Numeric.SpecFunctions:logGamma from math-functions-0.1.5.2

Percentage Accurate: 56.1% → 83.0%

Time: 8.5s

Alternatives: 15

Speedup: 2.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 56.1% 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: 83.0% accurate, 0.4× speedup

Math

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

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 y x z) y 27464.7644705) y 230661.510616) t_1)
      (/ t t_1))
     (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x))))

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(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), (t / t_1));
	} else {
		tmp = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	}
	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, Float64(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616) / t_1), Float64(t / t_1));
	else
		tmp = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x);
	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[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $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 90.6%

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

   4. Applied rewrites 91.3%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 69.7

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

   5. Applied rewrites 69.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(\left(\left(x \cdot y + z\right) \cdot y + 27464.7644705\right) \cdot y + 230661.510616\right) \cdot y + t}{\left(\left(\left(y + a\right) \cdot y + b\right) \cdot y + c\right) \cdot y + i}\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{+298}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1
         (/
          (+
           (* (+ (* (+ (* (+ (* x y) z) y) 27464.7644705) y) 230661.510616) y)
           t)
          (+ (* (+ (* (+ (* (+ y a) y) b) y) c) y) i))))
   (if (<= t_1 2e+298) t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = ((((((((x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / (((((((y + a) * y) + b) * y) + c) * y) + i);
	double tmp;
	if (t_1 <= 2e+298) {
		tmp = t_1;
	} else {
		tmp = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * y) + z) * y) + 27464.7644705) * y) + 230661.510616) * y) + t) / Float64(Float64(Float64(Float64(Float64(Float64(Float64(y + a) * y) + b) * y) + c) * y) + i))
	tmp = 0.0
	if (t_1 <= 2e+298)
		tmp = t_1;
	else
		tmp = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(N[(N[(N[(N[(N[(N[(x * y), $MachinePrecision] + z), $MachinePrecision] * y), $MachinePrecision] + 27464.7644705), $MachinePrecision] * y), $MachinePrecision] + 230661.510616), $MachinePrecision] * y), $MachinePrecision] + t), $MachinePrecision] / N[(N[(N[(N[(N[(N[(N[(y + a), $MachinePrecision] * y), $MachinePrecision] + b), $MachinePrecision] * y), $MachinePrecision] + c), $MachinePrecision] * y), $MachinePrecision] + i), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 2e+298], t$95$1, N[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $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)) < 1.9999999999999999e298

   1. Initial program 92.1%

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

   if 1.9999999999999999e298 < (/.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 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 y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 67.3

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

   5. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 78.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right)\\ \mathbf{if}\;y \leq -2.3 \cdot 10^{+84}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-23}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(y, x, z\right), y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{y}}{t\_1}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(t\_1, y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (fma (+ a y) y b) y c)))
   (if (<= y -2.3e+84)
     x
     (if (<= y -2.1e-23)
       (/
        (/ (fma (fma (fma (fma y x z) y 27464.7644705) y 230661.510616) y t) y)
        t_1)
       (if (<= y 1.15e+56)
         (/
          (fma (fma (fma z y 27464.7644705) y 230661.510616) y t)
          (fma t_1 y i))
         (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(fma((a + y), y, b), y, c);
	double tmp;
	if (y <= -2.3e+84) {
		tmp = x;
	} else if (y <= -2.1e-23) {
		tmp = (fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / y) / t_1;
	} else if (y <= 1.15e+56) {
		tmp = fma(fma(fma(z, y, 27464.7644705), y, 230661.510616), y, t) / fma(t_1, y, i);
	} else {
		tmp = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(fma(Float64(a + y), y, b), y, c)
	tmp = 0.0
	if (y <= -2.3e+84)
		tmp = x;
	elseif (y <= -2.1e-23)
		tmp = Float64(Float64(fma(fma(fma(fma(y, x, z), y, 27464.7644705), y, 230661.510616), y, t) / y) / t_1);
	elseif (y <= 1.15e+56)
		tmp = Float64(fma(fma(fma(z, y, 27464.7644705), y, 230661.510616), y, t) / fma(t_1, y, i));
	else
		tmp = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision]}, If[LessEqual[y, -2.3e+84], x, If[LessEqual[y, -2.1e-23], N[(N[(N[(N[(N[(N[(y * x + z), $MachinePrecision] * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / y), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y, 1.15e+56], N[(N[(N[(N[(z * y + 27464.7644705), $MachinePrecision] * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / N[(t$95$1 * y + i), $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -2.2999999999999999e84

