Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 55.5% → 86.0%

Time: 18.6s

Alternatives: 21

Speedup: 1.7×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.5% 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: 86.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(a + y\right) \cdot y + b\right) \cdot y + c\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ t_3 := \frac{y}{t\_2}\\ t_4 := \frac{t}{t\_2}\\ t_5 := {t\_1}^{2}\\ \mathbf{if}\;y \leq -5.6 \cdot 10^{+75}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq -2200000000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-i, \left(\left(\frac{x \cdot y}{t\_5} + \frac{z}{t\_5}\right) + \frac{230661.510616}{\left(y \cdot y\right) \cdot t\_5}\right) + \frac{27464.7644705}{t\_5 \cdot y}, \frac{\left(x \cdot y + z\right) \cdot y + 27464.7644705}{t\_1} + \frac{230661.510616}{t\_1 \cdot y}\right), t\_4\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \left(t\_3 \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right) \cdot t\_3 + t\_4\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ (* (+ (* (+ a y) y) b) y) c))
        (t_2 (fma (fma (fma (+ a y) y b) y c) y i))
        (t_3 (/ y t_2))
        (t_4 (/ t t_2))
        (t_5 (pow t_1 2.0)))
   (if (<= y -5.6e+75)
     (- x (/ (- (/ -27464.7644705 y) z) y))
     (if (<= y -2200000000.0)
       (fma
        y
        (fma
         (- i)
         (+
          (+ (+ (/ (* x y) t_5) (/ z t_5)) (/ 230661.510616 (* (* y y) t_5)))
          (/ 27464.7644705 (* t_5 y)))
         (+
          (/ (+ (* (+ (* x y) z) y) 27464.7644705) t_1)
          (/ 230661.510616 (* t_1 y))))
        t_4)
       (if (<= y 2.3e+51)
         (fma
          (* y y)
          (* (* t_3 y) x)
          (+ (* (fma (fma z y 27464.7644705) y 230661.510616) t_3) t_4))
         (- x (/ (* (- (/ a y) 1.0) z) y)))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((((a + y) * y) + b) * y) + c;
	double t_2 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double t_3 = y / t_2;
	double t_4 = t / t_2;
	double t_5 = pow(t_1, 2.0);
	double tmp;
	if (y <= -5.6e+75) {
		tmp = x - (((-27464.7644705 / y) - z) / y);
	} else if (y <= -2200000000.0) {
		tmp = fma(y, fma(-i, (((((x * y) / t_5) + (z / t_5)) + (230661.510616 / ((y * y) * t_5))) + (27464.7644705 / (t_5 * y))), ((((((x * y) + z) * y) + 27464.7644705) / t_1) + (230661.510616 / (t_1 * y)))), t_4);
	} else if (y <= 2.3e+51) {
		tmp = fma((y * y), ((t_3 * y) * x), ((fma(fma(z, y, 27464.7644705), y, 230661.510616) * t_3) + t_4));
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(a + y) * y) + b) * y) + c)
	t_2 = fma(fma(fma(Float64(a + y), y, b), y, c), y, i)
	t_3 = Float64(y / t_2)
	t_4 = Float64(t / t_2)
	t_5 = t_1 ^ 2.0
	tmp = 0.0
	if (y <= -5.6e+75)
		tmp = Float64(x - Float64(Float64(Float64(-27464.7644705 / y) - z) / y));
	elseif (y <= -2200000000.0)
		tmp = fma(y, fma(Float64(-i), Float64(Float64(Float64(Float64(Float64(x * y) / t_5) + Float64(z / t_5)) + Float64(230661.510616 / Float64(Float64(y * y) * t_5))) + Float64(27464.7644705 / Float64(t_5 * y))), Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) / t_1) + Float64(230661.510616 / Float64(t_1 * y)))), t_4);
	elseif (y <= 2.3e+51)
		tmp = fma(Float64(y * y), Float64(Float64(t_3 * y) * x), Float64(Float64(fma(fma(z, y, 27464.7644705), y, 230661.510616) * t_3) + t_4));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, Block[{t$95$3 = N[(y / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t / t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$1, 2.0], $MachinePrecision]}, If[LessEqual[y, -5.6e+75], N[(x - N[(N[(N[(-27464.7644705 / y), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -2200000000.0], N[(y * N[((-i) * N[(N[(N[(N[(N[(x * y), $MachinePrecision] / t$95$5), $MachinePrecision] + N[(z / t$95$5), $MachinePrecision]), $MachinePrecision] + N[(230661.510616 / N[(N[(y * y), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(27464.7644705 / N[(t$95$5 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(230661.510616 / N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision], If[LessEqual[y, 2.3e+51], N[(N[(y * y), $MachinePrecision] * N[(N[(t$95$3 * y), $MachinePrecision] * x), $MachinePrecision] + N[(N[(N[(N[(z * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * t$95$3), $MachinePrecision] + t$95$4), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.60000000000000023e75

   1. Initial program 0.2%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 60.4%

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

   6. Taylor expanded in a around 0

      \[\leadsto x - \frac{-1 \cdot \frac{\frac{54929528941}{2000000} - b \cdot x}{y} - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 69.6%

