Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 55.4% → 81.5%

Time: 13.9s

Alternatives: 13

Speedup: 2.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    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: 55.4% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    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: 81.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{27464.7644705 + \mathsf{fma}\left(230661.510616, \frac{1}{y}, \mathsf{fma}\left(y, z + x \cdot y, \frac{t}{y \cdot y}\right)\right)}{b}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{t\_2}, \frac{t}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y)))
        (t_2 (fma (fma (fma (+ a y) y b) y c) y i)))
   (if (<= y -4e+78)
     t_1
     (if (<= y -3.6e+25)
       (/
        (+
         27464.7644705
         (fma 230661.510616 (/ 1.0 y) (fma y (+ z (* x y)) (/ t (* y y)))))
        b)
       (if (<= y 5.6e+28)
         (fma
          y
          (/ (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) t_2)
          (/ t t_2))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = x + ((z - (a * x)) / y);
	double t_2 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double tmp;
	if (y <= -4e+78) {
		tmp = t_1;
	} else if (y <= -3.6e+25) {
		tmp = (27464.7644705 + fma(230661.510616, (1.0 / y), fma(y, (z + (x * y)), (t / (y * y))))) / b;
	} else if (y <= 5.6e+28) {
		tmp = fma(y, (fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_2), (t / t_2));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(a * x)) / y))
	t_2 = fma(fma(fma(Float64(a + y), y, b), y, c), y, i)
	tmp = 0.0
	if (y <= -4e+78)
		tmp = t_1;
	elseif (y <= -3.6e+25)
		tmp = Float64(Float64(27464.7644705 + fma(230661.510616, Float64(1.0 / y), fma(y, Float64(z + Float64(x * y)), Float64(t / Float64(y * y))))) / b);
	elseif (y <= 5.6e+28)
		tmp = fma(y, Float64(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_2), Float64(t / t_2));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, If[LessEqual[y, -4e+78], t$95$1, If[LessEqual[y, -3.6e+25], N[(N[(27464.7644705 + N[(230661.510616 * N[(1.0 / y), $MachinePrecision] + N[(y * N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 5.6e+28], N[(y * N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(t / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.00000000000000003e78 or 5.6000000000000003e28 < y

   1. Initial program 1.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   if -4.00000000000000003e78 < y < -3.60000000000000015e25

   1. Initial program 4.1%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 4.1%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 83.4

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

   7. Applied rewrites 83.4%

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

   if -3.60000000000000015e25 < y < 5.6000000000000003e28

   1. Initial program 98.3%

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

      1. 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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+25}:\\ \;\;\;\;\frac{27464.7644705 + \mathsf{fma}\left(230661.510616, \frac{1}{y}, \mathsf{fma}\left(y, z + x \cdot y, \frac{t}{y \cdot y}\right)\right)}{b}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}, \frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.4% accurate, 0.4× speedup

Math

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

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)
         (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
        INFINITY)
     (fma
      y
      (fma (fma (fma x y z) y 27464.7644705) (/ y t_1) (/ 230661.510616 t_1))
      (/ t t_1))
     (+ x (/ (- z (* a x)) 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) / (((((((y + a) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		tmp = fma(y, fma(fma(fma(x, y, z), y, 27464.7644705), (y / t_1), (230661.510616 / t_1)), (t / t_1));
	} else {
		tmp = x + ((z - (a * x)) / 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(y + a) * y) + b) * y) + c) * y) + i)) <= Inf)
		tmp = fma(y, fma(fma(fma(x, y, z), y, 27464.7644705), Float64(y / t_1), Float64(230661.510616 / t_1)), Float64(t / t_1));
	else
		tmp = Float64(x + Float64(Float64(z - Float64(a * x)) / 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[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], N[(y * N[(N[(N[(x * y + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * N[(y / t$95$1), $MachinePrecision] + N[(230661.510616 / t$95$1), $MachinePrecision]), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $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 88.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 88.6%

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

      1. lift-/.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. div-add N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-/.f64 90.9

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

   6. Applied rewrites 90.9%

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

   if +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 0

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

      1. lower-/.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

4. Final simplification 82.8%

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

Alternative 3: 78.7% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<=
      (/
       (+
        (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
        t)
       (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))
      INFINITY)
   (/
    (fma (fma (fma z y 27464.7644705) y 230661.510616) y t)
    (fma (fma (fma (+ a y) y b) y c) y i))
   (+ x (/ (- z (* a x)) 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) / (((((((y + a) * y) + b) * y) + c) * y) + i)) <= ((double) INFINITY)) {
		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 + ((z - (a * x)) / 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(y + a) * y) + b) * y) + c) * y) + i)) <= Inf)
		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(z - Float64(a * x)) / 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[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision], Infinity], 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[(z - N[(a * x), $MachinePrecision]), $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 88.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-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 83.9%

