Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 56.5% → 83.2%

Time: 21.6s

Alternatives: 22

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 56.5% 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: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)\\ t_2 := \frac{z}{y} + \frac{27464.7644705}{y \cdot y}\\ \mathbf{if}\;y \leq -1.12 \cdot 10^{+77}:\\ \;\;\;\;t\_2 + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), \frac{y}{t\_1}, \mathsf{fma}\left(x, \frac{y \cdot \left(y \cdot \left(y \cdot y\right)\right)}{t\_1}, \frac{t}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(x - a \cdot \left(\frac{x}{y} + \frac{z - x \cdot a}{y \cdot y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y (fma y (fma y (+ y a) b) c) i))
        (t_2 (+ (/ z y) (/ 27464.7644705 (* y y)))))
   (if (<= y -1.12e+77)
     (+ t_2 (- x (/ (* x b) (* y y))))
     (if (<= y 1.2e+43)
       (fma
        (fma y (fma y z 27464.7644705) 230661.510616)
        (/ y t_1)
        (fma x (/ (* y (* y (* y y))) t_1) (/ t t_1)))
       (+ t_2 (- x (* a (+ (/ x y) (/ (- z (* x a)) (* y y))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, fma(y, fma(y, (y + a), b), c), i);
	double t_2 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -1.12e+77) {
		tmp = t_2 + (x - ((x * b) / (y * y)));
	} else if (y <= 1.2e+43) {
		tmp = fma(fma(y, fma(y, z, 27464.7644705), 230661.510616), (y / t_1), fma(x, ((y * (y * (y * y))) / t_1), (t / t_1)));
	} else {
		tmp = t_2 + (x - (a * ((x / y) + ((z - (x * a)) / (y * y)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, fma(y, fma(y, Float64(y + a), b), c), i)
	t_2 = Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y)))
	tmp = 0.0
	if (y <= -1.12e+77)
		tmp = Float64(t_2 + Float64(x - Float64(Float64(x * b) / Float64(y * y))));
	elseif (y <= 1.2e+43)
		tmp = fma(fma(y, fma(y, z, 27464.7644705), 230661.510616), Float64(y / t_1), fma(x, Float64(Float64(y * Float64(y * Float64(y * y))) / t_1), Float64(t / t_1)));
	else
		tmp = Float64(t_2 + Float64(x - Float64(a * Float64(Float64(x / y) + Float64(Float64(z - Float64(x * a)) / Float64(y * y))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.12e+77], N[(t$95$2 + N[(x - N[(N[(x * b), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+43], N[(N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision] + N[(x * N[(N[(y * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(x - N[(a * N[(N[(x / y), $MachinePrecision] + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1199999999999999e77

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.6%

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

   if -1.1199999999999999e77 < y < 1.20000000000000012e43

   1. Initial program 91.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

   5. Applied rewrites 95.0%

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

   7. Applied rewrites 95.7%

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

   if 1.20000000000000012e43 < y

   1. Initial program 5.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - a \cdot \color{blue}{\left(\left(\frac{x}{y} + \frac{z}{{y}^{2}}\right) - \frac{a \cdot x}{{y}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 71.3%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - a \cdot \color{blue}{\left(\frac{x}{y} + \frac{z - a \cdot x}{y \cdot y}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{+77}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), \frac{y}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \mathsf{fma}\left(x, \frac{y \cdot \left(y \cdot \left(y \cdot y\right)\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - a \cdot \left(\frac{x}{y} + \frac{z - x \cdot a}{y \cdot y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)\\ t_2 := \frac{z}{y} + \frac{27464.7644705}{y \cdot y}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;t\_2 + \left(x - \frac{x \cdot a}{y}\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{t\_1}, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \frac{y \cdot y}{t\_1}, \frac{t}{t\_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(x - a \cdot \left(\frac{x}{y} + \frac{z - x \cdot a}{y \cdot y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y (fma y (fma y (+ y a) b) c) i))
        (t_2 (+ (/ z y) (/ 27464.7644705 (* y y)))))
   (if (<= y -1.6e+91)
     (+ t_2 (- x (/ (* x a) y)))
     (if (<= y 1.2e+43)
       (fma
        y
        (/ (fma y (fma y z 27464.7644705) 230661.510616) t_1)
        (fma x (* (* y y) (/ (* y y) t_1)) (/ t t_1)))
       (+ t_2 (- x (* a (+ (/ x y) (/ (- z (* x a)) (* y y))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, fma(y, fma(y, (y + a), b), c), i);
	double t_2 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -1.6e+91) {
		tmp = t_2 + (x - ((x * a) / y));
	} else if (y <= 1.2e+43) {
		tmp = fma(y, (fma(y, fma(y, z, 27464.7644705), 230661.510616) / t_1), fma(x, ((y * y) * ((y * y) / t_1)), (t / t_1)));
	} else {
		tmp = t_2 + (x - (a * ((x / y) + ((z - (x * a)) / (y * y)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, fma(y, fma(y, Float64(y + a), b), c), i)
	t_2 = Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y)))
	tmp = 0.0
	if (y <= -1.6e+91)
		tmp = Float64(t_2 + Float64(x - Float64(Float64(x * a) / y)));
	elseif (y <= 1.2e+43)
		tmp = fma(y, Float64(fma(y, fma(y, z, 27464.7644705), 230661.510616) / t_1), fma(x, Float64(Float64(y * y) * Float64(Float64(y * y) / t_1)), Float64(t / t_1)));
	else
		tmp = Float64(t_2 + Float64(x - Float64(a * Float64(Float64(x / y) + Float64(Float64(z - Float64(x * a)) / Float64(y * y))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+91], N[(t$95$2 + N[(x - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+43], N[(y * N[(N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x * N[(N[(y * y), $MachinePrecision] * N[(N[(y * y), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(x - N[(a * N[(N[(x / y), $MachinePrecision] + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.59999999999999995e91