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

   9. Taylor expanded in b around 0

      \[\leadsto x - \frac{\frac{\frac{-54929528941}{2000000}}{y} - z}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 79.7%

       \[\leadsto x - \frac{\frac{-27464.7644705}{y} - z}{y} \]

   if -5.60000000000000023e75 < y < -2.2e9

   1. Initial program 19.6%

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

   3. Taylor expanded in 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 31.5%

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

   6. Taylor expanded in i around 0

      \[\leadsto \mathsf{fma}\left(y, -1 \cdot \left(i \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{y \cdot {\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \left(\frac{28832688827}{125000} \cdot \frac{1}{{y}^{2} \cdot {\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \left(\frac{z}{{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}} + \frac{x \cdot y}{{\left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}^{2}}\right)\right)\right)\right) + \color{blue}{\left(\frac{28832688827}{125000} \cdot \frac{1}{y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)} + \frac{\frac{54929528941}{2000000} + y \cdot \left(z + x \cdot y\right)}{c + y \cdot \left(b + y \cdot \left(a + y\right)\right)}\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right) \]

   8. Applied rewrites 70.7%

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

   if -2.2e9 < y < 2.30000000000000005e51

   1. Initial program 95.5%

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

   3. Taylor expanded in 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 95.5%

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

   7. Applied rewrites 95.5%

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

   9. Applied rewrites 96.8%

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

   if 2.30000000000000005e51 < y

   1. Initial program 3.8%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 69.5%

      \[\leadsto x - \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+75}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq -2200000000:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(-i, \left(\left(\frac{x \cdot y}{{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right)}^{2}} + \frac{z}{{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right)}^{2}}\right) + \frac{230661.510616}{\left(y \cdot y\right) \cdot {\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right)}^{2}}\right) + \frac{27464.7644705}{{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right)}^{2} \cdot y}, \frac{\left(x \cdot y + z\right) \cdot y + 27464.7644705}{\left(\left(a + y\right) \cdot y + b\right) \cdot y + c} + \frac{230661.510616}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y}\right), \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot y, \left(\frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \cdot y\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right) \cdot \frac{y}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} + \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.9%

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

   3. Taylor expanded in 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 89.6%

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

   7. Applied rewrites 89.8%

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

   9. Applied rewrites 90.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.1%

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

4. Final simplification 82.5%

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

Alternative 3: 85.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.9%

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

   3. Taylor expanded in 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 89.6%

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

   7. Applied rewrites 89.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.1%

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

4. Final simplification 81.8%

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

Alternative 4: 85.1% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 87.9%

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

   3. Taylor expanded in 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 89.0%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 52.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 70.1%

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

4. Final simplification 81.4%

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

Alternative 5: 83.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (/
          (+
           (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
           t)
          (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))))
   (if (<= t_1 5e+293) t_1 (- x (/ (* (- (/ a y) 1.0) z) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i);
	double tmp;
	if (t_1 <= 5e+293) {
		tmp = t_1;
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / 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 = ((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)
    if (t_1 <= 5d+293) then
        tmp = t_1
    else
        tmp = x - ((((a / y) - 1.0d0) * z) / 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 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i);
	double tmp;
	if (t_1 <= 5e+293) {
		tmp = t_1;
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)
	tmp = 0
	if t_1 <= 5e+293:
		tmp = t_1
	else:
		tmp = x - ((((a / y) - 1.0) * z) / y)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = 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(a + y) * y) + b) * y) + c) * y) + i))
	tmp = 0.0
	if (t_1 <= 5e+293)
		tmp = t_1;
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	t_1 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i);
	tmp = 0.0;
	if (t_1 <= 5e+293)
		tmp = t_1;
	else
		tmp = x - ((((a / y) - 1.0) * z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = 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[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 5e+293], t$95$1, N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5.00000000000000033e293

   1. Initial program 89.0%

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

   if 5.00000000000000033e293 < (/.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 0.1%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 68.9%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
        t)
       (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))
      5e+293)
   (/
    1.0
    (/
     (fma (fma (fma (+ a y) y b) y c) y i)
     (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)))
   (- x (/ (* (- (/ a y) 1.0) z) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293) {
		tmp = 1.0 / (fma(fma(fma((a + y), y, b), y, c), y, i) / fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t));
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (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(a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293)
		tmp = Float64(1.0 / Float64(fma(fma(fma(Float64(a + y), y, b), y, c), y, i) / fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t)));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], 5e+293], N[(1.0 / N[(N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision] / N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5.00000000000000033e293

   1. Initial program 89.0%

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

      1. 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 88.8

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

   4. Applied rewrites 88.8%

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

   if 5.00000000000000033e293 < (/.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 0.1%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 68.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \cdot \left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
        t)
       (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))
      5e+293)
   (*
    (/ -1.0 (fma (fma (fma (+ a y) y b) y c) y i))
    (- (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t)))
   (- x (/ (* (- (/ a y) 1.0) z) y))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293) {
		tmp = (-1.0 / fma(fma(fma((a + y), y, b), y, c), y, i)) * -fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t);
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (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(a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293)
		tmp = Float64(Float64(-1.0 / fma(fma(fma(Float64(a + y), y, b), y, c), y, i)) * Float64(-fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t)));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[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[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], 5e+293], N[(N[(-1.0 / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision] * (-N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision])), $MachinePrecision], N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 x y) z) y) #s(literal 54929528941/2000000 binary64)) y) #s(literal 28832688827/125000 binary64)) y) t) (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 (*.f64 (+.f64 y a) y) b) y) c) y) i)) < 5.00000000000000033e293