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

   if +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 0

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

      1. lower-/.f64 3.0

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

   5. Applied rewrites 3.0%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 70.8

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

   8. Applied rewrites 70.8%

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

4. Final simplification 78.6%

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

Alternative 4: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.88 \cdot 10^{+25}:\\ \;\;\;\;\frac{27464.7644705 + \mathsf{fma}\left(230661.510616, \frac{1}{y}, \mathsf{fma}\left(y, z + x \cdot y, \frac{t}{y \cdot y}\right)\right)}{b}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right) \cdot y, y, 230661.510616 \cdot y\right) + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y))))
   (if (<= y -4e+78)
     t_1
     (if (<= y -1.88e+25)
       (/
        (+
         27464.7644705
         (fma 230661.510616 (/ 1.0 y) (fma y (+ z (* x y)) (/ t (* y y)))))
        b)
       (if (<= y 5.6e+28)
         (/
          (+
           (fma (* (fma (fma y x z) y 27464.7644705) y) y (* 230661.510616 y))
           t)
          (+ (* (+ (* (+ (* (+ y a) 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 + ((z - (a * x)) / y);
	double tmp;
	if (y <= -4e+78) {
		tmp = t_1;
	} else if (y <= -1.88e+25) {
		tmp = (27464.7644705 + fma(230661.510616, (1.0 / y), fma(y, (z + (x * y)), (t / (y * y))))) / b;
	} else if (y <= 5.6e+28) {
		tmp = (fma((fma(fma(y, x, z), y, 27464.7644705) * y), y, (230661.510616 * y)) + t) / (((((((y + a) * 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(z - Float64(a * x)) / y))
	tmp = 0.0
	if (y <= -4e+78)
		tmp = t_1;
	elseif (y <= -1.88e+25)
		tmp = Float64(Float64(27464.7644705 + fma(230661.510616, Float64(1.0 / y), fma(y, Float64(z + Float64(x * y)), Float64(t / Float64(y * y))))) / b);
	elseif (y <= 5.6e+28)
		tmp = Float64(Float64(fma(Float64(fma(fma(y, x, z), y, 27464.7644705) * y), y, Float64(230661.510616 * y)) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * 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[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+78], t$95$1, If[LessEqual[y, -1.88e+25], N[(N[(27464.7644705 + N[(230661.510616 * N[(1.0 / y), $MachinePrecision] + N[(y * N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 5.6e+28], N[(N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] * y + N[(230661.510616 * y), $MachinePrecision]), $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], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.00000000000000003e78 or 5.6000000000000003e28 < y

   1. Initial program 1.5%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   if -4.00000000000000003e78 < y < -1.88e25

   1. Initial program 4.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if -1.88e25 < y < 5.6000000000000003e28

   1. Initial program 99.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-*.f64 99.0

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

   4. Applied rewrites 99.0%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{elif}\;y \leq -1.88 \cdot 10^{+25}:\\ \;\;\;\;\frac{27464.7644705 + \mathsf{fma}\left(230661.510616, \frac{1}{y}, \mathsf{fma}\left(y, z + x \cdot y, \frac{t}{y \cdot y}\right)\right)}{b}\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{+28}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right) \cdot y, y, 230661.510616 \cdot y\right) + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 35.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 88.0%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 49.7

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

   5. Applied rewrites 49.7%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-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 0.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 17.7%

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

Alternative 6: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.88 \cdot 10^{+25}:\\ \;\;\;\;\frac{27464.7644705 + \mathsf{fma}\left(230661.510616, \frac{1}{y}, \mathsf{fma}\left(y, z + x \cdot y, \frac{t}{y \cdot y}\right)\right)}{b}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y))))
   (if (<= y -4e+78)
     t_1
     (if (<= y -1.88e+25)
       (/
        (+
         27464.7644705
         (fma 230661.510616 (/ 1.0 y) (fma y (+ z (* x y)) (/ t (* y y)))))
        b)
       (if (<= y 1.2e+25)
         (/
          (fma (fma (fma z y 27464.7644705) y 230661.510616) y 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 + ((z - (a * x)) / y);
	double tmp;
	if (y <= -4e+78) {
		tmp = t_1;
	} else if (y <= -1.88e+25) {
		tmp = (27464.7644705 + fma(230661.510616, (1.0 / y), fma(y, (z + (x * y)), (t / (y * y))))) / b;
	} else if (y <= 1.2e+25) {
		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 = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(a * x)) / y))
	tmp = 0.0
	if (y <= -4e+78)
		tmp = t_1;
	elseif (y <= -1.88e+25)
		tmp = Float64(Float64(27464.7644705 + fma(230661.510616, Float64(1.0 / y), fma(y, Float64(z + Float64(x * y)), Float64(t / Float64(y * y))))) / b);
	elseif (y <= 1.2e+25)
		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 = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+78], t$95$1, If[LessEqual[y, -1.88e+25], N[(N[(27464.7644705 + N[(230661.510616 * N[(1.0 / y), $MachinePrecision] + N[(y * N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(t / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 1.2e+25], 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], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.00000000000000003e78 or 1.19999999999999998e25 < y