   1. Initial program 0.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 61.1%

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

   if -2.2999999999999999e84 < y < -2.1000000000000001e-23

   1. Initial program 55.7%

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

   3. Taylor expanded in i around 0

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

      1. associate-/r* N/A

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

      2. lower-/.f64 N/A

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

   5. Applied rewrites 49.6%

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

   if -2.1000000000000001e-23 < y < 1.15000000000000007e56

   1. Initial program 96.0%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 91.9%

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

   if 1.15000000000000007e56 < y

   1. Initial program 2.2%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 70.1

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

   5. Applied rewrites 70.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ t_2 := \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(27464.7644705, y, 230661.510616\right), y, t\right)}{t\_2}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+64}:\\ \;\;\;\;\left(\left(y \cdot y\right) \cdot y\right) \cdot \frac{z}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x))
        (t_2 (fma (fma (fma (+ a y) y b) y c) y i)))
   (if (<= y -2.2e+23)
     t_1
     (if (<= y 1.22e-12)
       (/ (fma (fma 27464.7644705 y 230661.510616) y t) t_2)
       (if (<= y 9.6e+64) (* (* (* y y) y) (/ z 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 = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	double t_2 = fma(fma(fma((a + y), y, b), y, c), y, i);
	double tmp;
	if (y <= -2.2e+23) {
		tmp = t_1;
	} else if (y <= 1.22e-12) {
		tmp = fma(fma(27464.7644705, y, 230661.510616), y, t) / t_2;
	} else if (y <= 9.6e+64) {
		tmp = ((y * y) * y) * (z / t_2);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x)
	t_2 = fma(fma(fma(Float64(a + y), y, b), y, c), y, i)
	tmp = 0.0
	if (y <= -2.2e+23)
		tmp = t_1;
	elseif (y <= 1.22e-12)
		tmp = Float64(fma(fma(27464.7644705, y, 230661.510616), y, t) / t_2);
	elseif (y <= 9.6e+64)
		tmp = Float64(Float64(Float64(y * y) * y) * Float64(z / 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[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]}, If[LessEqual[y, -2.2e+23], t$95$1, If[LessEqual[y, 1.22e-12], N[(N[(N[(27464.7644705 * y + 230661.510616), $MachinePrecision] * y + t), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y, 9.6e+64], N[(N[(N[(y * y), $MachinePrecision] * y), $MachinePrecision] * N[(z / t$95$2), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000008e23 or 9.59999999999999997e64 < y

   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 y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -2.20000000000000008e23 < y < 1.2200000000000001e-12

   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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 94.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 88.5%

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

   if 1.2200000000000001e-12 < y < 9.59999999999999997e64

   1. Initial program 60.5%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 26.5%

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

Alternative 5: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \mathbf{if}\;y \leq -2.35 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(z, y, 27464.7644705\right), y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x)))
   (if (<= y -2.35e+23)
     t_1
     (if (<= y 1.15e+56)
       (/
        (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 = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	double tmp;
	if (y <= -2.35e+23) {
		tmp = t_1;
	} else if (y <= 1.15e+56) {
		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 = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x)
	tmp = 0.0
	if (y <= -2.35e+23)
		tmp = t_1;
	elseif (y <= 1.15e+56)
		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[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]}, If[LessEqual[y, -2.35e+23], t$95$1, If[LessEqual[y, 1.15e+56], 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 2 regimes

