%

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

      2. associate-/l* 59.6%

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

   4. Simplified 59.6%

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

   if -8.2000000000000003e40 < y < 7.6e12

   1. Initial program 97.6%

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

   2. Taylor expanded in y around 0 94.2%

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

      1. *-commutative 94.2%

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

   4. Simplified 94.2%

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

   5. Taylor expanded in t around inf 71.7%

      \[\leadsto \color{blue}{\frac{t}{i + y \cdot \left(c + b \cdot y\right)}} \]

   if 7.6e12 < y

   1. Initial program 4.5%

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

   2. Taylor expanded in x around inf 2.6%

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

      1. associate-/l* 9.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 9.2%

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

      7. fma-udef 9.2%

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

      8. fma-udef 9.2%

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

   4. Simplified 9.2%

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

   5. Taylor expanded in y around inf 65.4%

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

      1. +-commutative 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+40}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 7600000000000:\\ \;\;\;\;\frac{t}{i + y \cdot \left(c + y \cdot b\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \end{array} \]

Alternative 13: 57.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{+17}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \frac{a}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -3.2e+18)
   (+ x (- (/ z y) (/ a (/ y x))))
   (if (<= y 1.18e+17) (/ t i) (/ x (+ 1.0 (/ a y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -3.2e+18) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 1.18e+17) {
		tmp = t / i;
	} else {
		tmp = x / (1.0 + (a / y));
	}
	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 <= (-3.2d+18)) then
        tmp = x + ((z / y) - (a / (y / x)))
    else if (y <= 1.18d+17) then
        tmp = t / i
    else
        tmp = x / (1.0d0 + (a / y))
    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 <= -3.2e+18) {
		tmp = x + ((z / y) - (a / (y / x)));
	} else if (y <= 1.18e+17) {
		tmp = t / i;
	} else {
		tmp = x / (1.0 + (a / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -3.2e+18:
		tmp = x + ((z / y) - (a / (y / x)))
	elif y <= 1.18e+17:
		tmp = t / i
	else:
		tmp = x / (1.0 + (a / y))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -3.2e+18)
		tmp = Float64(x + Float64(Float64(z / y) - Float64(a / Float64(y / x))));
	elseif (y <= 1.18e+17)
		tmp = Float64(t / i);
	else
		tmp = Float64(x / Float64(1.0 + Float64(a / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -3.2e+18)
		tmp = x + ((z / y) - (a / (y / x)));
	elseif (y <= 1.18e+17)
		tmp = t / i;
	else
		tmp = x / (1.0 + (a / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -3.2e+18], N[(x + N[(N[(z / y), $MachinePrecision] - N[(a / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.18e+17], N[(t / i), $MachinePrecision], N[(x / N[(1.0 + N[(a / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2e18

   1. Initial program 5.8%

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

   2. Taylor expanded in y around inf 53.9%

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

      1. associate--l+ 53.9%

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

      2. associate-/l* 56.9%

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

   4. Simplified 56.9%

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

   if -3.2e18 < y < 1.18e17

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 52.6%

      \[\leadsto \color{blue}{\frac{t}{i}} \]

   if 1.18e17 < y

   1. Initial program 4.5%

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

   2. Taylor expanded in x around inf 2.6%

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

      1. associate-/l* 9.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 9.2%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 9.2%

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

      7. fma-udef 9.2%

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

      8. fma-udef 9.2%

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

   4. Simplified 9.2%

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

   5. Taylor expanded in y around inf 65.4%

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

      1. +-commutative 65.4%

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

   7. Simplified 65.4%

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

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+18}:\\ \;\;\;\;x + \left(\frac{z}{y} - \frac{a}{\frac{y}{x}}\right)\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{+17}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + \frac{a}{y}}\\ \end{array} \]