   1. Initial program 89.0%

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

      1. 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. frac-2neg N/A

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

      3. div-inv N/A

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

      4. lower-*.f64 N/A

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

   4. Applied rewrites 88.8%

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

   if 5.00000000000000033e293 < (/.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 0.1%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 68.9%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)} \cdot \left(-\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
        t)
       (+ (* (+ (* (+ (* (+ a y) y) b) y) c) y) i))
      5e+293)
   (/ 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 ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((((((((((x * y) + z) * y) + 27464.7644705d0) * y) + 230661.510616d0) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= 5d+293) then
        tmp = t / i
    else
        tmp = z / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293:
		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(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(a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293)
		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 ((((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((a + y) * y) + b) * y) + c) * y) + i)) <= 5e+293)
		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[(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[(a + y), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], 5e+293], 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)) < 5.00000000000000033e293

   1. Initial program 89.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 43.5

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

   5. Applied rewrites 43.5%

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

   if 5.00000000000000033e293 < (/.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 0.1%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 12.1%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(a + y\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+65}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.16e+65)
   (- x (/ (- (/ -27464.7644705 y) z) y))
   (if (<= y 2.25e+38)
     (/
      (fma (fma (fma z y 27464.7644705) y 230661.510616) y t)
      (fma (fma (fma (+ a y) y b) y c) y i))
     (- x (/ (* (- (/ a y) 1.0) z) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.16e+65) {
		tmp = x - (((-27464.7644705 / y) - z) / y);
	} else if (y <= 2.25e+38) {
		tmp = fma(fma(fma(z, y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.16e+65)
		tmp = Float64(x - Float64(Float64(Float64(-27464.7644705 / y) - z) / y));
	elseif (y <= 2.25e+38)
		tmp = Float64(fma(fma(fma(z, y, 27464.7644705), y, 230661.510616), y, t) / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.16e+65], N[(x - N[(N[(N[(-27464.7644705 / y), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.25e+38], N[(N[(N[(N[(z * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15999999999999997e65

   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

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto x - \frac{-1 \cdot \frac{\frac{54929528941}{2000000} - b \cdot x}{y} - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in b around 0

      \[\leadsto x - \frac{\frac{\frac{-54929528941}{2000000}}{y} - z}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.1%

       \[\leadsto x - \frac{\frac{-27464.7644705}{y} - z}{y} \]

   if -1.15999999999999997e65 < y < 2.2499999999999999e38

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 86.0%

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

   if 2.2499999999999999e38 < y

   1. Initial program 3.8%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.3%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+65}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+38}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+65}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(27464.7644705 \cdot y + 230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -1.16e+65)
   (- x (/ (- (/ -27464.7644705 y) z) y))
   (if (<= y 4.3e+37)
     (/
      (fma (+ (* 27464.7644705 y) 230661.510616) y t)
      (fma (fma (fma (+ a y) y b) y c) y i))
     (- x (/ (* (- (/ a y) 1.0) z) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -1.16e+65) {
		tmp = x - (((-27464.7644705 / y) - z) / y);
	} else if (y <= 4.3e+37) {
		tmp = fma(((27464.7644705 * y) + 230661.510616), y, t) / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -1.16e+65)
		tmp = Float64(x - Float64(Float64(Float64(-27464.7644705 / y) - z) / y));
	elseif (y <= 4.3e+37)
		tmp = Float64(fma(Float64(Float64(27464.7644705 * y) + 230661.510616), y, t) / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -1.16e+65], N[(x - N[(N[(N[(-27464.7644705 / y), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.3e+37], N[(N[(N[(N[(27464.7644705 * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.15999999999999997e65

   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

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto x - \frac{-1 \cdot \frac{\frac{54929528941}{2000000} - b \cdot x}{y} - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.7%

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

   9. Taylor expanded in b around 0

      \[\leadsto x - \frac{\frac{\frac{-54929528941}{2000000}}{y} - z}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 77.1%

       \[\leadsto x - \frac{\frac{-27464.7644705}{y} - z}{y} \]

   if -1.15999999999999997e65 < y < 4.2999999999999997e37

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{t + y \cdot \left(\frac{28832688827}{125000} + y \cdot \left(\frac{54929528941}{2000000} + y \cdot z\right)\right)}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 86.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 79.4%

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

   if 4.2999999999999997e37 < y

   1. Initial program 3.8%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+65}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 4.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(27464.7644705 \cdot y + 230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -5.2e+53)
   (- x (/ (- (/ -27464.7644705 y) z) y))
   (if (<= y 2.2e-59)
     (/ t (fma (fma (fma (+ a y) y b) y c) y i))
     (if (<= y 2.95e+34)
       (/ (fma (fma (fma (fma x y z) y 27464.7644705) y 230661.510616) y t) i)
       (- x (/ (* (- (/ a y) 1.0) z) y))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -5.2e+53) {
		tmp = x - (((-27464.7644705 / y) - z) / y);
	} else if (y <= 2.2e-59) {
		tmp = t / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else if (y <= 2.95e+34) {
		tmp = fma(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616), y, t) / i;
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -5.2e+53)
		tmp = Float64(x - Float64(Float64(Float64(-27464.7644705 / y) - z) / y));
	elseif (y <= 2.2e-59)
		tmp = Float64(t / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	elseif (y <= 2.95e+34)
		tmp = Float64(fma(fma(fma(fma(x, y, z), y, 27464.7644705), y, 230661.510616), y, t) / i);
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -5.2e+53], N[(x - N[(N[(N[(-27464.7644705 / y), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-59], N[(t / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.95e+34], N[(N[(N[(N[(N[(x * y + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / i), $MachinePrecision], N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -5.19999999999999996e53