   1. Initial program 2.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

   if -4.00000000000000003e78 < y < -1.88e25

   1. Initial program 4.0%

      \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
   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. lift-+.f64 N/A

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

      3. div-add N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 85.6

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

   7. Applied rewrites 85.6%

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

   if -1.88e25 < y < 1.19999999999999998e25

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 95.1%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{elif}\;y \leq -1.88 \cdot 10^{+25}:\\ \;\;\;\;\frac{27464.7644705 + \mathsf{fma}\left(230661.510616, \frac{1}{y}, \mathsf{fma}\left(y, z + x \cdot y, \frac{t}{y \cdot y}\right)\right)}{b}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.88 \cdot 10^{+25}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\frac{x}{b} + \frac{z}{b \cdot y}\right)\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y))))
   (if (<= y -4e+78)
     t_1
     (if (<= y -1.88e+25)
       (* (* y y) (+ (/ x b) (/ z (* b y))))
       (if (<= y -7.2e-51)
         (/ (fma (* (* y y) z) y t) (fma (fma (fma (+ a y) y b) y c) y i))
         (if (<= y 4.6e+23)
           (/
            (fma 230661.510616 y t)
            (+ (* (+ (* (+ (* (+ y a) 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 + ((z - (a * x)) / y);
	double tmp;
	if (y <= -4e+78) {
		tmp = t_1;
	} else if (y <= -1.88e+25) {
		tmp = (y * y) * ((x / b) + (z / (b * y)));
	} else if (y <= -7.2e-51) {
		tmp = fma(((y * y) * z), y, t) / fma(fma(fma((a + y), y, b), y, c), y, i);
	} else if (y <= 4.6e+23) {
		tmp = fma(230661.510616, y, t) / (((((((y + a) * 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(z - Float64(a * x)) / y))
	tmp = 0.0
	if (y <= -4e+78)
		tmp = t_1;
	elseif (y <= -1.88e+25)
		tmp = Float64(Float64(y * y) * Float64(Float64(x / b) + Float64(z / Float64(b * y))));
	elseif (y <= -7.2e-51)
		tmp = Float64(fma(Float64(Float64(y * y) * z), y, t) / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	elseif (y <= 4.6e+23)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * 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[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+78], t$95$1, If[LessEqual[y, -1.88e+25], N[(N[(y * y), $MachinePrecision] * N[(N[(x / b), $MachinePrecision] + N[(z / N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -7.2e-51], N[(N[(N[(N[(y * y), $MachinePrecision] * z), $MachinePrecision] * y + t), $MachinePrecision] / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+23], N[(N[(230661.510616 * y + 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], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -4.00000000000000003e78 or 4.6000000000000001e23 < y