2. if y < -2.3499999999999999e23 or 1.15000000000000007e56 < y

   1. Initial program 4.3%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.1

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

   5. Applied rewrites 65.1%

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

   if -2.3499999999999999e23 < y < 1.15000000000000007e56

   1. Initial program 95.6%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 90.2%

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

Alternative 6: 74.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \mathbf{if}\;y \leq -4.7 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{27464.7644705 + \mathsf{fma}\left(230661.510616, \frac{1}{y}, y \cdot \left(z + x \cdot y\right)\right)}{b}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x)))
   (if (<= y -4.7e+60)
     t_1
     (if (<= y -2.1e-17)
       (/
        (+ 27464.7644705 (fma 230661.510616 (/ 1.0 y) (* y (+ z (* x y)))))
        b)
       (if (<= y 6.2e+36)
         (/ (fma 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 = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	double tmp;
	if (y <= -4.7e+60) {
		tmp = t_1;
	} else if (y <= -2.1e-17) {
		tmp = (27464.7644705 + fma(230661.510616, (1.0 / y), (y * (z + (x * y))))) / b;
	} else if (y <= 6.2e+36) {
		tmp = fma(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 = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x)
	tmp = 0.0
	if (y <= -4.7e+60)
		tmp = t_1;
	elseif (y <= -2.1e-17)
		tmp = Float64(Float64(27464.7644705 + fma(230661.510616, Float64(1.0 / y), Float64(y * Float64(z + Float64(x * y))))) / b);
	elseif (y <= 6.2e+36)
		tmp = Float64(fma(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[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]}, If[LessEqual[y, -4.7e+60], t$95$1, If[LessEqual[y, -2.1e-17], N[(N[(27464.7644705 + N[(230661.510616 * N[(1.0 / y), $MachinePrecision] + N[(y * N[(z + N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 6.2e+36], N[(N[(230661.510616 * 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 < -4.6999999999999998e60 or 6.1999999999999999e36 < y

   1. Initial program 3.3%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -4.6999999999999998e60 < y < -2.09999999999999992e-17

   1. Initial program 64.5%

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

   4. Applied rewrites 66.8%

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

   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

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in t around 0

      \[\leadsto \frac{\frac{54929528941}{2000000} + \mathsf{fma}\left(\frac{28832688827}{125000}, \frac{1}{y}, y \cdot \left(z + x \cdot y\right)\right)}{b} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 19.6

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

   10. Applied rewrites 19.6%

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

   if -2.09999999999999992e-17 < y < 6.1999999999999999e36

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 86.9%

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

Alternative 7: 74.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(y \cdot y\right) \cdot \left(x + \frac{z}{y}\right)}{b}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{\mathsf{fma}\left(230661.510616, y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x)))
   (if (<= y -1.8e+60)
     t_1
     (if (<= y -2.1e-17)
       (/ (* (* y y) (+ x (/ z y))) b)
       (if (<= y 6.2e+36)
         (/ (fma 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 = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	double tmp;
	if (y <= -1.8e+60) {
		tmp = t_1;
	} else if (y <= -2.1e-17) {
		tmp = ((y * y) * (x + (z / y))) / b;
	} else if (y <= 6.2e+36) {
		tmp = fma(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 = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x)
	tmp = 0.0
	if (y <= -1.8e+60)
		tmp = t_1;
	elseif (y <= -2.1e-17)
		tmp = Float64(Float64(Float64(y * y) * Float64(x + Float64(z / y))) / b);
	elseif (y <= 6.2e+36)
		tmp = Float64(fma(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[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]}, If[LessEqual[y, -1.8e+60], t$95$1, If[LessEqual[y, -2.1e-17], N[(N[(N[(y * y), $MachinePrecision] * N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 6.2e+36], N[(N[(230661.510616 * 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 < -1.79999999999999984e60 or 6.1999999999999999e36 < y