   1. Initial program 0.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 79.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 84.2%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]

   if -1.59999999999999995e91 < y < 1.20000000000000012e43

   1. Initial program 89.2%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

   5. Applied rewrites 93.0%

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

   7. Applied rewrites 93.1%

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

   if 1.20000000000000012e43 < y

   1. Initial program 5.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - a \cdot \color{blue}{\left(\left(\frac{x}{y} + \frac{z}{{y}^{2}}\right) - \frac{a \cdot x}{{y}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 71.3%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - a \cdot \color{blue}{\left(\frac{x}{y} + \frac{z - a \cdot x}{y \cdot y}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \mathsf{fma}\left(x, \left(y \cdot y\right) \cdot \frac{y \cdot y}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - a \cdot \left(\frac{x}{y} + \frac{z - x \cdot a}{y \cdot y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)\\ t_2 := \frac{z}{y} + \frac{27464.7644705}{y \cdot y}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+62}:\\ \;\;\;\;t\_2 + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right)}{t\_1}, \frac{t}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 + \left(x - a \cdot \left(\frac{x}{y} + \frac{z - x \cdot a}{y \cdot y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma y (fma y (fma y (+ y a) b) c) i))
        (t_2 (+ (/ z y) (/ 27464.7644705 (* y y)))))
   (if (<= y -4.8e+62)
     (+ t_2 (- x (/ (* x b) (* y y))))
     (if (<= y 1.12e+43)
       (fma
        y
        (/ (fma y (fma y (fma y x z) 27464.7644705) 230661.510616) t_1)
        (/ t t_1))
       (+ t_2 (- x (* a (+ (/ x y) (/ (- z (* x a)) (* y y))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(y, fma(y, fma(y, (y + a), b), c), i);
	double t_2 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -4.8e+62) {
		tmp = t_2 + (x - ((x * b) / (y * y)));
	} else if (y <= 1.12e+43) {
		tmp = fma(y, (fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616) / t_1), (t / t_1));
	} else {
		tmp = t_2 + (x - (a * ((x / y) + ((z - (x * a)) / (y * y)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(y, fma(y, fma(y, Float64(y + a), b), c), i)
	t_2 = Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y)))
	tmp = 0.0
	if (y <= -4.8e+62)
		tmp = Float64(t_2 + Float64(x - Float64(Float64(x * b) / Float64(y * y))));
	elseif (y <= 1.12e+43)
		tmp = fma(y, Float64(fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616) / t_1), Float64(t / t_1));
	else
		tmp = Float64(t_2 + Float64(x - Float64(a * Float64(Float64(x / y) + Float64(Float64(z - Float64(x * a)) / Float64(y * y))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+62], N[(t$95$2 + N[(x - N[(N[(x * b), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.12e+43], N[(y * N[(N[(y * N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(t$95$2 + N[(x - N[(a * N[(N[(x / y), $MachinePrecision] + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e62

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.6%

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

   if -4.8e62 < y < 1.12e43

   1. Initial program 91.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 91.7%

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

   if 1.12e43 < y

   1. Initial program 5.5%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - a \cdot \color{blue}{\left(\left(\frac{x}{y} + \frac{z}{{y}^{2}}\right) - \frac{a \cdot x}{{y}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 71.3%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - a \cdot \color{blue}{\left(\frac{x}{y} + \frac{z - a \cdot x}{y \cdot y}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+62}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.12 \cdot 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}, \frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - a \cdot \left(\frac{x}{y} + \frac{z - x \cdot a}{y \cdot y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{y} + \frac{27464.7644705}{y \cdot y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, y + a, b\right), y, y \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(x - a \cdot \left(\frac{x}{y} + \frac{z - x \cdot a}{y \cdot y}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ z y) (/ 27464.7644705 (* y y)))))
   (if (<= y -4e+60)
     (+ t_1 (- x (/ (* x b) (* y y))))
     (if (<= y 1.25e+20)
       (/
        (+
         t
         (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* y x))))))))
        (+ i (fma (* y (fma y (+ y a) b)) y (* y c))))
       (+ t_1 (- x (* a (+ (/ x y) (/ (- z (* x a)) (* y y))))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -4e+60) {
		tmp = t_1 + (x - ((x * b) / (y * y)));
	} else if (y <= 1.25e+20) {
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + fma((y * fma(y, (y + a), b)), y, (y * c)));
	} else {
		tmp = t_1 + (x - (a * ((x / y) + ((z - (x * a)) / (y * y)))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y)))
	tmp = 0.0
	if (y <= -4e+60)
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * b) / Float64(y * y))));
	elseif (y <= 1.25e+20)
		tmp = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(y * x)))))))) / Float64(i + fma(Float64(y * fma(y, Float64(y + a), b)), y, Float64(y * c))));
	else
		tmp = Float64(t_1 + Float64(x - Float64(a * Float64(Float64(x / y) + Float64(Float64(z - Float64(x * a)) / Float64(y * y))))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+60], N[(t$95$1 + N[(x - N[(N[(x * b), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+20], 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[(N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * y + N[(y * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(x - N[(a * N[(N[(x / y), $MachinePrecision] + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e60