   1. Initial program 2.3%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto x - \frac{-1 \cdot \frac{\frac{54929528941}{2000000} - b \cdot x}{y} - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto x - \frac{\frac{\frac{-54929528941}{2000000}}{y} - z}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.4%

       \[\leadsto x - \frac{\frac{-27464.7644705}{y} - z}{y} \]

   if -5.19999999999999996e53 < y < 2.1999999999999999e-59

   1. Initial program 92.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if 2.1999999999999999e-59 < y < 2.9499999999999999e34

   1. Initial program 79.8%

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

   3. Taylor expanded in i around inf

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 53.5

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

   5. Applied rewrites 53.5%

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

   if 2.9499999999999999e34 < y

   1. Initial program 3.8%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.3%

      \[\leadsto x - \frac{z \cdot \left(\frac{a}{y} - 1\right)}{y} \]
3. Recombined 4 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-59}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x, y, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{i}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -5.2e+53)
   (- x (/ (- (/ -27464.7644705 y) z) y))
   (if (<= y 2.95e+34)
     (/ (fma 230661.510616 y t) (fma (fma (fma (+ a y) y b) y c) y i))
     (- x (/ (* (- (/ a y) 1.0) z) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -5.2e+53) {
		tmp = x - (((-27464.7644705 / y) - z) / y);
	} else if (y <= 2.95e+34) {
		tmp = fma(230661.510616, y, t) / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -5.2e+53)
		tmp = Float64(x - Float64(Float64(Float64(-27464.7644705 / y) - z) / y));
	elseif (y <= 2.95e+34)
		tmp = Float64(fma(230661.510616, y, t) / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -5.2e+53], N[(x - N[(N[(N[(-27464.7644705 / y), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.95e+34], N[(N[(230661.510616 * y + t), $MachinePrecision] / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.19999999999999996e53

   1. Initial program 2.3%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto x - \frac{-1 \cdot \frac{\frac{54929528941}{2000000} - b \cdot x}{y} - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto x - \frac{\frac{\frac{-54929528941}{2000000}}{y} - z}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.4%

       \[\leadsto x - \frac{\frac{-27464.7644705}{y} - z}{y} \]

   if -5.19999999999999996e53 < y < 2.9499999999999999e34

   1. Initial program 90.4%

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

   3. Taylor expanded in x around 0

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

      1. 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 79.8%

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

   if 2.9499999999999999e34 < y

   1. Initial program 3.8%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.3%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+34}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b} \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(27464.7644705 \cdot y + 230661.510616\right) \cdot y + t}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- x (/ (- (/ -27464.7644705 y) z) y))))
   (if (<= y -4.8e+76)
     t_1
     (if (<= y -5.5e-50)
       (* (/ z b) y)
       (if (<= y 1.1e-5)
         (/ (+ (* (+ (* 27464.7644705 y) 230661.510616) y) 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 - (((-27464.7644705 / y) - z) / y);
	double tmp;
	if (y <= -4.8e+76) {
		tmp = t_1;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 1.1e-5) {
		tmp = ((((27464.7644705 * y) + 230661.510616) * y) + 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 - ((((-27464.7644705d0) / y) - z) / y)
    if (y <= (-4.8d+76)) then
        tmp = t_1
    else if (y <= (-5.5d-50)) then
        tmp = (z / b) * y
    else if (y <= 1.1d-5) then
        tmp = ((((27464.7644705d0 * y) + 230661.510616d0) * y) + 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 - (((-27464.7644705 / y) - z) / y);
	double tmp;
	if (y <= -4.8e+76) {
		tmp = t_1;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 1.1e-5) {
		tmp = ((((27464.7644705 * y) + 230661.510616) * y) + t) / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x - (((-27464.7644705 / y) - z) / y)
	tmp = 0
	if y <= -4.8e+76:
		tmp = t_1
	elif y <= -5.5e-50:
		tmp = (z / b) * y
	elif y <= 1.1e-5:
		tmp = ((((27464.7644705 * y) + 230661.510616) * y) + 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(Float64(-27464.7644705 / y) - z) / y))
	tmp = 0.0
	if (y <= -4.8e+76)
		tmp = t_1;
	elseif (y <= -5.5e-50)
		tmp = Float64(Float64(z / b) * y);
	elseif (y <= 1.1e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(27464.7644705 * y) + 230661.510616) * y) + 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 - (((-27464.7644705 / y) - z) / y);
	tmp = 0.0;
	if (y <= -4.8e+76)
		tmp = t_1;
	elseif (y <= -5.5e-50)
		tmp = (z / b) * y;
	elseif (y <= 1.1e-5)
		tmp = ((((27464.7644705 * y) + 230661.510616) * y) + 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[(N[(-27464.7644705 / y), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+76], t$95$1, If[LessEqual[y, -5.5e-50], N[(N[(z / b), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.1e-5], N[(N[(N[(N[(N[(27464.7644705 * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e76 or 1.1e-5 < y

   1. Initial program 7.4%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in a around 0

      \[\leadsto x - \frac{-1 \cdot \frac{\frac{54929528941}{2000000} - b \cdot x}{y} - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.1%