   1. Initial program 3.3%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -1.79999999999999984e60 < y < -2.09999999999999992e-17

   1. Initial program 64.6%

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

   4. Applied rewrites 66.9%

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

   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

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{{y}^{2} \cdot \left(x + \frac{z}{y}\right)}{b} \]

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 17.3

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

   10. Applied rewrites 17.3%

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

   if -2.09999999999999992e-17 < y < 6.1999999999999999e36

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 86.9%

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

Alternative 8: 75.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{+42}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(27464.7644705, y, 230661.510616\right), y, t\right)}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x)))
   (if (<= y -2.2e+23)
     t_1
     (if (<= y 2.05e+42)
       (/
        (fma (fma 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 = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	double tmp;
	if (y <= -2.2e+23) {
		tmp = t_1;
	} else if (y <= 2.05e+42) {
		tmp = fma(fma(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 = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x)
	tmp = 0.0
	if (y <= -2.2e+23)
		tmp = t_1;
	elseif (y <= 2.05e+42)
		tmp = Float64(fma(fma(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[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]}, If[LessEqual[y, -2.2e+23], t$95$1, If[LessEqual[y, 2.05e+42], N[(N[(N[(27464.7644705 * 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 2 regimes

2. if y < -2.20000000000000008e23 or 2.05e42 < y

   1. Initial program 5.2%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 63.9

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

   5. Applied rewrites 63.9%

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

   if -2.20000000000000008e23 < y < 2.05e42

   1. Initial program 97.0%

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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

   5. Applied rewrites 91.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 84.3%

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

Alternative 9: 67.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+60}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.05 \cdot 10^{-17}:\\ \;\;\;\;\frac{\left(y \cdot y\right) \cdot \left(x + \frac{z}{y}\right)}{b}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+36}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x)))
   (if (<= y -1.8e+60)
     t_1
     (if (<= y -2.05e-17)
       (/ (* (* y y) (+ x (/ z y))) b)
       (if (<= y 9e+36) (/ 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 = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	double tmp;
	if (y <= -1.8e+60) {
		tmp = t_1;
	} else if (y <= -2.05e-17) {
		tmp = ((y * y) * (x + (z / y))) / b;
	} else if (y <= 9e+36) {
		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 = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x)
	tmp = 0.0
	if (y <= -1.8e+60)
		tmp = t_1;
	elseif (y <= -2.05e-17)
		tmp = Float64(Float64(Float64(y * y) * Float64(x + Float64(z / y))) / b);
	elseif (y <= 9e+36)
		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[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]}, If[LessEqual[y, -1.8e+60], t$95$1, If[LessEqual[y, -2.05e-17], N[(N[(N[(y * y), $MachinePrecision] * N[(x + N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[y, 9e+36], 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 < -1.79999999999999984e60 or 8.99999999999999994e36 < y

   1. Initial program 3.3%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if -1.79999999999999984e60 < y < -2.05e-17

   1. Initial program 64.6%

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

   4. Applied rewrites 66.9%

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

   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

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

   7. Applied rewrites 24.1%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \frac{{y}^{2} \cdot \left(x + \frac{z}{y}\right)}{b} \]

      2. pow2 N/A

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

      3. lift-*.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 17.3

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

   10. Applied rewrites 17.3%

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

   if -2.05e-17 < y < 8.99999999999999994e36

   1. Initial program 97.9%

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

   3. Taylor expanded in t around 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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 74.3