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.6%

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

   if -3.9999999999999998e60 < y < 1.25e20

   1. Initial program 92.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 92.8

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

   4. Applied rewrites 92.8%

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

   if 1.25e20 < y

   1. Initial program 6.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in b around 0

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - a \cdot \color{blue}{\left(\left(\frac{x}{y} + \frac{z}{{y}^{2}}\right) - \frac{a \cdot x}{{y}^{2}}\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.1%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - a \cdot \color{blue}{\left(\frac{x}{y} + \frac{z - a \cdot x}{y \cdot y}\right)}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, y + a, b\right), y, y \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - a \cdot \left(\frac{x}{y} + \frac{z - x \cdot a}{y \cdot y}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (((t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))))) <= Double.POSITIVE_INFINITY) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if ((t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))))) <= math.inf:
		tmp = t / i
	else:
		tmp = z / y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (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)))))))) <= Inf)
		tmp = Float64(t / i);
	else
		tmp = Float64(z / y);
	end
	return tmp
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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], Infinity], N[(t / i), $MachinePrecision], N[(z / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 86.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t}{i}} \]
   4. Step-by-step derivation

      1. lower-/.f64 44.0

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

   5. Applied rewrites 44.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 1.2

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

   5. Applied rewrites 1.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{z}{\color{blue}{y}} \]
   7. Step-by-step derivation

   8. Applied rewrites 28.3%

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

4. Final simplification 37.8%

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

Alternative 6: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{y} + \frac{27464.7644705}{y \cdot y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, y + a, b\right), y, y \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ z y) (/ 27464.7644705 (* y y)))))
   (if (<= y -4e+60)
     (+ t_1 (- x (/ (* x b) (* y y))))
     (if (<= y 1.25e+20)
       (/
        (+
         t
         (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* y x))))))))
        (+ i (fma (* y (fma y (+ y a) b)) y (* y c))))
       (+ t_1 (- x (/ (* x a) y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -4e+60) {
		tmp = t_1 + (x - ((x * b) / (y * y)));
	} else if (y <= 1.25e+20) {
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + fma((y * fma(y, (y + a), b)), y, (y * c)));
	} else {
		tmp = t_1 + (x - ((x * a) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y)))
	tmp = 0.0
	if (y <= -4e+60)
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * b) / Float64(y * y))));
	elseif (y <= 1.25e+20)
		tmp = Float64(Float64(t + Float64(y * Float64(230661.510616 + Float64(y * Float64(27464.7644705 + Float64(y * Float64(z + Float64(y * x)))))))) / Float64(i + fma(Float64(y * fma(y, Float64(y + a), b)), y, Float64(y * c))));
	else
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * a) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+60], N[(t$95$1 + N[(x - N[(N[(x * b), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+20], 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[(N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision]), $MachinePrecision] * y + N[(y * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(x - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e60