   1. Initial program 2.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

   if -4.00000000000000003e78 < y < -1.88e25

   1. Initial program 4.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 61.6%

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

   if -1.88e25 < y < -7.2000000000000001e-51

   1. Initial program 99.3%

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

   3. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 83.9

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

   5. Applied rewrites 83.9%

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

   7. Applied rewrites 83.9%

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

   if -7.2000000000000001e-51 < y < 4.6000000000000001e23

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 89.4

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

   5. Applied rewrites 89.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(230661.510616, y, t\right)}}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
3. Recombined 4 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{elif}\;y \leq -1.88 \cdot 10^{+25}:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\frac{x}{b} + \frac{z}{b \cdot y}\right)\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(y \cdot y\right) \cdot z, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.88 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(230661.510616, \frac{1}{b \cdot y}, \frac{\mathsf{fma}\left(y, z, 27464.7644705 + x \cdot \left(y \cdot y\right)\right)}{b}\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y))))
   (if (<= y -4e+78)
     t_1
     (if (<= y -1.88e+25)
       (fma
        230661.510616
        (/ 1.0 (* b y))
        (/ (fma y z (+ 27464.7644705 (* x (* y y)))) b))
       (if (<= y 1.2e+25)
         (/
          (fma (fma (fma z y 27464.7644705) y 230661.510616) y 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 + ((z - (a * x)) / y);
	double tmp;
	if (y <= -4e+78) {
		tmp = t_1;
	} else if (y <= -1.88e+25) {
		tmp = fma(230661.510616, (1.0 / (b * y)), (fma(y, z, (27464.7644705 + (x * (y * y)))) / b));
	} else if (y <= 1.2e+25) {
		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 = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(x + Float64(Float64(z - Float64(a * x)) / y))
	tmp = 0.0
	if (y <= -4e+78)
		tmp = t_1;
	elseif (y <= -1.88e+25)
		tmp = fma(230661.510616, Float64(1.0 / Float64(b * y)), Float64(fma(y, z, Float64(27464.7644705 + Float64(x * Float64(y * y)))) / b));
	elseif (y <= 1.2e+25)
		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 = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+78], t$95$1, If[LessEqual[y, -1.88e+25], N[(230661.510616 * N[(1.0 / N[(b * y), $MachinePrecision]), $MachinePrecision] + N[(N[(y * z + N[(27464.7644705 + N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+25], 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], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.00000000000000003e78 or 1.19999999999999998e25 < y

   1. Initial program 2.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

   if -4.00000000000000003e78 < y < -1.88e25

   1. Initial program 4.0%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 58.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 58.8%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 72.1%

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

   if -1.88e25 < y < 1.19999999999999998e25

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 95.1%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{elif}\;y \leq -1.88 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(230661.510616, \frac{1}{b \cdot y}, \frac{\mathsf{fma}\left(y, z, 27464.7644705 + x \cdot \left(y \cdot y\right)\right)}{b}\right)\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6000000000:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\frac{x}{b} + \frac{z}{b \cdot y}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y))))
   (if (<= y -4e+78)
     t_1
     (if (<= y -6000000000.0)
       (* (* y y) (+ (/ x b) (/ z (* b y))))
       (if (<= y 4.6e+23)
         (/
          (fma 230661.510616 y t)
          (+ (* (+ (* (+ (* (+ y a) 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 + ((z - (a * x)) / y);
	double tmp;
	if (y <= -4e+78) {
		tmp = t_1;
	} else if (y <= -6000000000.0) {
		tmp = (y * y) * ((x / b) + (z / (b * y)));
	} else if (y <= 4.6e+23) {
		tmp = fma(230661.510616, y, t) / (((((((y + a) * 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(z - Float64(a * x)) / y))
	tmp = 0.0
	if (y <= -4e+78)
		tmp = t_1;
	elseif (y <= -6000000000.0)
		tmp = Float64(Float64(y * y) * Float64(Float64(x / b) + Float64(z / Float64(b * y))));
	elseif (y <= 4.6e+23)
		tmp = Float64(fma(230661.510616, y, t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * 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[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+78], t$95$1, If[LessEqual[y, -6000000000.0], N[(N[(y * y), $MachinePrecision] * N[(N[(x / b), $MachinePrecision] + N[(z / N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+23], N[(N[(230661.510616 * y + 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], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.00000000000000003e78 or 4.6000000000000001e23 < y

   1. Initial program 2.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

   if -4.00000000000000003e78 < y < -6e9

   1. Initial program 32.5%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 44.4%

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

   if -6e9 < y < 4.6000000000000001e23

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 85.2

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

   5. Applied rewrites 85.2%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{elif}\;y \leq -6000000000:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\frac{x}{b} + \frac{z}{b \cdot y}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{z - a \cdot x}{y}\\ \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6000000000:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\frac{x}{b} + \frac{z}{b \cdot y}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (+ x (/ (- z (* a x)) y))))
   (if (<= y -4e+78)
     t_1
     (if (<= y -6000000000.0)
       (* (* y y) (+ (/ x b) (/ z (* b y))))
       (if (<= y 4.6e+23) (/ 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 + ((z - (a * x)) / y);
	double tmp;
	if (y <= -4e+78) {
		tmp = t_1;
	} else if (y <= -6000000000.0) {
		tmp = (y * y) * ((x / b) + (z / (b * y)));
	} else if (y <= 4.6e+23) {
		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(z - Float64(a * x)) / y))
	tmp = 0.0
	if (y <= -4e+78)
		tmp = t_1;
	elseif (y <= -6000000000.0)
		tmp = Float64(Float64(y * y) * Float64(Float64(x / b) + Float64(z / Float64(b * y))));
	elseif (y <= 4.6e+23)
		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[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4e+78], t$95$1, If[LessEqual[y, -6000000000.0], N[(N[(y * y), $MachinePrecision] * N[(N[(x / b), $MachinePrecision] + N[(z / N[(b * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.6e+23], 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 3 regimes