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

   9. Taylor expanded in b around 0

      \[\leadsto x - \frac{\frac{\frac{-54929528941}{2000000}}{y} - z}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

       \[\leadsto x - \frac{\frac{-27464.7644705}{y} - z}{y} \]

   if -4.8e76 < y < -5.49999999999999975e-50

   1. Initial program 50.9%

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

   3. Taylor expanded in 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 57.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{54929528941}{2000000} + \left(\frac{28832688827}{125000} \cdot \frac{1}{y} + \left(y \cdot \left(z + x \cdot y\right) + \frac{t}{{y}^{2}}\right)\right)}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.4%

       \[\leadsto \frac{y \cdot z}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.9%

       \[\leadsto \color{blue}{\frac{z}{b} \cdot y} \]

   if -5.49999999999999975e-50 < y < 1.1e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{28832688827}{125000} \cdot \frac{1}{i} + y \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{i} - \left(\frac{b \cdot t}{{i}^{2}} + \frac{c \cdot \left(\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}\right)}{i}\right)\right)\right) - \frac{c \cdot t}{{i}^{2}}\right) + \frac{t}{i}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{28832688827}{125000} \cdot \frac{1}{i} + y \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{i} - \left(\frac{b \cdot t}{{i}^{2}} + \frac{c \cdot \left(\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}\right)}{i}\right)\right)\right) - \frac{c \cdot t}{{i}^{2}}\right) \cdot y} + \frac{t}{i} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{28832688827}{125000} \cdot \frac{1}{i} + y \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{i} - \left(\frac{b \cdot t}{{i}^{2}} + \frac{c \cdot \left(\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}\right)}{i}\right)\right)\right) - \frac{c \cdot t}{{i}^{2}}, y, \frac{t}{i}\right)} \]

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 69.4%

      \[\leadsto \frac{t + y \cdot \left(230661.510616 + 27464.7644705 \cdot y\right)}{\color{blue}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b} \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(27464.7644705 \cdot y + 230661.510616\right) \cdot y + t}{i}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{a \cdot x - z}{y}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b} \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(27464.7644705 \cdot y + 230661.510616\right) \cdot y + t}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- x (/ (- (* a x) z) y))))
   (if (<= y -4.8e+76)
     t_1
     (if (<= y -5.5e-50)
       (* (/ z b) y)
       (if (<= y 1.1e-5)
         (/ (+ (* (+ (* 27464.7644705 y) 230661.510616) y) 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 - (((a * x) - z) / y);
	double tmp;
	if (y <= -4.8e+76) {
		tmp = t_1;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 1.1e-5) {
		tmp = ((((27464.7644705 * y) + 230661.510616) * y) + 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 - (((a * x) - z) / y)
    if (y <= (-4.8d+76)) then
        tmp = t_1
    else if (y <= (-5.5d-50)) then
        tmp = (z / b) * y
    else if (y <= 1.1d-5) then
        tmp = ((((27464.7644705d0 * y) + 230661.510616d0) * y) + 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 - (((a * x) - z) / y);
	double tmp;
	if (y <= -4.8e+76) {
		tmp = t_1;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 1.1e-5) {
		tmp = ((((27464.7644705 * y) + 230661.510616) * y) + t) / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x - (((a * x) - z) / y)
	tmp = 0
	if y <= -4.8e+76:
		tmp = t_1
	elif y <= -5.5e-50:
		tmp = (z / b) * y
	elif y <= 1.1e-5:
		tmp = ((((27464.7644705 * y) + 230661.510616) * y) + 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(Float64(a * x) - z) / y))
	tmp = 0.0
	if (y <= -4.8e+76)
		tmp = t_1;
	elseif (y <= -5.5e-50)
		tmp = Float64(Float64(z / b) * y);
	elseif (y <= 1.1e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(27464.7644705 * y) + 230661.510616) * y) + 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 - (((a * x) - z) / y);
	tmp = 0.0;
	if (y <= -4.8e+76)
		tmp = t_1;
	elseif (y <= -5.5e-50)
		tmp = (z / b) * y;
	elseif (y <= 1.1e-5)
		tmp = ((((27464.7644705 * y) + 230661.510616) * y) + 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[(N[(a * x), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+76], t$95$1, If[LessEqual[y, -5.5e-50], N[(N[(z / b), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 1.1e-5], N[(N[(N[(N[(N[(27464.7644705 * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e76 or 1.1e-5 < y

   1. Initial program 7.4%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot x - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.8%

      \[\leadsto x - \frac{a \cdot x - z}{y} \]

   if -4.8e76 < y < -5.49999999999999975e-50

   1. Initial program 50.9%

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

   3. Taylor expanded in 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 57.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{54929528941}{2000000} + \left(\frac{28832688827}{125000} \cdot \frac{1}{y} + \left(y \cdot \left(z + x \cdot y\right) + \frac{t}{{y}^{2}}\right)\right)}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.4%

       \[\leadsto \frac{y \cdot z}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.9%

       \[\leadsto \color{blue}{\frac{z}{b} \cdot y} \]

   if -5.49999999999999975e-50 < y < 1.1e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{y \cdot \left(\left(\frac{28832688827}{125000} \cdot \frac{1}{i} + y \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{i} - \left(\frac{b \cdot t}{{i}^{2}} + \frac{c \cdot \left(\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}\right)}{i}\right)\right)\right) - \frac{c \cdot t}{{i}^{2}}\right) + \frac{t}{i}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{28832688827}{125000} \cdot \frac{1}{i} + y \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{i} - \left(\frac{b \cdot t}{{i}^{2}} + \frac{c \cdot \left(\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}\right)}{i}\right)\right)\right) - \frac{c \cdot t}{{i}^{2}}\right) \cdot y} + \frac{t}{i} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\frac{28832688827}{125000} \cdot \frac{1}{i} + y \cdot \left(\frac{54929528941}{2000000} \cdot \frac{1}{i} - \left(\frac{b \cdot t}{{i}^{2}} + \frac{c \cdot \left(\frac{28832688827}{125000} \cdot \frac{1}{i} - \frac{c \cdot t}{{i}^{2}}\right)}{i}\right)\right)\right) - \frac{c \cdot t}{{i}^{2}}, y, \frac{t}{i}\right)} \]