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.6%

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

   if -3.9999999999999998e60 < y < 1.25e20

   1. Initial program 92.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 92.8

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

   4. Applied rewrites 92.8%

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

   if 1.25e20 < y

   1. Initial program 6.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + \mathsf{fma}\left(y \cdot \mathsf{fma}\left(y, y + a, b\right), y, y \cdot c\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{y} + \frac{27464.7644705}{y \cdot y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ z y) (/ 27464.7644705 (* y y)))))
   (if (<= y -4e+60)
     (+ t_1 (- x (/ (* x b) (* y y))))
     (if (<= y 1.25e+20)
       (/
        (+
         t
         (* y (+ 230661.510616 (* y (+ 27464.7644705 (* y (+ z (* y x))))))))
        (+ i (* y (+ c (* y (+ b (* y (+ y a))))))))
       (+ t_1 (- x (/ (* x a) y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -4e+60) {
		tmp = t_1 + (x - ((x * b) / (y * y)));
	} else if (y <= 1.25e+20) {
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	} else {
		tmp = t_1 + (x - ((x * 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) :: t_1
    real(8) :: tmp
    t_1 = (z / y) + (27464.7644705d0 / (y * y))
    if (y <= (-4d+60)) then
        tmp = t_1 + (x - ((x * b) / (y * y)))
    else if (y <= 1.25d+20) then
        tmp = (t + (y * (230661.510616d0 + (y * (27464.7644705d0 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))))
    else
        tmp = t_1 + (x - ((x * 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 t_1 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -4e+60) {
		tmp = t_1 + (x - ((x * b) / (y * y)));
	} else if (y <= 1.25e+20) {
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	} else {
		tmp = t_1 + (x - ((x * a) / y));
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = (z / y) + (27464.7644705 / (y * y))
	tmp = 0
	if y <= -4e+60:
		tmp = t_1 + (x - ((x * b) / (y * y)))
	elif y <= 1.25e+20:
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))))
	else:
		tmp = t_1 + (x - ((x * a) / y))
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y)))
	tmp = 0.0
	if (y <= -4e+60)
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * b) / Float64(y * y))));
	elseif (y <= 1.25e+20)
		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 * Float64(b + Float64(y * Float64(y + a))))))));
	else
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * a) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = (z / y) + (27464.7644705 / (y * y));
	tmp = 0.0;
	if (y <= -4e+60)
		tmp = t_1 + (x - ((x * b) / (y * y)));
	elseif (y <= 1.25e+20)
		tmp = (t + (y * (230661.510616 + (y * (27464.7644705 + (y * (z + (y * x)))))))) / (i + (y * (c + (y * (b + (y * (y + a)))))));
	else
		tmp = t_1 + (x - ((x * a) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+60], N[(t$95$1 + N[(x - N[(N[(x * b), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+20], 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], N[(t$95$1 + N[(x - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e60

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.6%

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

   if -3.9999999999999998e60 < y < 1.25e20

   1. Initial program 92.8%

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

   if 1.25e20 < y

   1. Initial program 6.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\frac{t + y \cdot \left(230661.510616 + y \cdot \left(27464.7644705 + y \cdot \left(z + y \cdot x\right)\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(y + a\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{y} + \frac{27464.7644705}{y \cdot y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ z y) (/ 27464.7644705 (* y y)))))
   (if (<= y -4e+60)
     (+ t_1 (- x (/ (* x b) (* y y))))
     (if (<= y 1.25e+20)
       (*
        (fma y (fma y (fma y (fma x y z) 27464.7644705) 230661.510616) t)
        (/ 1.0 (fma y (fma y (fma y (+ y a) b) c) i)))
       (+ t_1 (- x (/ (* x a) y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -4e+60) {
		tmp = t_1 + (x - ((x * b) / (y * y)));
	} else if (y <= 1.25e+20) {
		tmp = fma(y, fma(y, fma(y, fma(x, y, z), 27464.7644705), 230661.510616), t) * (1.0 / fma(y, fma(y, fma(y, (y + a), b), c), i));
	} else {
		tmp = t_1 + (x - ((x * a) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y)))
	tmp = 0.0
	if (y <= -4e+60)
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * b) / Float64(y * y))));
	elseif (y <= 1.25e+20)
		tmp = Float64(fma(y, fma(y, fma(y, fma(x, y, z), 27464.7644705), 230661.510616), t) * Float64(1.0 / fma(y, fma(y, fma(y, Float64(y + a), b), c), i)));
	else
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * a) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+60], N[(t$95$1 + N[(x - N[(N[(x * b), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.25e+20], N[(N[(y * N[(y * N[(y * N[(x * y + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] * N[(1.0 / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(x - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e60

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.6%

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

   if -3.9999999999999998e60 < y < 1.25e20

   1. Initial program 92.8%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. lower-*.f64 N/A

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

   4. Applied rewrites 92.7%

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

   if 1.25e20 < y

   1. Initial program 6.9%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 62.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 70.0%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(x, y, z\right), 27464.7644705\right), 230661.510616\right), t\right) \cdot \frac{1}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{y} + \frac{27464.7644705}{y \cdot y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y + a, c\right), i\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (/ z y) (/ 27464.7644705 (* y y)))))
   (if (<= y -4e+60)
     (+ t_1 (- x (/ (* x b) (* y y))))
     (if (<= y -5e-44)
       (/
        (fma y (fma y (fma y (fma y x z) 27464.7644705) 230661.510616) t)
        (fma y (fma (* y y) (+ y a) c) i))
       (if (<= y 4.8e+14)
         (/
          (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
          (fma y (fma y (fma y (+ y a) b) c) i))
         (+ t_1 (- x (/ (* x a) y))))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = (z / y) + (27464.7644705 / (y * y));
	double tmp;
	if (y <= -4e+60) {
		tmp = t_1 + (x - ((x * b) / (y * y)));
	} else if (y <= -5e-44) {
		tmp = fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / fma(y, fma((y * y), (y + a), c), i);
	} else if (y <= 4.8e+14) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1 + (x - ((x * a) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y)))
	tmp = 0.0
	if (y <= -4e+60)
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * b) / Float64(y * y))));
	elseif (y <= -5e-44)
		tmp = Float64(fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / fma(y, fma(Float64(y * y), Float64(y + a), c), i));
	elseif (y <= 4.8e+14)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = Float64(t_1 + Float64(x - Float64(Float64(x * a) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+60], N[(t$95$1 + N[(x - N[(N[(x * b), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5e-44], N[(N[(y * N[(y * N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(N[(y * y), $MachinePrecision] * N[(y + a), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+14], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(x - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.9999999999999998e60