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

   5. Applied rewrites 74.3%

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

Alternative 10: 64.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(c, y, i\right)}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+65}:\\ \;\;\;\;\frac{y \cdot z}{b + y \cdot \left(a + y\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x)))
   (if (<= y -2.2e+23)
     t_1
     (if (<= y 4.7e-12)
       (/ t (fma c y i))
       (if (<= y 2.55e+65) (/ (* y z) (+ b (* y (+ a y)))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double t_1 = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	double tmp;
	if (y <= -2.2e+23) {
		tmp = t_1;
	} else if (y <= 4.7e-12) {
		tmp = t / fma(c, y, i);
	} else if (y <= 2.55e+65) {
		tmp = (y * z) / (b + (y * (a + y)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	t_1 = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x)
	tmp = 0.0
	if (y <= -2.2e+23)
		tmp = t_1;
	elseif (y <= 4.7e-12)
		tmp = Float64(t / fma(c, y, i));
	elseif (y <= 2.55e+65)
		tmp = Float64(Float64(y * z) / Float64(b + Float64(y * Float64(a + y))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := Block[{t$95$1 = N[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]}, If[LessEqual[y, -2.2e+23], t$95$1, If[LessEqual[y, 4.7e-12], N[(t / N[(c * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.55e+65], N[(N[(y * z), $MachinePrecision] / N[(b + N[(y * N[(a + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000008e23 or 2.54999999999999994e65 < y

   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 y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 65.6

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

   5. Applied rewrites 65.6%

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

   if -2.20000000000000008e23 < y < 4.69999999999999976e-12

   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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.1%

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

   if 4.69999999999999976e-12 < y < 2.54999999999999994e65

   1. Initial program 59.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 18.8

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

   8. Applied rewrites 18.8%

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

   9. Taylor expanded in c around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-+.f64 19.3

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

   11. Applied rewrites 19.3%

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

Alternative 11: 59.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{-12}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(c, y, i\right)}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+65}:\\ \;\;\;\;\frac{y \cdot z}{b + y \cdot \left(a + y\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -2.2e+23)
   x
   (if (<= y 4.7e-12)
     (/ t (fma c y i))
     (if (<= y 2.6e+65) (/ (* y z) (+ b (* y (+ a y)))) x))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -2.2e+23) {
		tmp = x;
	} else if (y <= 4.7e-12) {
		tmp = t / fma(c, y, i);
	} else if (y <= 2.6e+65) {
		tmp = (y * z) / (b + (y * (a + y)));
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -2.2e+23)
		tmp = x;
	elseif (y <= 4.7e-12)
		tmp = Float64(t / fma(c, y, i));
	elseif (y <= 2.6e+65)
		tmp = Float64(Float64(y * z) / Float64(b + Float64(y * Float64(a + y))));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -2.2e+23], x, If[LessEqual[y, 4.7e-12], N[(t / N[(c * y + i), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.6e+65], N[(N[(y * z), $MachinePrecision] / N[(b + N[(y * N[(a + y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.20000000000000008e23 or 2.60000000000000003e65 < y

   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 y around inf

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

   5. Applied rewrites 54.2%

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

   if -2.20000000000000008e23 < y < 4.69999999999999976e-12

   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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.1%

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

   if 4.69999999999999976e-12 < y < 2.60000000000000003e65

   1. Initial program 59.8%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto {y}^{3} \cdot \color{blue}{\frac{z}{i + y \cdot \left(c + y \cdot \left(b + y \cdot \left(a + y\right)\right)\right)}} \]