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in i around inf

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

   8. Applied rewrites 69.4%

      \[\leadsto \frac{t + y \cdot \left(230661.510616 + 27464.7644705 \cdot y\right)}{\color{blue}{i}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;x - \frac{a \cdot x - z}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b} \cdot y\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(27464.7644705 \cdot y + 230661.510616\right) \cdot y + t}{i}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{a \cdot x - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 69.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+34}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -5.2e+53)
   (- x (/ (- (/ -27464.7644705 y) z) y))
   (if (<= y 2.95e+34)
     (/ t (fma (fma (fma (+ a y) y b) y c) y i))
     (- x (/ (* (- (/ a y) 1.0) z) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -5.2e+53) {
		tmp = x - (((-27464.7644705 / y) - z) / y);
	} else if (y <= 2.95e+34) {
		tmp = t / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = x - ((((a / y) - 1.0) * z) / y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -5.2e+53)
		tmp = Float64(x - Float64(Float64(Float64(-27464.7644705 / y) - z) / y));
	elseif (y <= 2.95e+34)
		tmp = Float64(t / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(a / y) - 1.0) * z) / y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -5.2e+53], N[(x - N[(N[(N[(-27464.7644705 / y), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.95e+34], N[(t / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(a / y), $MachinePrecision] - 1.0), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.19999999999999996e53

   1. Initial program 2.3%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in a around 0

      \[\leadsto x - \frac{-1 \cdot \frac{\frac{54929528941}{2000000} - b \cdot x}{y} - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.4%

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

   9. Taylor expanded in b around 0

      \[\leadsto x - \frac{\frac{\frac{-54929528941}{2000000}}{y} - z}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 74.4%

       \[\leadsto x - \frac{\frac{-27464.7644705}{y} - z}{y} \]

   if -5.19999999999999996e53 < y < 2.9499999999999999e34

   1. Initial program 90.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 66.4

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

   5. Applied rewrites 66.4%

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

   if 2.9499999999999999e34 < y

   1. Initial program 3.8%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.3%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+34}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\left(\frac{a}{y} - 1\right) \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 70.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\frac{-27464.7644705}{y} - z}{y}\\ \mathbf{if}\;y \leq -5.2 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{+33}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- x (/ (- (/ -27464.7644705 y) z) y))))
   (if (<= y -5.2e+53)
     t_1
     (if (<= y 1.22e+33) (/ t (fma (fma (fma (+ a y) y b) y c) y 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 - (((-27464.7644705 / y) - z) / y);
	double tmp;
	if (y <= -5.2e+53) {
		tmp = t_1;
	} else if (y <= 1.22e+33) {
		tmp = t / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x - Float64(Float64(Float64(-27464.7644705 / y) - z) / y))
	tmp = 0.0
	if (y <= -5.2e+53)
		tmp = t_1;
	elseif (y <= 1.22e+33)
		tmp = Float64(t / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x - N[(N[(N[(-27464.7644705 / y), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.2e+53], t$95$1, If[LessEqual[y, 1.22e+33], N[(t / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999996e53 or 1.22000000000000005e33 < y

   1. Initial program 3.1%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 53.8%

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

   6. Taylor expanded in a around 0

      \[\leadsto x - \frac{-1 \cdot \frac{\frac{54929528941}{2000000} - b \cdot x}{y} - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.5%

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

   9. Taylor expanded in b around 0

      \[\leadsto x - \frac{\frac{\frac{-54929528941}{2000000}}{y} - z}{y} \]
   10. Step-by-step derivation

   11. Applied rewrites 70.6%

       \[\leadsto x - \frac{\frac{-27464.7644705}{y} - z}{y} \]