Alternative 14: 57.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-10} \lor \neg \left(y \leq 1.15 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= y -2.8e-10) (not (<= y 1.15e+14))) (/ x (+ (/ a y) 1.0)) (/ t i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((y <= -2.8e-10) || !(y <= 1.15e+14)) {
		tmp = x / ((a / y) + 1.0);
	} else {
		tmp = t / i;
	}
	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 <= (-2.8d-10)) .or. (.not. (y <= 1.15d+14))) then
        tmp = x / ((a / y) + 1.0d0)
    else
        tmp = t / i
    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 <= -2.8e-10) || !(y <= 1.15e+14)) {
		tmp = x / ((a / y) + 1.0);
	} else {
		tmp = t / i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (y <= -2.8e-10) or not (y <= 1.15e+14):
		tmp = x / ((a / y) + 1.0)
	else:
		tmp = t / i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((y <= -2.8e-10) || !(y <= 1.15e+14))
		tmp = Float64(x / Float64(Float64(a / y) + 1.0));
	else
		tmp = Float64(t / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((y <= -2.8e-10) || ~((y <= 1.15e+14)))
		tmp = x / ((a / y) + 1.0);
	else
		tmp = t / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[y, -2.8e-10], N[Not[LessEqual[y, 1.15e+14]], $MachinePrecision]], N[(x / N[(N[(a / y), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(t / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.80000000000000015e-10 or 1.15e14 < y

   1. Initial program 6.7%

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

   2. Taylor expanded in x around inf 3.2%

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

      1. associate-/l* 7.9%

         \[\leadsto \color{blue}{\frac{x}{\frac{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}{{y}^{4}}}} \]

      2. +-commutative 7.9%

         \[\leadsto \frac{x}{\frac{\color{blue}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right) + i}}{{y}^{4}}} \]

      3. +-commutative 7.9%

         \[\leadsto \frac{x}{\frac{y \cdot \color{blue}{\left(y \cdot \left(b + y \cdot \left(a + y\right)\right) + c\right)} + i}{{y}^{4}}} \]

      4. +-commutative 7.9%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \color{blue}{\left(y \cdot \left(a + y\right) + b\right)} + c\right) + i}{{y}^{4}}} \]

      5. +-commutative 7.9%

         \[\leadsto \frac{x}{\frac{y \cdot \left(y \cdot \left(y \cdot \color{blue}{\left(y + a\right)} + b\right) + c\right) + i}{{y}^{4}}} \]

      6. fma-udef 7.9%

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

      7. fma-udef 7.9%

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

      8. fma-udef 7.9%

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

   4. Simplified 7.9%

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

   5. Taylor expanded in y around inf 59.5%

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

      1. +-commutative 59.5%

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

   7. Simplified 59.5%

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

   if -2.80000000000000015e-10 < y < 1.15e14

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 53.3%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-10} \lor \neg \left(y \leq 1.15 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{x}{\frac{a}{y} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i}\\ \end{array} \]

Alternative 15: 50.8% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+29}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.9e+29) x (if (<= y 7.8e+15) (/ 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 <= -1.9e+29) {
		tmp = x;
	} else if (y <= 7.8e+15) {
		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 <= (-1.9d+29)) then
        tmp = x
    else if (y <= 7.8d+15) 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 <= -1.9e+29) {
		tmp = x;
	} else if (y <= 7.8e+15) {
		tmp = t / i;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -1.9e+29:
		tmp = x
	elif y <= 7.8e+15:
		tmp = t / i
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.9e+29)
		tmp = x;
	elseif (y <= 7.8e+15)
		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 <= -1.9e+29)
		tmp = x;
	elseif (y <= 7.8e+15)
		tmp = t / i;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.9e+29], x, If[LessEqual[y, 7.8e+15], N[(t / i), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -1.89999999999999985e29 or 7.8e15 < y

   1. Initial program 5.2%

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

   2. Taylor expanded in y around inf 47.1%

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

   if -1.89999999999999985e29 < y < 7.8e15

   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. Taylor expanded in y around 0 52.2%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{+29}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 16: 26.2% accurate, 33.0× 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 55.0%

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

2. Taylor expanded in y around inf 23.8%

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

3. Final simplification 23.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023290 
(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)))