   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. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 79.6%

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

   if -3.9999999999999998e60 < y < -5.00000000000000039e-44

   1. Initial program 68.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -5.00000000000000039e-44 < y < 4.8e14

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if 4.8e14 < y

   1. Initial program 8.3%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 69.1%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot b}{y \cdot y}\right)\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y + a, c\right), i\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y + a, c\right), i\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (/ z y) (/ 27464.7644705 (* y y))) (- x (/ (* x a) y)))))
   (if (<= y -4e+60)
     t_1
     (if (<= y -5e-44)
       (/
        (fma y (fma y (fma y (fma y x z) 27464.7644705) 230661.510616) t)
        (fma y (fma (* y y) (+ y a) c) i))
       (if (<= y 4.8e+14)
         (/
          (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
          (fma y (fma y (fma y (+ y a) b) c) i))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z / y) + (27464.7644705 / (y * y))) + (x - ((x * a) / y));
	double tmp;
	if (y <= -4e+60) {
		tmp = t_1;
	} else if (y <= -5e-44) {
		tmp = fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / fma(y, fma((y * y), (y + a), c), i);
	} else if (y <= 4.8e+14) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y))) + Float64(x - Float64(Float64(x * a) / y)))
	tmp = 0.0
	if (y <= -4e+60)
		tmp = t_1;
	elseif (y <= -5e-44)
		tmp = Float64(fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / fma(y, fma(Float64(y * y), Float64(y + a), c), i));
	elseif (y <= 4.8e+14)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+60], t$95$1, If[LessEqual[y, -5e-44], N[(N[(y * N[(y * N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(N[(y * y), $MachinePrecision] * N[(y + a), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+14], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e60 or 4.8e14 < y

   1. Initial program 4.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 73.8%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]

   if -3.9999999999999998e60 < y < -5.00000000000000039e-44

   1. Initial program 68.8%

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

   3. Taylor expanded in b around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. unpow2 N/A

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

      16. lower-*.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 56.4

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

   5. Applied rewrites 56.4%

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

   if -5.00000000000000039e-44 < y < 4.8e14

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \mathbf{elif}\;y \leq -5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y \cdot y, y + a, c\right), i\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \mathbf{if}\;y \leq -3 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (/ z y) (/ 27464.7644705 (* y y))) (- x (/ (* x a) y)))))
   (if (<= y -3e+60)
     t_1
     (if (<= y -2.25e-42)
       (/
        (fma y (fma y (fma y (fma y x z) 27464.7644705) 230661.510616) t)
        (fma y (fma y (fma y y b) c) i))
       (if (<= y 4.8e+14)
         (/
          (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
          (fma y (fma y (fma y (+ y a) b) c) i))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z / y) + (27464.7644705 / (y * y))) + (x - ((x * a) / y));
	double tmp;
	if (y <= -3e+60) {
		tmp = t_1;
	} else if (y <= -2.25e-42) {
		tmp = fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, y, b), c), i);
	} else if (y <= 4.8e+14) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y))) + Float64(x - Float64(Float64(x * a) / y)))
	tmp = 0.0
	if (y <= -3e+60)
		tmp = t_1;
	elseif (y <= -2.25e-42)
		tmp = Float64(fma(y, fma(y, fma(y, fma(y, x, z), 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, y, b), c), i));
	elseif (y <= 4.8e+14)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e+60], t$95$1, If[LessEqual[y, -2.25e-42], N[(N[(y * N[(y * N[(y * N[(y * x + z), $MachinePrecision] + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * y + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+14], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.9999999999999998e60 or 4.8e14 < y

   1. Initial program 4.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 73.8%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]

   if -2.9999999999999998e60 < y < -2.25e-42

   1. Initial program 68.8%

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

   3. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. unpow2 N/A

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

      17. lower-fma.f64 56.0

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

   5. Applied rewrites 56.0%

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

   if -2.25e-42 < y < 4.8e14

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 98.8

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

   5. Applied rewrites 98.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+60}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \mathbf{elif}\;y \leq -2.25 \cdot 10^{-42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, x, z\right), 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y, b\right), c\right), i\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 75.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - x \cdot a}{y}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, c + y \cdot b, i\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z \cdot \left(y \cdot y\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* x a)) y))))
   (if (<= y -1.6e+91)
     t_1
     (if (<= y 6.4e-21)
       (/
        (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
        (fma y (+ c (* y b)) i))
       (if (<= y 4.8e+14)
         (/ (fma y (* z (* y y)) t) (fma y (fma y (fma y (+ y a) b) c) i))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (x * a)) / y);
	double tmp;
	if (y <= -1.6e+91) {
		tmp = t_1;
	} else if (y <= 6.4e-21) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, (c + (y * b)), i);
	} else if (y <= 4.8e+14) {
		tmp = fma(y, (z * (y * y)), t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(x * a)) / y))
	tmp = 0.0
	if (y <= -1.6e+91)
		tmp = t_1;
	elseif (y <= 6.4e-21)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, Float64(c + Float64(y * b)), i));
	elseif (y <= 4.8e+14)
		tmp = Float64(fma(y, Float64(z * Float64(y * y)), t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+91], t$95$1, If[LessEqual[y, 6.4e-21], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(c + N[(y * b), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.8e+14], N[(N[(y * N[(z * N[(y * y), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.59999999999999995e91 or 4.8e14 < y