   if -5.19999999999999996e53 < y < 1.22000000000000005e33

   1. Initial program 90.4%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 66.4

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

   5. Applied rewrites 66.4%

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

Alternative 17: 55.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{a \cdot x - z}{y}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+76}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b} \cdot y\\ \mathbf{elif}\;y \leq 27000000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (- x (/ (- (* a x) z) y))))
   (if (<= y -4.8e+76)
     t_1
     (if (<= y -5.5e-50) (* (/ z b) y) (if (<= y 27000000.0) (/ 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 - (((a * x) - z) / y);
	double tmp;
	if (y <= -4.8e+76) {
		tmp = t_1;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 27000000.0) {
		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 - (((a * x) - z) / y)
    if (y <= (-4.8d+76)) then
        tmp = t_1
    else if (y <= (-5.5d-50)) then
        tmp = (z / b) * y
    else if (y <= 27000000.0d0) 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 - (((a * x) - z) / y);
	double tmp;
	if (y <= -4.8e+76) {
		tmp = t_1;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 27000000.0) {
		tmp = t / i;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	t_1 = x - (((a * x) - z) / y)
	tmp = 0
	if y <= -4.8e+76:
		tmp = t_1
	elif y <= -5.5e-50:
		tmp = (z / b) * y
	elif y <= 27000000.0:
		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(Float64(a * x) - z) / y))
	tmp = 0.0
	if (y <= -4.8e+76)
		tmp = t_1;
	elseif (y <= -5.5e-50)
		tmp = Float64(Float64(z / b) * y);
	elseif (y <= 27000000.0)
		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 - (((a * x) - z) / y);
	tmp = 0.0;
	if (y <= -4.8e+76)
		tmp = t_1;
	elseif (y <= -5.5e-50)
		tmp = (z / b) * y;
	elseif (y <= 27000000.0)
		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[(N[(a * x), $MachinePrecision] - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.8e+76], t$95$1, If[LessEqual[y, -5.5e-50], N[(N[(z / b), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 27000000.0], N[(t / i), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.8e76 or 2.7e7 < y

   1. Initial program 6.6%

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

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{x + -1 \cdot \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}\right)\right)} \]

      2. unsub-neg N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

      4. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{\left(-1 \cdot z + -1 \cdot \frac{\frac{54929528941}{2000000} - \left(-1 \cdot \left(a \cdot \left(-1 \cdot z - -1 \cdot \left(a \cdot x\right)\right)\right) + b \cdot x\right)}{y}\right) - -1 \cdot \left(a \cdot x\right)}{y}} \]

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot x - z}{y} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

      \[\leadsto x - \frac{a \cdot x - z}{y} \]

   if -4.8e76 < y < -5.49999999999999975e-50

   1. Initial program 50.9%

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

   3. Taylor expanded in 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 57.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{54929528941}{2000000} + \left(\frac{28832688827}{125000} \cdot \frac{1}{y} + \left(y \cdot \left(z + x \cdot y\right) + \frac{t}{{y}^{2}}\right)\right)}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.4%

       \[\leadsto \frac{y \cdot z}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.9%

       \[\leadsto \color{blue}{\frac{z}{b} \cdot y} \]

   if -5.49999999999999975e-50 < y < 2.7e7

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.3

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

   5. Applied rewrites 57.3%

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

Alternative 18: 35.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{z}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b} \cdot y\\ \mathbf{elif}\;y \leq 2900000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y} \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -7.8e+77)
   (/ z y)
   (if (<= y -5.5e-50)
     (* (/ z b) y)
     (if (<= y 2900000.0) (/ t i) (* (/ -1.0 y) (- z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -7.8e+77) {
		tmp = z / y;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 2900000.0) {
		tmp = t / i;
	} else {
		tmp = (-1.0 / y) * -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-7.8d+77)) then
        tmp = z / y
    else if (y <= (-5.5d-50)) then
        tmp = (z / b) * y
    else if (y <= 2900000.0d0) then
        tmp = t / i
    else
        tmp = ((-1.0d0) / y) * -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -7.8e+77) {
		tmp = z / y;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 2900000.0) {
		tmp = t / i;
	} else {
		tmp = (-1.0 / y) * -z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -7.8e+77:
		tmp = z / y
	elif y <= -5.5e-50:
		tmp = (z / b) * y
	elif y <= 2900000.0:
		tmp = t / i
	else:
		tmp = (-1.0 / y) * -z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -7.8e+77)
		tmp = Float64(z / y);
	elseif (y <= -5.5e-50)
		tmp = Float64(Float64(z / b) * y);
	elseif (y <= 2900000.0)
		tmp = Float64(t / i);
	else
		tmp = Float64(Float64(-1.0 / y) * Float64(-z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -7.8e+77)
		tmp = z / y;
	elseif (y <= -5.5e-50)
		tmp = (z / b) * y;
	elseif (y <= 2900000.0)
		tmp = t / i;
	else
		tmp = (-1.0 / y) * -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -7.8e+77], N[(z / y), $MachinePrecision], If[LessEqual[y, -5.5e-50], N[(N[(z / b), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2900000.0], N[(t / i), $MachinePrecision], N[(N[(-1.0 / y), $MachinePrecision] * (-z)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.7999999999999995e77

   1. Initial program 0.2%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 0.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 15.7%

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

   if -7.7999999999999995e77 < y < -5.49999999999999975e-50

   1. Initial program 50.9%

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

   3. Taylor expanded in 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 57.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{54929528941}{2000000} + \left(\frac{28832688827}{125000} \cdot \frac{1}{y} + \left(y \cdot \left(z + x \cdot y\right) + \frac{t}{{y}^{2}}\right)\right)}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.4%

       \[\leadsto \frac{y \cdot z}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.9%

       \[\leadsto \color{blue}{\frac{z}{b} \cdot y} \]

   if -5.49999999999999975e-50 < y < 2.9e6

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.3

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

   5. Applied rewrites 57.3%

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

   if 2.9e6 < y

   1. Initial program 10.8%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 5.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 12.5%