      3. unpow3 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 26.1%

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

   6. Taylor expanded in i around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-+.f64 18.8

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

   8. Applied rewrites 18.8%

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

   9. Taylor expanded in c around 0

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-+.f64 N/A

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

       5. lift-+.f64 19.3

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

   11. Applied rewrites 19.3%

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

Alternative 12: 67.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{-1 \cdot \left(z - a \cdot x\right)}{y}, -1, x\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+36}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(a + y, y, b\right), y, c\right), y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (let* ((t_1 (fma (/ (* -1.0 (- z (* a x))) y) -1.0 x)))
   (if (<= y -2.2e+23)
     t_1
     (if (<= y 9e+36) (/ 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 = fma(((-1.0 * (z - (a * x))) / y), -1.0, x);
	double tmp;
	if (y <= -2.2e+23) {
		tmp = t_1;
	} else if (y <= 9e+36) {
		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 = fma(Float64(Float64(-1.0 * Float64(z - Float64(a * x))) / y), -1.0, x)
	tmp = 0.0
	if (y <= -2.2e+23)
		tmp = t_1;
	elseif (y <= 9e+36)
		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[(N[(N[(-1.0 * N[(z - N[(a * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -1.0 + x), $MachinePrecision]}, If[LessEqual[y, -2.2e+23], t$95$1, If[LessEqual[y, 9e+36], N[(t / N[(N[(N[(N[(a + y), $MachinePrecision] * y + b), $MachinePrecision] * y + c), $MachinePrecision] * y + i), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000008e23 or 8.99999999999999994e36 < y

   1. Initial program 5.5%

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

   3. Taylor expanded in y around -inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-*.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if -2.20000000000000008e23 < y < 8.99999999999999994e36

   1. Initial program 97.3%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 71.4

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

   5. Applied rewrites 71.4%

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

Alternative 13: 58.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+23}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+36}:\\ \;\;\;\;\frac{t}{\mathsf{fma}\left(c, y, i\right)}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -2.2e+23) x (if (<= y 9e+36) (/ t (fma c y i)) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -2.2e+23) {
		tmp = x;
	} else if (y <= 9e+36) {
		tmp = t / fma(c, y, i);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -2.2e+23)
		tmp = x;
	elseif (y <= 9e+36)
		tmp = Float64(t / fma(c, y, i));
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -2.2e+23], x, If[LessEqual[y, 9e+36], N[(t / N[(c * y + i), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -2.20000000000000008e23 or 8.99999999999999994e36 < y

   1. Initial program 5.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 52.0%

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

   if -2.20000000000000008e23 < y < 8.99999999999999994e36

   1. Initial program 97.3%

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

   3. Taylor expanded in 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

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. +-commutative N/A

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

      13. lower-+.f64 71.4

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

   5. Applied rewrites 71.4%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 64.0%

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

Alternative 14: 50.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -24000000000:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 440000000000:\\ \;\;\;\;\frac{t}{i}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i)
 :precision binary64
 (if (<= y -24000000000.0) x (if (<= y 440000000000.0) (/ t i) x)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -24000000000.0) {
		tmp = x;
	} else if (y <= 440000000000.0) {
		tmp = t / i;
	} else {
		tmp = x;
	}
	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 <= (-24000000000.0d0)) then
        tmp = x
    else if (y <= 440000000000.0d0) then
        tmp = t / i
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	double tmp;
	if (y <= -24000000000.0) {
		tmp = x;
	} else if (y <= 440000000000.0) {
		tmp = t / i;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i):
	tmp = 0
	if y <= -24000000000.0:
		tmp = x
	elif y <= 440000000000.0:
		tmp = t / i
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i)
	tmp = 0.0
	if (y <= -24000000000.0)
		tmp = x;
	elseif (y <= 440000000000.0)
		tmp = Float64(t / i);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i)
	tmp = 0.0;
	if (y <= -24000000000.0)
		tmp = x;
	elseif (y <= 440000000000.0)
		tmp = t / i;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := If[LessEqual[y, -24000000000.0], x, If[LessEqual[y, 440000000000.0], N[(t / i), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if y < -2.4e10 or 4.4e11 < y

   1. Initial program 8.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 49.2%

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

   if -2.4e10 < y < 4.4e11

   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 y around 0

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

      1. lower-/.f64 52.2

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

   5. Applied rewrites 52.2%

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

Alternative 15: 25.4% accurate, 71.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t a b c i) :precision binary64 x)

C

double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return x;
}

Fortran

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
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i) {
	return x;
}

Python

def code(x, y, z, t, a, b, c, i):
	return x

Julia

function code(x, y, z, t, a, b, c, i)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_] := x

Derivation

1. Initial program 56.1%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 25.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025092 
(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)))