   1. Initial program 4.8%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. cancel-sign-sub N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -1.59999999999999995e91 < y < 6.4000000000000003e-21

   1. Initial program 90.2%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.5%

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

   if 6.4000000000000003e-21 < y < 4.8e14

   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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto \frac{\mathsf{fma}\left(y, {y}^{2} \cdot z, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 88.5%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{-21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, c + y \cdot b, i\right)}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z \cdot \left(y \cdot y\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (+ (/ z y) (/ 27464.7644705 (* y y))) (- x (/ (* x a) y)))))
   (if (<= y -1.6e+91)
     t_1
     (if (<= y 4.8e+14)
       (/
        (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
        (fma y (fma y (fma y (+ y a) b) c) i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((z / y) + (27464.7644705 / (y * y))) + (x - ((x * a) / y));
	double tmp;
	if (y <= -1.6e+91) {
		tmp = t_1;
	} else if (y <= 4.8e+14) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(z / y) + Float64(27464.7644705 / Float64(y * y))) + Float64(x - Float64(Float64(x * a) / y)))
	tmp = 0.0
	if (y <= -1.6e+91)
		tmp = t_1;
	elseif (y <= 4.8e+14)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(z / y), $MachinePrecision] + N[(27464.7644705 / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x - N[(N[(x * a), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+91], t$95$1, If[LessEqual[y, 4.8e+14], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.59999999999999995e91 or 4.8e14 < y

   1. Initial program 4.8%

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

   3. Taylor expanded in y around inf

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

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + x\right)} - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right) \]

      2. associate--l+ N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      3. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{\frac{54929528941}{2000000}}{{y}^{2}} + \frac{z}{y}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      5. lower-+.f64 N/A

         \[\leadsto \color{blue}{\left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right)} + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \left(\color{blue}{\frac{z}{y}} + \frac{\frac{54929528941}{2000000}}{{y}^{2}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \color{blue}{\frac{\frac{54929528941}{2000000}}{{y}^{2}}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{\color{blue}{y \cdot y}}\right) + \left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \color{blue}{\left(x - \left(\frac{a \cdot x}{y} + \left(\frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}} + \frac{b \cdot x}{{y}^{2}}\right)\right)\right)} \]

      11. associate-+r+ N/A

          \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \color{blue}{\left(\left(\frac{a \cdot x}{y} + \frac{a \cdot \left(z - a \cdot x\right)}{{y}^{2}}\right) + \frac{b \cdot x}{{y}^{2}}\right)}\right) \]

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \left(\frac{z}{y} + \frac{\frac{54929528941}{2000000}}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 75.7%

      \[\leadsto \left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{a \cdot x}{\color{blue}{y}}\right) \]

   if -1.59999999999999995e91 < y < 4.8e14

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{z}{y} + \frac{27464.7644705}{y \cdot y}\right) + \left(x - \frac{x \cdot a}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 78.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - x \cdot a}{y}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* x a)) y))))
   (if (<= y -1.6e+91)
     t_1
     (if (<= y 4.8e+14)
       (/
        (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
        (fma y (fma y (fma y (+ y a) b) c) i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (x * a)) / y);
	double tmp;
	if (y <= -1.6e+91) {
		tmp = t_1;
	} else if (y <= 4.8e+14) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(x * a)) / y))
	tmp = 0.0
	if (y <= -1.6e+91)
		tmp = t_1;
	elseif (y <= 4.8e+14)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+91], t$95$1, If[LessEqual[y, 4.8e+14], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.59999999999999995e91 or 4.8e14 < y

   1. Initial program 4.8%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. cancel-sign-sub N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -1.59999999999999995e91 < y < 4.8e14

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - x \cdot a}{y}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, c + y \cdot b, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* x a)) y))))
   (if (<= y -1.6e+91)
     t_1
     (if (<= y 4.8e+14)
       (/
        (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
        (fma y (+ c (* y b)) i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (x * a)) / y);
	double tmp;
	if (y <= -1.6e+91) {
		tmp = t_1;
	} else if (y <= 4.8e+14) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, (c + (y * b)), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(x * a)) / y))
	tmp = 0.0
	if (y <= -1.6e+91)
		tmp = t_1;
	elseif (y <= 4.8e+14)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / fma(y, Float64(c + Float64(y * b)), i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.6e+91], t$95$1, If[LessEqual[y, 4.8e+14], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(y * N[(c + N[(y * b), $MachinePrecision]), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.59999999999999995e91 or 4.8e14 < y

   1. Initial program 4.8%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. cancel-sign-sub N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -1.59999999999999995e91 < y < 4.8e14