2. if y < -4.00000000000000003e78 or 4.6000000000000001e23 < y

   1. Initial program 2.3%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 67.3

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

   8. Applied rewrites 67.3%

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

   if -4.00000000000000003e78 < y < -6e9

   1. Initial program 32.5%

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

   3. Taylor expanded in b around inf

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 44.4%

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

   if -6e9 < y < 4.6000000000000001e23

   1. Initial program 99.0%

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

   3. Taylor expanded in 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 76.5

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

   5. Applied rewrites 76.5%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+78}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{elif}\;y \leq -6000000000:\\ \;\;\;\;\left(y \cdot y\right) \cdot \left(\frac{x}{b} + \frac{z}{b \cdot y}\right)\\ \mathbf{elif}\;y \leq 4.6 \cdot 10^{+23}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+83} \lor \neg \left(y \leq 4.6 \cdot 10^{+23}\right):\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= y -1.15e+83) (not (<= y 4.6e+23)))
   (+ x (/ (- z (* a x)) y))
   (/ t (fma (fma (fma (+ a y) 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) {
	double tmp;
	if ((y <= -1.15e+83) || !(y <= 4.6e+23)) {
		tmp = x + ((z - (a * x)) / y);
	} else {
		tmp = t / fma(fma(fma((a + y), y, b), y, c), y, i);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((y <= -1.15e+83) || !(y <= 4.6e+23))
		tmp = Float64(x + Float64(Float64(z - Float64(a * x)) / y));
	else
		tmp = Float64(t / fma(fma(fma(Float64(a + y), y, b), y, c), y, i));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[y, -1.15e+83], N[Not[LessEqual[y, 4.6e+23]], $MachinePrecision]], N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(t / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.14999999999999997e83 or 4.6000000000000001e23 < y

   1. Initial program 2.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.1

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

   5. Applied rewrites 3.1%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 67.8

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

   8. Applied rewrites 67.8%

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

   if -1.14999999999999997e83 < y < 4.6000000000000001e23

   1. Initial program 93.5%

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

   3. Taylor expanded in 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 70.8

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

   5. Applied rewrites 70.8%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.15 \cdot 10^{+83} \lor \neg \left(y \leq 4.6 \cdot 10^{+23}\right):\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 55.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+29} \lor \neg \left(y \leq 8.6 \cdot 10^{-18}\right):\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (or (<= y -3.2e+29) (not (<= y 8.6e-18)))
   (+ x (/ (- z (* a x)) y))
   (/ t i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((y <= -3.2e+29) || !(y <= 8.6e-18)) {
		tmp = x + ((z - (a * x)) / y);
	} else {
		tmp = t / i;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8) :: tmp
    if ((y <= (-3.2d+29)) .or. (.not. (y <= 8.6d-18))) then
        tmp = x + ((z - (a * x)) / y)
    else
        tmp = t / i
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if ((y <= -3.2e+29) || !(y <= 8.6e-18)) {
		tmp = x + ((z - (a * x)) / y);
	} else {
		tmp = t / i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if (y <= -3.2e+29) or not (y <= 8.6e-18):
		tmp = x + ((z - (a * x)) / y)
	else:
		tmp = t / i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if ((y <= -3.2e+29) || !(y <= 8.6e-18))
		tmp = Float64(x + Float64(Float64(z - Float64(a * x)) / y));
	else
		tmp = Float64(t / i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if ((y <= -3.2e+29) || ~((y <= 8.6e-18)))
		tmp = x + ((z - (a * x)) / y);
	else
		tmp = t / i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[Or[LessEqual[y, -3.2e+29], N[Not[LessEqual[y, 8.6e-18]], $MachinePrecision]], N[(x + N[(N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(t / i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999987e29 or 8.6000000000000005e-18 < y

   1. Initial program 7.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 3.8

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

   5. Applied rewrites 3.8%

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

   6. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-/.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 61.6

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

   8. Applied rewrites 61.6%

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

   if -3.19999999999999987e29 < y < 8.6000000000000005e-18

   1. Initial program 98.2%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+29} \lor \neg \left(y \leq 8.6 \cdot 10^{-18}\right):\\ \;\;\;\;x + \frac{z - a \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{i}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 11.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i)
use fmin_fmax_functions
    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 52.6%

   \[\frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i} \]
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 50.3%

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

6. Taylor expanded in y around inf

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

8. Applied rewrites 10.8%

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

Reproduce

herbie shell --seed 2025016 
(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)))