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

   10. Applied rewrites 12.5%

       \[\leadsto \left(-z\right) \cdot \frac{-1}{\color{blue}{y}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{z}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b} \cdot y\\ \mathbf{elif}\;y \leq 2900000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{y} \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{z}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{z}{b} \cdot y\\ \mathbf{elif}\;y \leq 2900000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -7.8e+77)
   (/ z y)
   (if (<= y -5.5e-50) (* (/ z b) y) (if (<= y 2900000.0) (/ 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 (y <= -7.8e+77) {
		tmp = z / y;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 2900000.0) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-7.8d+77)) then
        tmp = z / y
    else if (y <= (-5.5d-50)) then
        tmp = (z / b) * y
    else if (y <= 2900000.0d0) then
        tmp = t / i
    else
        tmp = z / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -7.8e+77) {
		tmp = z / y;
	} else if (y <= -5.5e-50) {
		tmp = (z / b) * y;
	} else if (y <= 2900000.0) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -7.8e+77:
		tmp = z / y
	elif y <= -5.5e-50:
		tmp = (z / b) * y
	elif y <= 2900000.0:
		tmp = t / i
	else:
		tmp = z / y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -7.8e+77)
		tmp = Float64(z / y);
	elseif (y <= -5.5e-50)
		tmp = Float64(Float64(z / b) * y);
	elseif (y <= 2900000.0)
		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 (y <= -7.8e+77)
		tmp = z / y;
	elseif (y <= -5.5e-50)
		tmp = (z / b) * y;
	elseif (y <= 2900000.0)
		tmp = t / i;
	else
		tmp = z / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -7.8e+77], N[(z / y), $MachinePrecision], If[LessEqual[y, -5.5e-50], N[(N[(z / b), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2900000.0], N[(t / i), $MachinePrecision], N[(z / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999995e77 or 2.9e6 < y

   1. Initial program 6.6%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 13.7%

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

   if -7.7999999999999995e77 < y < -5.49999999999999975e-50

   1. Initial program 50.9%

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

   3. Taylor expanded in 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 57.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{54929528941}{2000000} + \left(\frac{28832688827}{125000} \cdot \frac{1}{y} + \left(y \cdot \left(z + x \cdot y\right) + \frac{t}{{y}^{2}}\right)\right)}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.4%

       \[\leadsto \frac{y \cdot z}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.9%

       \[\leadsto \color{blue}{\frac{z}{b} \cdot y} \]

   if -5.49999999999999975e-50 < y < 2.9e6

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.3

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

   5. Applied rewrites 57.3%

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

Alternative 20: 35.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{z}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{b} \cdot z\\ \mathbf{elif}\;y \leq 2900000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -7.8e+77)
   (/ z y)
   (if (<= y -5.5e-50) (* (/ y b) z) (if (<= y 2900000.0) (/ 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 (y <= -7.8e+77) {
		tmp = z / y;
	} else if (y <= -5.5e-50) {
		tmp = (y / b) * z;
	} else if (y <= 2900000.0) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a, b, c, i)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if (y <= (-7.8d+77)) then
        tmp = z / y
    else if (y <= (-5.5d-50)) then
        tmp = (y / b) * z
    else if (y <= 2900000.0d0) then
        tmp = t / i
    else
        tmp = z / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -7.8e+77) {
		tmp = z / y;
	} else if (y <= -5.5e-50) {
		tmp = (y / b) * z;
	} else if (y <= 2900000.0) {
		tmp = t / i;
	} else {
		tmp = z / y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -7.8e+77:
		tmp = z / y
	elif y <= -5.5e-50:
		tmp = (y / b) * z
	elif y <= 2900000.0:
		tmp = t / i
	else:
		tmp = z / y
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -7.8e+77)
		tmp = Float64(z / y);
	elseif (y <= -5.5e-50)
		tmp = Float64(Float64(y / b) * z);
	elseif (y <= 2900000.0)
		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 (y <= -7.8e+77)
		tmp = z / y;
	elseif (y <= -5.5e-50)
		tmp = (y / b) * z;
	elseif (y <= 2900000.0)
		tmp = t / i;
	else
		tmp = z / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -7.8e+77], N[(z / y), $MachinePrecision], If[LessEqual[y, -5.5e-50], N[(N[(y / b), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 2900000.0], N[(t / i), $MachinePrecision], N[(z / y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.7999999999999995e77 or 2.9e6 < y

   1. Initial program 6.6%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 3.6%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 13.7%

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

   if -7.7999999999999995e77 < y < -5.49999999999999975e-50

   1. Initial program 50.9%

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

   3. Taylor expanded in 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 57.6%

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

   6. Taylor expanded in b around inf

      \[\leadsto \frac{\frac{54929528941}{2000000} + \left(\frac{28832688827}{125000} \cdot \frac{1}{y} + \left(y \cdot \left(z + x \cdot y\right) + \frac{t}{{y}^{2}}\right)\right)}{\color{blue}{b}} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{b} \]
   10. Step-by-step derivation

   11. Applied rewrites 16.4%

       \[\leadsto \frac{y \cdot z}{b} \]
   12. Step-by-step derivation

   13. Applied rewrites 22.8%

       \[\leadsto z \cdot \frac{y}{b} \]

   if -5.49999999999999975e-50 < y < 2.9e6

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 57.3

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

   5. Applied rewrites 57.3%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+77}:\\ \;\;\;\;\frac{z}{y}\\ \mathbf{elif}\;y \leq -5.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{b} \cdot z\\ \mathbf{elif}\;y \leq 2900000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 11.2% 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. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. +-commutative N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

5. Applied rewrites 49.4%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 8.0%

   \[\leadsto \frac{z}{\color{blue}{y}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024270 
(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)))