   1. Initial program 90.8%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 76.6%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.6 \cdot 10^{+91}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{\mathsf{fma}\left(y, c + y \cdot b, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - x \cdot a}{y}\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 230661.510616, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* x a)) y))))
   (if (<= y -8.8e+43)
     t_1
     (if (<= y 3.6e+14)
       (/ (fma y 230661.510616 t) (fma y (fma y (fma y (+ y a) b) c) i))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (x * a)) / y);
	double tmp;
	if (y <= -8.8e+43) {
		tmp = t_1;
	} else if (y <= 3.6e+14) {
		tmp = fma(y, 230661.510616, t) / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(x * a)) / y))
	tmp = 0.0
	if (y <= -8.8e+43)
		tmp = t_1;
	elseif (y <= 3.6e+14)
		tmp = Float64(fma(y, 230661.510616, t) / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.8e+43], t$95$1, If[LessEqual[y, 3.6e+14], N[(N[(y * 230661.510616 + t), $MachinePrecision] / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.80000000000000002e43 or 3.6e14 < y

   1. Initial program 6.3%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. cancel-sign-sub N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if -8.80000000000000002e43 < y < 3.6e14

   1. Initial program 94.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto \frac{\mathsf{fma}\left(y, \frac{28832688827}{125000}, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 76.8%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, 230661.510616, t\right)}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - x \cdot a}{y}\\ \mathbf{if}\;y \leq -8.2 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{i + y \cdot c}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* x a)) y))))
   (if (<= y -8.2e+43)
     t_1
     (if (<= y 0.00095)
       (/
        (fma y (fma y (fma y z 27464.7644705) 230661.510616) t)
        (+ i (* y c)))
       t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (x * a)) / y);
	double tmp;
	if (y <= -8.2e+43) {
		tmp = t_1;
	} else if (y <= 0.00095) {
		tmp = fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / (i + (y * c));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(x * a)) / y))
	tmp = 0.0
	if (y <= -8.2e+43)
		tmp = t_1;
	elseif (y <= 0.00095)
		tmp = Float64(fma(y, fma(y, fma(y, z, 27464.7644705), 230661.510616), t) / Float64(i + Float64(y * c)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.2e+43], t$95$1, If[LessEqual[y, 0.00095], N[(N[(y * N[(y * N[(y * z + 27464.7644705), $MachinePrecision] + 230661.510616), $MachinePrecision] + t), $MachinePrecision] / N[(i + N[(y * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2000000000000001e43 or 9.49999999999999998e-4 < y

   1. Initial program 11.4%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. cancel-sign-sub N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if -8.2000000000000001e43 < y < 9.49999999999999998e-4

   1. Initial program 93.7%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. +-commutative N/A

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

      15. lower-+.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 74.4%

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

4. Final simplification 70.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \mathbf{elif}\;y \leq 0.00095:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, z, 27464.7644705\right), 230661.510616\right), t\right)}{i + y \cdot c}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 67.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - x \cdot a}{y}\\ \mathbf{if}\;y \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* x a)) y))))
   (if (<= y -8.8e+43)
     t_1
     (if (<= y 3.6e+14) (/ t (fma y (fma y (fma y (+ y a) b) c) i)) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (x * a)) / y);
	double tmp;
	if (y <= -8.8e+43) {
		tmp = t_1;
	} else if (y <= 3.6e+14) {
		tmp = t / fma(y, fma(y, fma(y, (y + a), b), c), i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(x * a)) / y))
	tmp = 0.0
	if (y <= -8.8e+43)
		tmp = t_1;
	elseif (y <= 3.6e+14)
		tmp = Float64(t / fma(y, fma(y, fma(y, Float64(y + a), b), c), i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -8.8e+43], t$95$1, If[LessEqual[y, 3.6e+14], N[(t / N[(y * N[(y * N[(y * N[(y + a), $MachinePrecision] + b), $MachinePrecision] + c), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -8.80000000000000002e43 or 3.6e14 < y

   1. Initial program 6.3%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. cancel-sign-sub N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 70.3

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

   5. Applied rewrites 70.3%

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

   if -8.80000000000000002e43 < y < 3.6e14

   1. Initial program 94.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 67.2

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

   5. Applied rewrites 67.2%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+14}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \mathsf{fma}\left(y, y + a, b\right), c\right), i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 56.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - x \cdot a}{y}\\ \mathbf{if}\;y \leq -5.8 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* x a)) y))))
   (if (<= y -5.8e+43) t_1 (if (<= y 1.15e-17) (/ t i) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (x * a)) / y);
	double tmp;
	if (y <= -5.8e+43) {
		tmp = t_1;
	} else if (y <= 1.15e-17) {
		tmp = t / i;
	} else {
		tmp = t_1;
	}
	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 = x + ((z - (x * a)) / y)
    if (y <= (-5.8d+43)) then
        tmp = t_1
    else if (y <= 1.15d-17) then
        tmp = t / i
    else
        tmp = t_1
    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 = x + ((z - (x * a)) / y);
	double tmp;
	if (y <= -5.8e+43) {
		tmp = t_1;
	} else if (y <= 1.15e-17) {
		tmp = t / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x + ((z - (x * a)) / y)
	tmp = 0
	if y <= -5.8e+43:
		tmp = t_1
	elif y <= 1.15e-17:
		tmp = t / i
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(x * a)) / y))
	tmp = 0.0
	if (y <= -5.8e+43)
		tmp = t_1;
	elseif (y <= 1.15e-17)
		tmp = Float64(t / i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = x + ((z - (x * a)) / y);
	tmp = 0.0;
	if (y <= -5.8e+43)
		tmp = t_1;
	elseif (y <= 1.15e-17)
		tmp = t / i;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(x * a), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.8e+43], t$95$1, If[LessEqual[y, 1.15e-17], N[(t / i), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8000000000000004e43 or 1.15000000000000004e-17 < y

   1. Initial program 12.7%

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

   3. Taylor expanded in y around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-*r* N/A

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

      6. mul-1-neg N/A

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

      7. cancel-sign-sub N/A

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

      8. lower-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 65.7

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

   5. Applied rewrites 65.7%

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

   if -5.8000000000000004e43 < y < 1.15000000000000004e-17

   1. Initial program 93.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t}{i}} \]
   4. Step-by-step derivation

      1. lower-/.f64 53.8

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

   5. Applied rewrites 53.8%

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

4. Final simplification 59.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+43}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-17}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - x \cdot a}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{1}{x}}\\ \mathbf{if}\;y \leq -3.55 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-6}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (/ 1.0 (/ 1.0 x))))
   (if (<= y -3.55e+44) t_1 (if (<= y 1.75e-6) (/ t i) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = 1.0 / (1.0 / x);
	double tmp;
	if (y <= -3.55e+44) {
		tmp = t_1;
	} else if (y <= 1.75e-6) {
		tmp = t / i;
	} else {
		tmp = t_1;
	}
	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 = 1.0d0 / (1.0d0 / x)
    if (y <= (-3.55d+44)) then
        tmp = t_1
    else if (y <= 1.75d-6) then
        tmp = t / i
    else
        tmp = t_1
    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 = 1.0 / (1.0 / x);
	double tmp;
	if (y <= -3.55e+44) {
		tmp = t_1;
	} else if (y <= 1.75e-6) {
		tmp = t / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = 1.0 / (1.0 / x)
	tmp = 0
	if y <= -3.55e+44:
		tmp = t_1
	elif y <= 1.75e-6:
		tmp = t / i
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(1.0 / Float64(1.0 / x))
	tmp = 0.0
	if (y <= -3.55e+44)
		tmp = t_1;
	elseif (y <= 1.75e-6)
		tmp = Float64(t / i);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = 1.0 / (1.0 / x);
	tmp = 0.0;
	if (y <= -3.55e+44)
		tmp = t_1;
	elseif (y <= 1.75e-6)
		tmp = t / i;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(1.0 / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.55e+44], t$95$1, If[LessEqual[y, 1.75e-6], N[(t / i), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.55e44 or 1.74999999999999997e-6 < y

   1. Initial program 11.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 11.4

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

   4. Applied rewrites 11.4%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 48.0

         \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   7. Applied rewrites 48.0%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{x}}} \]

   if -3.55e44 < y < 1.74999999999999997e-6

   1. Initial program 93.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t}{i}} \]
   4. Step-by-step derivation

      1. lower-/.f64 53.0

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

   5. Applied rewrites 53.0%

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

Alternative 21: 10.4% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{z}{y} \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 (/ z y))

C

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

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 = z / y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(z / y), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-fma.f64 N/A

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

   12. +-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. +-commutative N/A

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

   15. lower-+.f64 47.5

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

5. Applied rewrites 47.5%

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

6. Taylor expanded in y around inf

   \[\leadsto \frac{z}{\color{blue}{y}} \]
7. Step-by-step derivation

8. Applied rewrites 14.9%

   \[\leadsto \frac{z}{\color{blue}{y}} \]
9. Add Preprocessing

Alternative 22: 3.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{27464.7644705}{b} \end{array} \]

FPCore

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

C

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

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 = 27464.7644705d0 / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := N[(27464.7644705 / b), $MachinePrecision]

Derivation

1. Initial program 52.2%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. +-commutative N/A

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

   3. associate-+l+ N/A

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

5. Applied rewrites 54.6%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-+.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-+.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-+.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 9.1

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

8. Applied rewrites 9.1%

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

9. Taylor expanded in y around 0

   \[\leadsto \frac{y \cdot \left(\frac{54929528941}{2000000} \cdot \frac{y}{b} + \frac{28832688827}{125000} \cdot \frac{1}{b}\right) + \frac{t}{b}}{\color{blue}{{y}^{2}}} \]
10. Step-by-step derivation

11. Applied rewrites 6.2%

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

12. Taylor expanded in y around inf

    \[\leadsto \frac{\frac{54929528941}{2000000}}{b} \]
13. Step-by-step derivation

14. Applied rewrites 3.3%

    \[\leadsto \frac{27464.7644705}{b} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024221 
(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)))